text:
\section{Introduction}

Many problems in classical projective geometry ask about the
nature  of special subvarieties of some given family of varieties,
e.g. how many isolated singular points can a surface of degree $d$
in $\mathbb{P}^3$ have? when is it true that the members of a certain
family of varieties contain a line? contain a linear space of any
positive dimension?  The reader can easily supply other examples
of such questions.

This is the kind of problem we consider in this paper: what  types
of complete intersection varieties of codimension $r$ in $\mathbb{P}^n$
can one find on the generic hypersurface of degree $d$?

In case $r=2$ it was known to Severi \cite{Severi} that for  $n
\geq 4$ the only complete intersections on a general hypersurface
are obtained by intersecting that hypersurface with another.

This observation was extended to $\mathbb{P}^3$ by Noether (and
Lefschetz)  \cite{Lefschetz, GH85} for general hypersurfaces of
degree $\geq 4$.  These ideas were further generalized by
Grothendieck \cite{SGA2}.

Our approach to the problem mentioned above uses a mix of
projective  geometry and commutative algebra and is much more
elementary and accesible than, for example, the approach of
Grothendieck.  We are able to give a complete answer to the
question we raised for complete intersections of codimension $r$
in $\mathbb{P}^n$ which lie on a general hypersurface of degree $d$
whenever $2r \leq n+2$.

The paper is organized in the following way: in the next section
(Section 2) we lay out the question we want to consider and
explain what are the interesting parameters for a response.

In Section 3 we collect some technical information we will  need
about varieties of reducible forms and their joins.  In order to
find the dimensions of these joins (using \lq\lq Terracini's
Lemma") we calculate the tangent space at a point of any variety
of reducible forms.  We also recall some information about
artinian complete intersection quotients of a polynomial ring.

In Section 4, we use the technical facts collected in Section 3
to reformulate our original question.  We illustrate the utility
of this reformulation to discuss complete intersections of
codimension $r$ in $\mathbb{P}^n$ on a general hypersurface when $2r <
n+1$.  We further use our approach to give a new proof for the
existence of a line on the general hypersextic of $\mathbb{P}^5$.

In Section 5 we state and prove our main theorem  which gives a
complete description of all complete intersections of codimension
$r$ in $\mathbb{P}^n$ which lie on a generic hypersurface when $2r \leq
n+2$.

\section{Question}\label{questionSECTION}

The  objects of study of this paper are complete intersection
subschemes of projective space.  Recall that $Y\subset\mathbb{P}^n$ is a
complete intersection scheme  if its ideal is generated by a
regular sequence, more precisely, $I(Y)=(F_1,\ldots,F_r)$, $F_i\in
S=\mathbb{C}[x_0,\ldots,x_n]$, and $F_1, \ldots , F_r$ form a
regular sequence in $S$. If $\deg F_i=a_i$ for all $i$, we will
say that such a $Y$ is a $CI(a_1,\ldots,a_r)$ and we will assume
$a_1\leq\ldots\leq a_r$; notice that $Y$ is unmixed of codimension
$r$ in $\mathbb{P}^n$. With this notation we can rephrase the statement

\begin{quote} the degree $d$ hypersurface $X$ contains a
$CI(a_1,\ldots,a_r)$
\end{quote}

in terms of ideals as follows

\begin{quote}
$I(X)=(F)\subset (F_1,\ldots,F_r)$ for forms $F_i$ forming a
regular sequence and such that $\deg F_i=a_i$ for all $i$.
\end{quote}

Clearly not all choices of the degrees are of interest for us,
e.g. if $a_i>d$ for all $i$, then no $CI(a_1,\ldots,a_r)$ can be
found on a degree $d$ hypersurface. On the other hand, any
hypersurface of degree $d$ contains a $CI(a_1,\ldots,a_r)$ if
$a_i=d$, for some $i$. Simply cut that hypersurface with general
hypersurfaces of degrees $a_j,\ j\neq i$.

So, one need only consider $CI(a_1,\ldots , a_r)$ where none of the $a_i = d$.

\begin{lem}\label{lessthand} Let $a_1 \leq \ldots \leq a_i < d < a_{i+1} \leq \ldots \leq a_r$ with $r \leq n$.
The following are equivalent facts:

\begin{itemize}
\item there is a $CI(a_1, \ldots, a_r)$ on the general
hypersurface of degree $d$ in $\mathbb{P}^n$;

\item there is a $CI(a_1, \ldots , a_i)$ on the general
hypersurface of degree $d$ in $\mathbb{P}^n$.
\end{itemize}
\end{lem}

\begin{proof}
Let $I(X)=(F)$, where $X$ is a general hypersurface of  degree $d$
in $\mathbb{P}^n$.  Let $I(Y)=(F_1,\ldots,F_r)$ be the ideal of a
$CI(a_1, \ldots , a_r)$, with degrees $a_i$ as above.

Then $X\supset Y$ if and only if
\[
F=\sum_{j=1}^i F_jG_j
\]
and hence, if and only if $X\supset Y^\prime$, where $Y^\prime$ is
the complete intersection defined by $F_1, \ldots , F_i$.
\end{proof}

From this Lemma it is clear that the basic question to be
considered is:
\begin{quote}{(Q): }{\em
For which degrees $a_1,\ldots, a_r<d$ does the generic degree $d$
hypersurface of $\mathbb{P}^n$ contain a $CI(a_1,\ldots,a_r)$?
}\end{quote}

If, rather than restricting to the generic case, we asked if {\em
some} hypersurface of degree $d$ contains a $CI(a_1, \ldots ,
a_r)$, then the answer is trivial. Indeed,  the ideal of any
$CI(a_1,\ldots,a_r)$ ($a_i < d$) always contains degree $d$
elements.

\section{Technical facts}

If $\lambda = (\lambda_1, \ldots , \lambda_s)$ is a  partition of
the integer $d$ (i.e. $\sum_{i=1}^s\lambda_i = d$ and $\lambda_1
\geq \ldots \geq \lambda _s > 0$) we write $\lambda\vdash d$.  For
each $\lambda \vdash d$ we define a subvariety $\mathbb{X}_\lambda
\subset \mathbb{P} (S_d) \simeq \mathbb{P}^N$ (where $N = {d+n \choose n} -1$)
as follows:
\[
\mathbb{X}_\lambda:= \{\ \ [F] \in S_d \ \vert \ F=F_1\cdots F_s, \ \deg F_i = \lambda_i \ \ \}.
\]
We call $\mathbb{X}_\lambda$ the {\it variety of reducible forms of type
$\lambda$}.  The dimension of $\mathbb{X}_\lambda$ is easily seen to be
$[\sum_{i=1}^s {\lambda_i+n\choose n}]-s$. (For other elementary
properties of $\mathbb{X}_\lambda$ see \cite{Mamma} and for the special
case $\lambda_1=\ldots=\lambda_s=1$ see \cite{Ca04JA},
\cite{Ca05Siena} or \cite{Chi02} for the $n=2$ case).

If $x_1, \ldots , x_r$ are independent points  of $\mathbb{P}^N$ we will
call the $\mathbb{P}^{r-1}$ spanned by these points the {\it join of the
points $x_1, \ldots x_r$} and write
\[
J(x_1, \ldots , x_r) := \langle x_1, \ldots , x_r \rangle .
\]

More generally, if $X_1, \ldots , X_r$ are  varieties in $\mathbb{P}^N$
then the {\it join of $X_1, \ldots , X_r$} is
\[
J(X_1, \ldots , X_r):= \overline{ \bigcup \{ J(x_1, \ldots , x_r)\ \vert \ x_i \in X_i, \{x_1, \ldots , x_r\}\ \hbox{independent} \}}
\]

In case $X_1 = \cdots =  X_r = X$ we write
\[
J(X_1, \ldots , X_r):= Sec_{r-1}(X)
\]
and call this the {\it $(r-1)^{st}$ (higher) secant variety of
$X$}.

Joins and secants of projective  varieties are important auxiliary
varieties which can help us better understand the geometry of the
original varieties  (see e.g.
\cite{CGG1,CGG2,ChCi06,Ci01,Ge,LM,AAd,CatJ}). One of the most
fundamental questions we can ask about joins and secants is: {\it
What are their dimensions?}

This is, in general, an extremely  difficult question to answer.
The famous {\it Lemma of Terracini} (which we recall below) is an
important observation which will aid us in answering this
question.

\begin{lem}
[Lemma of Terracini]\label{terracinilemma}   Let $X_1, \ldots ,
X_r$  be reduced subvarieties of $\mathbb{P}^N$ and let $p \in J = J(X_1,
\ldots , X_r)$ be a generic point of $J$.

Suppose that $p \in J(p_1, \ldots , p_r)$,  then the
(projectivized) tangent space to $J$ at $p$, i.e. $T_p(J)$, can be
described as follows:
\[
T_p(J) = \langle T_{p_1}(X_1), \ldots , T_{p_r}(X_r) \rangle .
\]

Consequently,
\[
\dim J = \dim \langle T_{p_1}(X_1), \ldots , T_{p_r}(X_r) \rangle .
\]

\end{lem}

We want to apply this Lemma in the case that  the $X_i$ are all of
the form $\mathbb{X}_{\lambda^{(i)}}, \ \lambda^{(i)} \vdash d, \ i = 1,
\ldots , r$.  A crucial first step in such an application is,
therefore, a calculation of $T_{p_i}(\mathbb{X}_{\lambda ^{(i)} })$ where
$p_i \in \mathbb{X}_{\lambda ^{(i)} }$.

\begin{prop}\label{tangentspace}
Let $\lambda \vdash d$, $\lambda = (\lambda_1, \ldots , \lambda_s)$ and let $p \in \mathbb{X}_\lambda$ be a generic point of $\mathbb{X}_\lambda$.

Write $p = [F_1\cdots F_s]$ where $\deg F_i = \lambda_i, i = 1,
\ldots , s$ and let $I_p\subset S=\mathbb{C}[x_0, \ldots , x_n]$ be the
ideal defined by:
\[
I_p = (F_2\cdots F_s, F_1F_3\cdots F_s, \ldots , F_1 \cdots F_{s-1} ).
\]

Then the tangent space to $\mathbb{X}_\lambda$ at the point $p$ is the projectivization of $(I_p)_d$ and hence has dimension
\[
\dim T_p(\mathbb{X}_\lambda) = \dim _\mathbb{C} (I_p)_d - 1.
\]
\end{prop}

\begin{proof}
Consider the map of affine spaces
\[
\Phi: S_{\lambda_1} \times \cdots \times  S_{\lambda_s} \rightarrow S_d
\]
defined by
\[
\Phi((A_1, \ldots , A_s)) =  A_1\cdots A_s .
\]

Let $P \in S_{\lambda_1} \times \cdots \times  S_{\lambda_s}$ be
the  point $P = (F_1, \ldots , F_s)$.  A tangent direction at $P$
is given by any vector of the form ${v} = (F_1^\prime, \ldots ,
F_s^\prime)$ and the line through $P$ in that direction is
\[
L_{{v}}: = (F_1 + \mu F_1^\prime, \ldots , F_s + \mu F_s^\prime),\
\mu \in \mathbb{C} .
\]

A simple calculation shows that the tangent  vector to
$\Phi(L_{{v}})$ at the point $\Phi(P)=p$ is exactly $\sum_{i=1}^s
F_1\cdots  {F_i}^\prime  \cdots F_s$ and that proves the
proposition.

\end{proof}

In view of Terracini's Lemma, the following corollary is immediate.

\begin{cor} \label{tangentjoin}

Let $\lambda^{(1)}, \ldots , \lambda^{(r)}$ all be partitions of $d$ where
\[
\lambda^{(i)} = (\lambda_{i1}, \lambda_{i2}) .
\]

Let
\[
I = (F_{11}, F_{12}, F_{21}, F_{22}, \ldots , F_{r1}, F_{r2} )
\]
be an ideal of $S$ generated by generic forms where
\[
\deg F_{ij} = \lambda_{ij}, \hbox{ for } 1 \leq i \leq r, j = 1,2.
\]

If
\[
J = J(\mathbb{X}_{\lambda^{(1)}}, \ldots , \mathbb{X}_{\lambda^{(r)}})
\]
then
\[
\dim J = \dim _\mathbb{C} I_d - 1.
\]
\end{cor}

\begin{rem} \label{genjoin} It is useful to note the
following facts:

\begin{enumerate}
\item In Proposition \ref{tangentspace} we are using the fact that
$\mathbb{C}$ has characteristic 0. The problem is that the
differential is not necessarily generically injective in
characteristic $p$.

\item\label{regseqrem1}   Observe that the generic  point in
$\mathbb{X}_\lambda$, $\lambda = (\lambda_1, \ldots , \lambda_s) \vdash
d$ can always be written as the product of $s$ irreducible forms
with the property that any $\ell$-subset of these $s$ forms ($\ell
\leq n+1$) is a regular sequence.

\item\label{regseqrem3}  This last can be extended  easily to
joins of varieties of reducible forms.  I.e. the generic point in
such a join can be written as a sum of elements with the property
that each summand is a point enjoying the property described in
\eqref{regseqrem1} above. Moreover, every $\ell$-subset ($\ell
\leq n+1$) of the set of all the irreducible factors of all of
these summands is also a regular sequence.

\item Fr\"{o}berg (see \cite{Fro}) has a conjecture about the
multiplicative  structure of rings $S/I$, where $S = \mathbb{C}[x_0,
\ldots , x_n]$ and $I$ is an ideal generated by a set of generic
forms.  This conjecture gives the Hilbert functions of such rings.
However, apart from the cases $n=1$ (proved several times by
various authors, see \cite{Fro, GeSc,
IaKa}) and $n=2$ (proved by \cite{Anick}) this conjecture
has resisted attempts to prove it.

Notice that in terms of the geometric  problem in Corollary
\ref{tangentjoin}, one need only consider Fr\"{o}berg's conjecture
for a strongly restricted collection of degrees.
\end{enumerate}
\end{rem}

We will need some specific information  about the Hilbert function
of some artinian complete intersections in polynomial rings.  The
following lemma summarizes the facts we shall use.

\begin{lem}\label{basiclemma} Let $r>1$ and $F_1,\ldots,F_r,G_1,\ldots,G_r$ be generic forms in $\mathbb{C}[y_1,\ldots,y_{2r-1}]$ having degrees
\[1<\deg F_1=a_1\leq\deg F_2=a_2\leq\ldots\leq\deg F_r=a_r\leq d/2\]
and
\[d/2\leq\deg G_r=d-a_r\leq\ldots\leq\deg G_1=d-a_1\]
for a non-negative integer $d$.

Consider the quotient
\[A=\mathbb{C}[y_1,\ldots,y_{2r-1}]/(F_1,\ldots,F_r,G_r,\ldots,G_{3},G_{2})\]
and its Hilbert function $H_A$. The following facts hold:
\begin{enumerate}

\item\label{one} $H_A$ is symmetric with respect to
$c={(r-1)d+a_1-2r+1\over 2}$;

\item\label{two} if $H_A(i)\geq H_A(i+1)$ then $H_A(j)$ is
non-increasing for $j\geq i$;

\item\label{six} the multiplication map on  $A_i$  given by
$\overline{G_1}$ (the class of ${G_1}$ in $A$) has maximal rank.

\end{enumerate}

If one of the following holds

\[r=2 \mbox{ and } a_1\geq 5, \mbox{ or}\]
\[r=3 \mbox{ and } a_1\geq 3, \mbox{ or}\]
\[r=3 \mbox{ and } a_1=2,d\neq 4, \mbox{ or}\]
\[r>3 \mbox{ and } a_1\geq 2,\]

we also have that:

\begin{enumerate}

\setcounter{enumi}{3}

\item\label{three} if $i\leq a_1$, then $H_A(i)<H_A(i+1)$.

\item\label{four} if $a_1<i\leq c$, then $H_A(a_1)<H_A(i)$.

\item\label{five} if $c<i$, then $H_A(a_1)>H_A(i)$ if and only if
$c-a_1<i-c$.

\end{enumerate}
\end{lem}

\begin{proof}
As $A$ is a Gorenstein graded ring \eqref{one}  follows
immediately, while \eqref{six} is a consequence of a theorem of
Stanley \cite{Stanley} and Watanabe \cite{Wat}.

To prove \eqref{two} we can use the Weak Lefschetz property, i.e.
multiplication by a general linear form  has maximal rank, e.g.
see \cite{MigMR}. The condition on $H_A$, coupled with the Weak
Lefschetz property, yields that every element of $A_{i+1}$ is the
product of a fixed linear form with a form of degree $i$. Now
consider an element of $A_{i+2}$, call it $M$ , then since $A$ is
a standard graded algebra, $M = \sum_{i=1}^{2r-1} y_iC_i$, where
$y_i$ is the class of $y_i$ in $A$, and $C_i$ is the class of a
form of degree $i+1$.  By what we have seen, $C_i = LD_i$ where $L
$ is the form we had earlier and the $D_i$ are forms of degree
$i$. Rewriting we get $M = L\sum_{i=1}^{2r-1}y_iD_i$.  But
$\sum_{i=1}^{2r-1}y_iD_i$ is in $A_{i+1}$ hence $A_{i+2} =
LA_{i+1}$ and hence the dimension cannot increase. Proceeding by
induction we prove the statement.

As for \eqref{three}, it is enough to give the proof for $i=a_1$
as there are no generators of degree smaller than $a_1$. Let $\bar
A$ be a quotient obtained when all the forms $F_i$ and $G_i$ have
the same degree $a = a_1 = \ldots =a_r=d-a_r=\ldots = d-a_2 $.
Notice that it is enough to show the result for $\bar A$. In fact,
whenever we pass from $\bar A$ to another quotient $A$ by
increasing the degrees of $s$ forms we obtain

\begin{equation}
\begin{array}{l}
\displaystyle H_A(a)=H_{\bar A}(a)+s\\ \\\displaystyle H_{\bar
A}(a+1)+s(2r-1)-s\leq H_A(a+1)
\end{array}
\label{HFestimates}
\end{equation}

and the inequality $H_{\bar A}(a)<H_{\bar A}(a +1)$ is preserved;
these Hilbert function estimates use the fact that the forms $F_i$
and $G_i$ do not have linear syzygies. By straightforward
computations one gets
\[H_{\bar A}(a)={a+2r-2\choose a}-2r+1 \]
and
\[H_{\bar A}(a+1)={a+2r-1\choose a+1}-(2r-1)^2.\]
Thus the inequality $H_{\bar A}(a)<H_{\bar A}(a+1)$ is equivalent
to

\begin{equation}
{a+2r-2\choose a+1}-(2r-1)(2r-2)>0. \label{star}
\end{equation}

Notice that if \eqref{star} holds for the pair $(a,r)$ then it
holds for all the pairs $(a+i,r)$ with $i\geq 0$. By direct
computations we verify that the inequality is satisfied for
$(a,r)=(5,2),(3,3)$ and for $a=2$ and $r>3$. Hence, \eqref{star}
holds for
\[r=2 \mbox{ and } a\geq 5, \mbox{ or }\]
\[r=3 \mbox{ and } a\geq 3, \mbox{ or }\]
\[r>3 \mbox{ and } a\geq 2.\]
To complete the proof of \eqref{three} it is enough to evaluate
\eqref{HFestimates} for $r=3,a=2$ in the case $d\neq 4$, i.e.
$s>0$.

To show \eqref{four}, notice that by \eqref{two}, if
\[H_A(a_1)>H_A(i), a_1<i\]
then $H_A$ is definitely non-increasing and hence it cannot be
symmetric with respect to $c$ by \eqref{three}.

To get \eqref{five} it is enough to use symmetry and \eqref{four}.

\end{proof}

\section{Equivalences}

In this section we give some equivalent formulations of our basic
question (Q), formulated at the end of Section
\ref{questionSECTION}.

Clearly, if $X\subset\mathbb{P}^n$ is a hypersurface of degree $d$ and
$Y\subset X$ is a $CI(a_1,\ldots,a_r)$, then the ideal inclusion
$I(X)=(F)\subset I(Y)=(F_1,\ldots,F_r)$ yields
\[F=F_1G_1+\ldots +F_rG_r \]
for forms $G_i$ of degrees $d-a_i$. But the converse is not true
in general. If $F=F_1G_1+\ldots +F_rG_r$ and the forms $F_i$ do
not form a regular sequence, then $(F_1,\ldots,F_r)$ is not the
ideal of a complete intersection. To produce an equivalence we
need to use joins:

\begin{lem}\label{joineqlem} The following are equivalent:
\begin{enumerate}
\item\label{joineqlem1} a generic hypersurface of degree $d$ of
$\mathbb{P}^n$ contains a $CI(a_1,\ldots,a_r)$, where $a_i<d$ for all
$i$;

\item\label{joineqlem2} the join of the varieties of reducible
forms
 $\mathbb{X}_{(a_i,d-a_i)},
i=1,\ldots,r$ fills the space of degree $d$ forms in $n+1$
variables, i.e.
\[J(\mathbb{X}_{(a_1,d-a_1)},\ldots,\mathbb{X}_{(a_r,d-a_r)})=\mathbb{P}(S_d).\]
\end{enumerate}
\end{lem}
\begin{proof}
The implication \eqref{joineqlem1}$\Rightarrow$\eqref{joineqlem2}
simply follows from the ideal inclusion argument above yielding
the presentation $F=\sum F_iG_i$ for the generic degree $d$ form,
where $[F_iG_i]\in \mathbb{X}_{(a_i,d-a_i)}$ for all $i$. The
implication \eqref{joineqlem2}$\Rightarrow$\eqref{joineqlem1} is
easily shown using the description of the generic element of the
join, see Remark \ref{genjoin} \eqref{regseqrem3}.
\end{proof}
\begin{rem}\label{switch} Notice that there is an equality of varieties
\[\mathbb{X}_{(i,j)}=\mathbb{X}_{(j,i)}\]
for all non negative integers $i$ and $j$. Hence, by Lemma
\ref{joineqlem}, the condition
\[J(\mathbb{X}_{(a_1,d-a_1)},\ldots,\mathbb{X}_{(a_r,d-a_r)})=\mathbb{P}(S_d)\]
is equivalent to the statement
\begin{quote}
a generic hypersurface of degree $d$ of $\mathbb{P}^n$ contains a
$CI(b_1,\ldots,b_r)$, where $b_i=a_i$ or $b_i=d-a_i$ for all $i$.
\end{quote}
It follows from these observations that we can further restrict
the range of the degrees in our basic question (Q), i.e. it is
enough to consider
\[a_1\leq\ldots\leq a_r\leq {d\over 2}.\]
\end{rem}

Now we exploit Terracini's Lemma and the tangent space description
given in Corollary \ref{tangentjoin} in order to produce another equivalent
formulation of question (Q).
\begin{lem}\label{algeqlem} The following are equivalent:
\begin{enumerate}
\item\label{algeqlem1} the generic hypersurface of degree $d$ of
$\mathbb{P}^n$ contains a $CI(a_1,\ldots,a_r)$, where $a_i<d$ for all
$i$;

\item\label{algeqlem2} let $F_1,\ldots,F_r$ and $G_1,\ldots,G_r$
be generic forms in $S=\mathbb{C}[x_0,\ldots,x_n]$ of degrees
$a_1\leq\ldots \leq a_r<d$ and $d-a_1,\ldots,d-a_r$ respectively,
then
\[H(S/(F_1,\ldots,F_r,G_r,\ldots,G_1),d)=0\]
where $H(\cdot,d)$ denotes the Hilbert function in degree $d$ of
the ring.
\end{enumerate}
\end{lem}
\begin{proof}
The condition on the join in Lemma \ref{joineqlem} can be read in
term of tangent spaces as equivalent to
\[\langle T_{P_1}(\mathbb{X}_{(a_1,d-a_1)}),\ldots,T_{P_r}(\mathbb{X}_{(a_r,d-a_r)}) \rangle=\mathbb{P}(S_d)\]
for generic points $P_1=[F_1G_1],\ldots,P_r=[F_rG_r]$. Using the
description of the tangent space to the variety of reducible forms
this is equivalent to saying
\[(F_1,G_1)_d + \ldots+(F_r,G_r)_d=S_d\]
where $S_d$ is the degree $d$ piece of the polynomial ring $S$ and
the forms $F_i$ and $G_i$ are generic of degrees $a_i$ and $d-a_i$
respectively.
\end{proof}

As a straightforward application we get the following result:
\begin{prop}\label{smallrPROP}
The generic degree $d$ hypersurface of $\mathbb{P}^n$ contains no
$CI(a_1,\ldots,a_r)$, $a_i<d$ for all $i$, when $2r<n+1$.
\end{prop}
\begin{proof}
We use Lemma \ref{algeqlem}. Consider in
$S=\mathbb{C}[x_0,\ldots,x_n]$ generic forms $F_1,\ldots,F_r$ and
$G_1,\ldots,G_r$ of degrees $a_i$ and $d-a_i$ respectively. If we
let $I$ be the ideal $(F_1,\ldots,F_r,G_1,\ldots,G_r)$, then we
want to show that $H(S/I,d)\neq 0$ and for that it is enough to
show that $S/I$ is not an artinian ring. As $I$ has height $2r$
and $2r<n+1$ the quotient cannot be zero dimensional and the
conclusion follows.
\end{proof}
\begin{rem} Using Lemma \ref{algeqlem} we can also recover many
classical results in an elegant and simple way. More precisely, we
can easily study the existence of complete intersections curves,
e.g. lines and conics, on hypersurfaces.

\end{rem}

\begin{ex} As an example we prove the following without using Schubert calculus:

{\em The generic hypersextic of $\mathbb{P}^5$ contains a line.}
\begin{proof} Let $S=\mathbb{C}[x_0,\ldots,x_5]$ and consider the
ideal

\[I=(L_1,\ldots ,L_4,G_1,\ldots,G_4)\]

where the forms $L_i$ are linear forms and the forms $G_i$ have
degree $5$. We want to show that $H(S/I,6)=0$.  Clearly
\[ S/I \simeq \mathbb{C}[x_0,x_1]/(\bar G_1,\ldots,\bar G_4).\]

It is well known \cite{GeSc, IaKa,
Fro}  that 4 general binary forms of degree 5 generate $\mathbb{C}
[x_0,x_1]_6$ and we are done.
\end{proof}

For more on this topic see Remark \ref{hochesterRem}.
\end{ex}

\section{The Theorem}

We are now ready to prove the main  theorem of this paper, a
description of all the possible complete intersections of
codimension $r$ that can be found on a general hypersurface of
degree $d$ in $\mathbb{P}^n$ when $2r \leq n+2$.

\begin{thm}\label{maintheorem}
Let $X\subset\mathbb{P}^n$ be a generic degree $d$ hypersurface, with
$n,d>1$. Then $X$ contains a $CI(a_1,\ldots,a_r)$, with $2r\leq
n+2$, and the $a_i$ all less than $d$, in the following (and only
in the following) instances:
\medskip

\begin{itemize}
\item $n=2$: then $r=2$, $d$ arbitrary and $a_1$ and $a_2$ can
assume any value less than $d$;

\item $n=3$, $r=2 $:  for $d \leq 3$ we have that $a_1$ and $a_2$
can assume any value less than $d$;

\item $n=4$, $r = 3$: for $d \leq 5$ we have that $a_1, a_2$ and
$a_3$ can assume any value less than $d$;

\item $n=6,r = 4$ or $n=8,r = 5$: for $d \leq 3$ we have that
$a_1, \ldots ,a_r$ can assume any value less than $d$;

\item $n=5,7$ or $n>8$, $2r=n+1$ or $2r = n+2$:  we have only
linear spaces on quadrics, i.e. $d = 2$ and $a_1 = \ldots =a_r =
1$.

\end{itemize}

\end{thm}
\begin{proof}
 Recall that from Lemma \ref{lessthand} and Remark \ref{switch} it
 is  sufficient to consider the existence of a $CI(a_1, \ldots ,
 a_r)$ on the generic hypersurface of degree $d$ when $a_1\leq
 \ldots \leq  a_r\leq d/2$.

When $2r<n+1$, by Proposition \ref{smallrPROP}, we know that no
complete intersection exists. Hence we have only to consider the
cases $2r=n+1$ and $2r=n+2$

In order to use Lemma \ref{algeqlem} we consider the generic forms
$F_1,\ldots,F_r$ and $G_1,\ldots,G_r$ of degrees $a_i$ and $d-a_i$
respectively. If we let $S=\mathbb{C}[x_0,\ldots,x_n]$ and
$I=(F_1,\ldots,F_r,G_r,\ldots,G_1)$ we want to check whether
$H(S/I,d)=0$ or not.

The case $2r=n+1$. In this case, $S/I$ is an artinian Gorenstein
ring and $e=r(d-2)+1$ is the first place where one has
$H(S/I,e)=0$. Thus, the generic degree $d$ hypersurface contains a
$CI(a_1,\ldots,a_r)$ if and only if $H(S/I,d)=0$ and this is
equivalent to the inequality
\[d\geq r(d-2)+1\]
which is never satisfied unless $d=2$ and $a_1=\ldots=a_r=1$.

The case $2r=n+2$ will be proved using Lemma \ref{basiclemma}. In
order to do this, we divide the proof into three parts:

\begin{itemize}

\item {\em the hyperplane case}: $a_1=1$ any $r$;

\item {\em the plane case}: $a_1=2,3,4$ for $r=2$, and hence
$n=2$;

\item {\em the four space case}: $a_1=a_2=a_3=2$ and $d=4$ for
$r=3$, and hence $n=4$, ;

\item {\em the general case}: one of the following holds

\begin{equation}
\begin{array}{l}
r=2 \mbox{ and } a_1\geq 5, \mbox{ or}\\
r=3 \mbox{ and } a_1\geq 3, \mbox{ or}\\
r=3, a_1=2 \mbox{ and } d\neq 4, \mbox{ or}\\
r>3 \mbox{ and } a_1\geq 2.
\end{array}
\label{generalcasecond}
\end{equation}

\end{itemize}

{\em The hyperplane case.} We need to study $CI(1,a_2,\ldots,
a_r)$ on the generic degree $d$ hypersurface of $\mathbb{P}^{2r-2}$. As
one of the generators of the complete intersection is a
hyperplane, we can reduce to a smaller dimensional case. In
algebraic terms, for  a generic linear form $L$, we consider the
surjective quotient map
\[S\longrightarrow S/(L)\]
to get the following:

\begin{quote}{\it
the generic element of $S_d$ can be decomposed as a product of
forms of degrees $1,a_2,\ldots, a_r$, i.e. it has the form
$\sum_{i=1}^r F_iG_i$ with $\deg F_1=1$ and $\deg F_i=a_i$,
$i=2,\ldots,r$}
\end{quote}

if and only if

\begin{quote}{\it
the generic element of $(S/(L))_d\simeq(
\mathbb{C}[x_0,\ldots,x_{n-1}])_d$ can be decomposed as a product
of forms of degrees $a_2,\ldots, a_r$, i.e. it has the form
$\sum_{i=2}^r \bar{F_i}\bar{G_i}$ with $\deg \bar F_i=a_i$,
$i=2,\ldots,r$.}
\end{quote}

Hence, we have to study $CI(a_2,\ldots, a_r)$ on the generic
degree $d$ hypersurface of $\mathbb{P}^{2r-3}$, i.e. codimension $r'=r-1$
complete intersections in $\mathbb{P}^{n'}$, $n'=2r-3$. As $2r'=n'+1$
this situation was treated before and the only case where the
complete intersections exist is for $d=2$ and $a_i=1$ for all $i$.

{\em The plane case.} We have to study $CI(a_1,a_2)$ on the
generic degree $d$ curve of $\mathbb{P}^2$ for $a_1=2,3,4$, any $a_2,d$
such that $a_1\leq a_2\leq d$. Now, $S=\mathbb{C}[x_0,x_1,x_2]$ and we
consider forms $F_1,F_2,G_2$ and $G_1$ of degrees, respectively,
$a_1,a_2,d-a_2$ and $d-a_1$. We want to study the ring
\[A=S/(F_1,F_2,G_2)\]
and to compare $H(A,a_1)$ and $H(A,d)$ in order to apply
\eqref{six} of Lemma \ref{basiclemma} to show that
\[H(A/(\bar G_1),d)=0.\] Using Lemma \ref{basiclemma} \eqref{one}, we see that the last non-zero value of $H_A$
occurs for
\[ d+a_1-3.\]
In particular, for $a_1=2$, $H(A,d)=0$ and a $CI(2,a_2)$ exists
for any $a_2$ and $d$, $2\leq a_2\leq d$. If $a_1=3$, then
$H(A,d)=1$ and the same conclusion holds for $CI(3,a_2)$. Finally,
if $a_1=4$, then $H(A,d)=H(A,1)$ and it is easy to see that
$H(A,1)\leq H(A,4)$. Hence, for $a_1=2,3,4$ and any $a_2,d$ such
that $a_1\leq a_2\leq d$, the generic degree $d$ plane curve
contains a $CI(a_1,a_2)$.

{\em The four space case.} We address the case of $CI(2,2,2)$ on
the generic degree 4 threefold in $\mathbb{P}^4$. Hence, we consider
$S=\mathbb{C}[x_0,\ldots,x_4]$ and generic quadratic forms
$F_1,F_2,F_3,G_3,G_2$ and $G_1$. Let $A$ be the quotient ring
\[S/(F_1,F_2,F_3,G_3,G_2)\]
and notice that by the vanishing of the lefthand side of
\eqref{star} in the proof of Lemma \ref{basiclemma} we have that
$H(A,2)=H(A,3)$. Applying \eqref{two} of Lemma \ref{basiclemma}
yields $H(A,2)=H(A,4)$ and hence the required $CI$ exists.

{\em The general case.} Consider the ring
\[A=\mathbb{C}[x_0,\ldots,x_{2r-2}]/(F_1,\ldots,F_r,G_r,\ldots,G_3,G_2)\]
and the multiplication map given by the degree $d-a_1$ form $\bar
G_1$
\[m:A_{a_1}\rightarrow A_d.\]
Clearly, with this notation, one has that the generic degree $d$
hypersurface contains a $CI(a_1,\ldots,a_r)$ if and only if
\[H(S/I,d)=H(A/(\overline{G_1}),d)=0\]
and this is equivalent to the surjectivity of $m$. We also recall
that by Lemma \ref{basiclemma} \eqref{six} $m$ has maximal rank.
Hence, to study the surjectivity we only have to compare
$H(A,a_1)=\dim A_{a_1}$ and $H(A,d)=\dim A_d$.

When $d=2$, all the degrees $a_i$ are equal to 1, and this was
treated in the hyperplane case.

Now we consider the $d>2$ case and we let $c={(r-1)d+a_1-2r+1\over
2}$. If $d\leq c$ then $\dim A_{a_1}<\dim A_{d}$ by Lemma
\ref{basiclemma}\eqref{four}, thus $m$ cannot be surjective.
Standard computations yield
\[d\leq c \Leftrightarrow 2d\leq (r-1)d+a_1-2r+1 \Leftrightarrow 2+{5-a_1\over r-3}\leq d \mbox{ if } r>3\]
while for $r=2$ the inequality $d\leq c$ never holds. Thus we get
that, when one of the conditions \eqref{generalcasecond} holds,
$m$ is not surjective if
\[r>3 \mbox{ and } d>7\]
or
\[r=3 \mbox{ and } a_1>5. \]

In the case $c<d$ we have to be more careful and the distances
$\alpha =c-a_1$ and $\beta =d-c$ have to be considered. When one
of the conditions \eqref{generalcasecond} holds, by
\ref{basiclemma}\eqref{five}, $m$ is surjective if and only if
$\alpha\leq \beta$. Thus we solve the inequality $\beta-\alpha\geq
0$. This is equivalent to
\[d\leq 2+{3\over r-2}\mbox{ if }r>2\]
while for $r=2$ we always have $\beta -\alpha\geq 0$.

Summing up all these facts one gets that when one of the
conditions \eqref{generalcasecond} holds:
\[r=2: m \mbox{ is surjective},\]
\[r=3: d>5, m \mbox{ is not surjective};d\leq 5, m \mbox{ is surjective};\]
\[r=4,5: d>3, m \mbox{ is not surjective};d\leq 3, m \mbox{ is surjective};\]
\[r\geq 6: d>2, m \mbox{ is not surjective};d\leq 2, m \mbox{ is surjective}\]
and using the treatment of the hyperplane, the plane and the four
space cases we obtain the final result.

\end{proof}

\begin{rem} The fact that a general hypersurface of degree $d\geq 6$ in $\mathbb{P}^4$ cannot
contain a complete intersection of any type with $a_1,a_2,a_3<d$
is also a consequence of a result about vector bundles proved by
Mohan Kumar, Rao and Ravindra (see \cite{MKRR}).

In $\mathbb{P}^n$ the existence statement for $d =  2$ is classical. The
$d=3$ cases in $\mathbb{P}^6$ and $\mathbb{P}^8$ can be obtained using Theorem
12.8 in \cite{Harris} (see also Proposition \ref{harrisPROP} in
this paper). In $\mathbb{P}^4$, for $d=3$ and $d=4$ the existence also
follows from the analysis of arithmetically Cohen-Macaulay rank
two bundles on hypersurfaces, contained in \cite{ArrCos} and
\cite{Mad}.

In $\mathbb{P}^4$, for the case $d=5$, when $\min\{a_i\}=2$, the result also follows from the existence of a canonical curve on the generic quintic threefold of $\mathbb{P}^4$, as was essentially proved in \cite{Kl}.
\end{rem}

\begin{rem}
For $n=2$ the theorem above states that the generic degree $d$
plane curve contains a $CI(a,b)$ for any $a,b<d$, but it does not
say that this is a set of $ab$ points. The complete intersection
scheme could very well not be reduced. Actually, we can show
reducedness and hence the following holds
\begin{quote}{\em
the generic degree $d$ plane curve contains $ab$ complete
intersection points for any $a,b<d$. }\end{quote}
\end{rem}

\begin{rem}
Again a remark in the case $n=2$. Taking $a_1=a_2=a$ the theorem
above states that $\mbox{Sec}_1(\mathbb{X}_{(a,d-a)})$ is the
whole space. Now, quite generally, the points of the variety of
secant lines either lie on a true secant line or on a tangent line
to $\mathbb{X}_{(a,d-a)}$.  We claim that Proposition \ref{tangentspace}
allows us to conclude that the points of the tangent lines are in
fact already on the true secant lines.  In fact, if
$p=[FG]\in\mathbb{X}_{(a,d-a)}$, then any point $q$ of a tangent
line to $p$ can be written as $[\alpha FG'+\beta F'G]$ for forms
$G',F'$ of degrees $d-a,a$ respectively, and scalars $\alpha$ and
$\beta$. Thus, $q$ lies on the secant line to
$\mathbb{X}_{(a,d-a)}$ joining $[FG']$ and $[GF']$. In conclusion,
we can rephrase the equality
$\mbox{Sec}_1(\mathbb{X}_{(a,d-a)})=\mathbb{P}(S_d)$ in terms of
polynomial decompositions
\begin{quote}{\em
let $a<d$, \underline{any} degree $d$ form in three variables $F$
can be written as $F=F_1G_1+F_2G_2$ for suitable forms $F_i$ of
degree $a$ and $G_i$ of degree $d-a$.}\end{quote}

This answers a question raised during a correspondence between
Zinovy Reichstein and the first author.
\end{rem}

\begin{rem}\label{hochesterRem} The restriction $2r\leq n+2$ in Theorem
\ref{maintheorem} is related to the fact that Fr\"{o}berg's
conjecture is only known to be true, in general, when the number
of forms does not exceed one more than the number of variables.
However, there are other partial results on this conjecture that
we can use to extend our theorem. E.g. in \cite{HoLak} Hochester
and Laksov showed that a piece of Fr\"{o}berg's conjecture holds.
More precisely they showed that if an ideal is generated by
generic forms of the same degree $d$ then the size of that ideal
in degree $d+1$ is exactly what is predicted by Fr\"{o}berg's
conjecture. Using this we can prove the following:
\begin{prop}\label{harrisPROP}
The generic degree $d>2$ hypersurface in $\mathbb{P}^n$ contains a
complete intersection of type $(a_1,\ldots,a_r)$ where $a_i=1$ or
$a_i=d-1$ for all i, if and only if
\[{n-r+d\choose d}\leq (n-r+1)r.\]
When $a_1=\ldots=a_r=1$ this is the well known result on the
non-emptyness of the Fano variety of $(n-r)$-planes on the generic
degree $d$ hypersurface of $\mathbb{P}^n$ (e.g. see \cite{Harris} Theorem
12.8).
\end{prop}
\begin{proof}
Using Lemma \ref{algeqlem} we have to show the vanishing, in
degree $d$, of the Hilbert function  of
\[A=\mathbb{C}[x_0,\ldots,x_n]/(L_1,\ldots,L_r,F_1,\ldots,F_r)\]
for generic linear forms $L_i$ and generic degree $d-1$ forms
$F_i$. Clearly, as the linear forms are generic, we have
\[A\simeq\mathbb{C}[x_0,\ldots,x_{n-r}]/(\overline{F_1},\ldots,\overline{F_r}).\]
Hence $A_d=0$ if and only if
$(\overline{F_1},\ldots,\overline{F_r})$ contains all the degree
$d$ forms. Using the result by Hochester and Laksov this is
equivalent to
\[{n-r+d\choose d}\leq (n-r+1)r.\]
\end{proof}

\end{rem}

\begin{ex} The variety $\mathbb{X}_{(1,3)}$ of reducible quartic hypersurfaces of $\mathbb{P}^3$ and its secant line variety provide interesting examples for several reasons.

First note that $\mathbb{X}_{(1,3)} \subset \mathbb{P}^{34}$ is a variety of
dimension $3 + 19 = 22$.  From Corollary \ref{tangentjoin} it is
easy to deduce that $\dim Sec_1(\mathbb{X}_{(1,3)}) = 33$.  Thus,
$\mathbb{X}_{(1,3)}$ is a defective variety whose virtual defect $e$ is,
$$e = 2\dim \mathbb{X}_{(1,3)} + 1 - \dim Sec_1(\mathbb{X}_{(1,3)}) = 12.$$

\medskip\noindent $1)$ \ Consider the Noether-Lefschetz locus of quartic hypersurfaces in $\mathbb{P}^3$ with Picard group $\neq \mathbb{Z}$.  The quartic hypersurfaces which contain a line are clearly in the Noether-Lefschetz locus.  If $\ell$ is a line defined by the linear forms $L_1, L_2$ then the form $F$, of degree 4, defines a hypersurface containing $\ell$ if and only if
$$
F = L_1G_1 + L_2G_2 \hbox{ where } \deg G_i = 3 .
$$

I.e. if and only if $[F] \in Sec_1(\mathbb{X}_{(1,3)})$.  Since, as we observed, $$\dim Sec_1(\mathbb{X}_{(1,3)}) = 33$$ this forces the secant variety to be a component of the Noether-Lefschetz locus.

We wonder how often joins of other varieties of reducible forms give components of the appropriate Noether-Lefschetz locus.

\medskip\noindent $2)$ \ Since $\mathbb{X}_{(1,3)}$ is defective for secant lines we have, by a theorem of \cite{ChCi06}, that for every two points on $\mathbb{X}_{(1,3)}$ there is a subvariety $\Sigma$,  containing those two points, whose linear span has dimension $ \leq 2 \dim \Sigma + 1 - e$, where $e$ is the defect of $\mathbb{X}_{(1,3)}$.

We now give a description of such $\Sigma$'s for the variety $\mathbb{X}_{(1,3)}$.

Let $[H_1F_1], [H_2F_2]$ be two points of $\mathbb{X}_{(1,3)}$ and let
$\ell$ be the line in $\mathbb{P}^3$ defined by $H_1 = 0 = H_2$. Consider
$\Sigma \subset \mathbb{X}_{(1,3)}$, the subvariety of reducible quartics
whose linear components contain $\ell$.  Clearly $\dim \Sigma = 1
+ 19 = 20$.  Notice that the linear span of $\Sigma$, $<\Sigma>$,
is contained in the subvariety of all quartics containing $\ell$
and that variety has dimension $34 - 5 = 29$.

Thus, $$\dim <\Sigma > \leq 29 = 2(20) + 1 - 12 = 2 \dim \Sigma + 1 - e,$$ as we wanted to show.

Notice that the existence of $\Sigma$, as above, gives another proof of the defectivity of $\mathbb{X}_{(1,3)}$.

\end{ex}

\bibliographystyle{alpha}


text:
\section{Introduction}
\label{sect_intro}

``Silicate carbon stars'' are characterized by oxygen-rich circumstellar
material as evident from the prominent silicate emission,
despite their carbon-rich photospheres (Little-Marenin
\cite{little-marenin86}; Willems \& de Jong \cite{willems86}).
The detection of masers from O-bearing molecules such as OH and \mbox{H$_2$O}\
from some of the silicate carbon stars confirms
the presence of oxygen-rich circumstellar gas (e.g.,
Nakada et al. \cite{nakada87}; Engels \cite{engels94};
Szymczak et al. \cite{szymczak01} and references therein).
Moreover, optical spectroscopic studies (e.g.,
Lloyd-Evans \cite{lloyd-evans90}; Chan \cite{chan93};
Ohnaka \& Tsuji \cite{ohnaka99}) reveal that silicate carbon stars
have \mbox{$^{12}$C/$^{13}$C}\ ratios as low as $\sim$4--5 (classified as ``J-type''),
which is difficult to explain by the present stellar evolution
theory.
Currently, silicate carbon stars are considered to have an unseen,
low-luminosity companion.
The oxygen-rich material which was shed by mass loss when the primary star
was an M giant can be stored in a circumbinary disk (Morris \cite{morris87};
Lloyd-Evans \cite{lloyd-evans90}) or in a circumstellar disk around the
companion (``circum-companion disk'', Yamamura et al. \cite{yamamura00}),
even after the primary star becomes a carbon star.
In the latter case, Yamamura et al. (\cite{yamamura00}) suggest that
the primary star drives an outflow from the circum-companion
disk, and the silicate emission emanates from this outflow.

Thanks to the recent progress of high-angular resolution observation
techniques,
there is growing observational evidence for the presence of
circumbinary or circum-companion disks around silicate carbon stars.
Our $N$-band spectro-interferometric observations of IRAS08002-3803
with the MID-infrared Interferometric instrument (MIDI) at
ESO's Very Large Telescope Interferometer (VLTI)
spatially resolved the dusty environment of a silicate carbon star
for the first time (Ohnaka et al. \cite{ohnaka06}), and
our radiative transfer modeling suggests the presence of an optically
thick circumbinary disk
in which amorphous silicate and a second grain species (amorphous carbon,
large silicate grains, or metallic iron) coexist.
The presence of a circumbinary disk with multiple grain populations
is also suggested for
another silicate carbon star IRAS18006-3213 by Deroo et al. (\cite{deroo07}).
The spatial distributions of the 22 GHz \mbox{H$_2$O}\ maser
emission toward two silicate carbon stars, V778~Cyg and EU~And, have
recently been resolved by Szczerba et al. (\cite{szczerba06}), Engels
(priv. comm.), and
Ohnaka \& Boboltz (\cite{ohnaka_boboltz08}) using MERLIN and the Very Long
Baseline Array (VLBA).
The observed spatio-kinematic structures are consistent with the presence of
circum-companion disks.
However, the formation mechanisms of these
disks are little understood, and the possible link between the disk formation
and the anomalously low \mbox{$^{12}$C/$^{13}$C}\ ratios is by no means clear,
either.

Direct detection of a companion toward a silicate carbon star is hampered
by the huge luminosity contrast between the primary carbon star
($10^3$--$10^4$~\mbox{$L_{\sun}$}) and the companion ($\la$1~\mbox{$L_{\sun}$}), as well as small
angular separations (e.g., Engels \& Leinert \cite{engels_leinert94}).
However, Izumiura (\cite{izumiura03}) and Izumiura et al. (\cite{izumiura08})
have
overcome the first problem by taking advantage of the fact that the violet
flux of a carbon star (primary star) is heavily suppressed due to
molecular and atomic absorption.  Using the High Dispersion Spectrograph (HDS)
on the Subaru Telescope, they discovered continuum emission below 4000~\AA\
and Balmer lines showing P~Cygni profiles with an outflow velocity of
$\ga$400~\mbox{km s$^{-1}$}\ toward the silicate carbon star BM~Gem.
These spectroscopic signatures strongly suggest the presence of an accretion
disk around an unseen companion,
although it could not be spatially resolved, and its nature---whether it is
a main-sequence star or a white dwarf---remains to be determined.
Comparison between the observationally derived accretion luminosity
($\sim$0.2~\mbox{$L_{\sun}$}) and the theoretical prediction from the Bondi-Hoyle-type
accretion resulted in estimated binary separations of $\sim$3--60~AU for a
main-sequence star companion and $\sim$100--500~AU for a white dwarf
companion.
These values translate into angular separations of 2.5--50~mas and
83--420~mas for a distance of 1.2~kpc adopted for BM~Gem by
Izumiura et al. (\cite{izumiura08}).

Given these estimated separations, it is challenging to resolve the
circum-companion disk toward BM~Gem with a single-dish telescope.
On the other hand, infrared interferometry provides us with a unique
opportunity to achieve sufficiently high spatial resolution.
In this paper, we present the results of mid-infrared spectro-interferometric
observations of BM~Gem with VLTI/MIDI to probe
the presence of the circum-companion disk with dust thermal emission.

\section{Observations}
\label{sect_obs}

\begin{table}
\begin{center}
\caption {MIDI observations of BM~Gem:
night, time of observation (Coordinated Universal Time=UTC), telescope
configuration (Tel.), projected baseline length $B_{\rm p}$, position angle
of the projected baseline on the sky (P.A.), and seeing in the visible
at the times of the
observations of BM~Gem and the calibrators listed in Table~\ref{table_calib}.
}
\begin{tabular}{r c c c r r l}
\hline
\hline
\# & Night & $t_{\rm obs}$ & Tel.      & $B_{\rm p}$ & P.A.  & Seeing \\
   &       & (UTC)         &            & (m)         & (\degr) & \\
\hline
1        & 2006 Dec. 29 & 07:29:47 & UT3-4 & 45.7 & 93.1  & 1\farcs0--1\farcs5\\
2        & 2006 Dec. 31 & 06:15:26 & UT3-4 & 55.2 & 99.1  & 0\farcs7--1\farcs0\\
3        & 2007 Jan. 04 & 06:31:59 & UT2-3 & 43.8 & 49.8  & 0\farcs8 \\
4        & 2007 Jan. 05 & 04:17:10 & UT3-4 & 62.1 & 108.0 & 0\farcs7--1\farcs0\\
\hline
\label{table_obs}
\end{tabular}
\end{center}
\end{table}

\begin{table}[!hbt]
\begin{center}
\caption {
MIDI observations of calibrators:
12~\mbox{$\mu$m}\ fluxes ($F_{12}$),
uniform-disk diameters ($d_{\rm{UD}}$),
night, and the time stamp ($t_{\rm obs}$).
The uniform-disk diameters were taken from the CalVin
list available at ESO
(http://www.eso.org/observing/etc/).
The data set used for spectrophotometric calibration
is marked with $\dagger$.
}
\begin{tabular}{l r r c l}
\hline
\hline
Calibrator & $F_{12}$ & $d_{\rm{UD}}$  & night & $t_{\rm obs}$ (UTC) \\
           &  (Jy)    &  (mas)         &      &                     \\ \hline
HD48433    & 8.5      & $2.10\pm 0.13$ & 2006 Dec. 29 & 07:52:09 \\
           &          &                & 2006 Dec. 31 & 06:42:34$^{\dagger}$ \\
           &          &                & 2007 Jan. 04 & 06:08:26 \\
HD47205    & 9.7      & $2.30\pm 0.12$ & 2007 Jan. 05 & 03:41:13 \\
HD50778    & 24.6     & $3.95\pm 0.22$ & 2006 Dec. 31 & 01:35:21 \\
           &          &                & 2006 Dec. 31 & 05:35:01 \\
HD98430    & 19.1     & $3.09\pm 0.16$ & 2007 Jan. 04 & 07:03:45 \\

\hline

\label{table_calib}
\end{tabular}
\end{center}
\end{table}

Our MIDI observations of BM~Gem and the calibrators are summarized in
Tables~\ref{table_obs} and \ref{table_calib}, respectively.
BM~Gem was observed
in the HIGH\_SENS mode using a prism ($\lambda/\Delta \lambda \simeq 30$
at 10~\mbox{$\mu$m}), and
four data sets were taken using the UT2-UT3 and UT3-UT4
configurations with projected baseline lengths of 44--62~m.
A detailed description of the instrument and the
observing procedure are given in
Przygodda et al. (\cite{przygodda03}), Leinert et al.\ (\cite{leinert04}),
and Chesneau et al. (\cite{chesneau05}).
We used the MIA+EWS package ver.1.5\footnote{Available at
http://www.strw.leidenuniv.nl/\textasciitilde nevec/MIDI}
to reduce the MIDI data (Leinert et al. \cite{leinert04};
Jaffe \cite{jaffe04}).
While both of MIA and EWS can extract visibility amplitude,
EWS can also extract the so-called
``differential phase'', which contains information on the wavelength
dependence of the object's phase.  However, it should be kept
in mind that two pieces of information on the object's phase are
lost in the derivation of differential phase as described in
Jaffe (\cite{jaffe04}): the absolute phase offset and the phase gradient
with respect to wavenumber.

We assessed the data quality and derived the calibrated visibilities as
described in Ohnaka et al. (\cite{ohnaka08}).
Since it is difficult to properly estimate the errors of the calibrated
visibilities from just 1--3 calibrator measurements,
we assumed a total relative error of 15\% as
adopted in Ohnaka et al. (\cite{ohnaka08}).
The visibilities obtained with MIA and EWS are in good agreement, and
we only show the results derived with EWS in the discussion
below.
We also extracted the absolutely calibrated $N$-band spectrum of BM~Gem
from our MIDI data as described in Ohnaka et al.
(\cite{ohnaka07}).  A calibrator observed at airmass similar to
our target was used for the spectrophotometric calibration and is marked with
$\dagger$ in Table~\ref{table_calib}.
The error of the absolutely calibrated MIDI spectrum is difficult to estimate
properly with only one calibrator, so we assumed errors of
10--20\% as in Ohnaka et al. (\cite{ohnaka07}).

\section{Results}
\label{sect_result}

Figure~\ref{obsspec_bmgem}a shows the MIDI spectrum of
BM~Gem, together with IRAS Low Resolution Spectrum
(LRS)\footnote{Downloaded from
http://staff.gemini.edu/\textasciitilde kvolk/getlrs\_plot.html
with the correction of the absolute calibration by
Cohen et al. (\cite{cohen92}).} and the photometric data from the
Midcourse Space Experiment (MSX, Egan et al. \cite{egan03}) and
the IRAS Point Source Catalog (PSC).
The MIDI spectrum and IRAS/LRS show that the flux level
has little changed over the last 23 years, and the flux variation longward
of 13~\mbox{$\mu$m}\ is also absent, as can be seen from the IRAS/LRS and MSX data
taken in 1996--1997.
This stability of the mid-infrared flux is the same as found for V778~Cyg
(Yamamura et al. \cite{yamamura00}), IRAS08002-3803
(Ohnaka et al. \cite{ohnaka06}), IRAS18006-3213 (Deroo et al. \cite{deroo07}),
and EU~And (Ohnaka \& Boboltz \cite{ohnaka_boboltz08}).

\begin{figure}[!hbt]
\resizebox{\hsize}{!}{\rotatebox{0}{\includegraphics{10229F1.ps}}}
\caption{
{\bf a:} $N$-band spectrum of BM~Gem extracted from the MIDI data, plotted
together with IRAS/LRS as well as the photometric data from the MSX and
IRAS PSC catalogs.  The width of each MSX band is shown with the horizontal
bar.
{\bf b:} Comparison of the $N$-band spectra of four silicate carbon stars:
BM~Gem, V778~Cyg, IRAS08002-3803, and IRAS18006-3213.
{\bf c:} SED of BM~Gem constructed from the (spectro)photometric data
available in the literature (Hipparcos \& Tycho Catalogs, ESA \cite{esa97};
2MASS, Cutri et al. \cite{cutri03}; Noguchi et al. \cite{noguchi81};
COBE/DIRBE, Smith et al. \cite{smith04}; MSX; IRAS/PSC; IRAS/LRS; MIDI).
The COBE/DIRBE data longward of 12~\mbox{$\mu$m}\ are not used, because of the
large errors.
The dashed line represents the stellar flux contribution, which is
approximated with a blackbody of 3000~K.
}
\label{obsspec_bmgem}
\end{figure}

In Fig.~\ref{obsspec_bmgem}b, we also plot the spectrum of V778~Cyg
obtained with the Short Wavelength Spectrometer (SWS) onboard the
Infrared Space Observatory (ISO) presented in
Yamamura et al. (\cite{yamamura00}) as well as those of IRAS08002-3803 and
IRAS18006-3213 obtained with MIDI by Ohnaka et al. (\cite{ohnaka06}) and
Deroo et al. (\cite{deroo07}), respectively.
BM~Gem and V778~Cyg show a well-defined peak at $\sim$9.8~\mbox{$\mu$m}, while
IRAS08002-3803 and IRAS18006-3213 show broader features.
Yamamura et al. (\cite{yamamura00}) show that the silicate feature of V778~Cyg
is characterized by optically thin emission from sub-micron-sized amorphous
silicate grains.  The close resemblance between BM~Gem and V778~Cyg
indicates the predominance of such small, amorphous silicate grains
around BM~Gem, and the silicate dust toward this object is optically thin.
This is also consistent with the spectral energy distribution
(SED) of BM~Gem plotted in Fig.~\ref{obsspec_bmgem}c, which shows only a
modest infrared excess longward of $\sim$8~\mbox{$\mu$m}.
As Yamamura et al. (\cite{yamamura00}) argue, such small grains in an optically
thin environment are blown away by the radiation pressure of the primary
(carbon-rich) star, suggesting that a circumbinary (or circum-primary) disk
cannot exist for a long period of time.  This argument
is in line with our interpretation of the $N$-band visibilities and
differential phases presented in Sect~\ref{sect_discuss}.

On the other hand, the broad $N$-band spectra of
IRAS08002-3803 and IRAS18006-3213 indicate the presence of grain species
other than small silicate.
In particular, featureless dust emission from large grains,
amorphous carbon, metallic iron, or iron sulfide is suggested for both
objects (Ohnaka et al. \cite{ohnaka06}; Deroo et al. \cite{deroo07}).
Also, the SEDs of these two silicate carbon stars show much larger infrared
excesses longward of 3--4~\mbox{$\mu$m}\ (see Fig.~5 in
Ohnaka et al. \cite{ohnaka06} and Deroo et al. \cite{deroo07}) than BM~Gem.
This suggests the presence of optically thick disks, which is actually
confirmed by the radiative transfer modeling of
Ohnaka et al. (\cite{ohnaka06}).
These differences in mineralogy and optical thickness
seem to affect the wavelength dependence of the observed $N$-band
visibilities and differential phases, as discussed below.

\begin{figure}[!hbt]
\resizebox{\hsize}{!}{\rotatebox{0}{\includegraphics{10229F2.ps}}}
\caption{
{\bf a:} $N$-band visibilities of BM~Gem obtained with MIDI.
The error bars are
shown only for the data sets \#1 and \#4 for visual clarity, but all
error bars are drawn in the inset, which shows the enlarged view of the
visibilities observed between 9.4 and 11.4~\mbox{$\mu$m}.  The data set numbers
are defined in Table~\ref{table_obs}.
{\bf b:} Calibrated differential phases obtained for BM~Gem.
{\bf c:} Instrumental differential phases as measured on the calibrators.
The open circles represent the phase observed on the same night as the
data set \#1, while the open squares and upside-down triangles represent
that observed together with \#3.
}
\label{obsvis_bmgem}
\end{figure}

The $N$-band visibilities observed toward BM~Gem shown
in Fig.~\ref{obsvis_bmgem}a
basically reflect the optically thin silicate emission: the visibilities are
the lowest at $\sim$10~\mbox{$\mu$m}, where the flux contribution of the extended
silicate emission is the highest.
The baseline dependence of the visibilities observed at $\la$9~\mbox{$\mu$m}\ and
at $\ga$11~\mbox{$\mu$m}\ is not significant in the baseline range from 44 to 62~m,
given the errors of the measurements.
This means that the extended silicate emission is completely resolved out
and that the sampled visibility component corresponds to the unresolved
central star (its angular diameter is estimated to be $\sim$2~mas as explained
below), for which the visibility is constant for the baselines used
in our MIDI observations.
In this case, the observed visibilities represent the fractional flux
contribution from the star.
On the other hand, the visibilities observed between 9 and 11~\mbox{$\mu$m}\
increase with baseline length from 44-46~m to 62~m,
as can be seen in
the data \#1, \#2, and \#4 taken at roughly the same
position angle.
Therefore, the observed visibilities contain some information on
the geometry of the silicate emitting region and will be modeled
in the next section.

Figure~\ref{obsvis_bmgem}b shows the calibrated differential phases of
BM~Gem.
Significant non-zero differential phases ($\sim \! -70$\degr\ at 10~\mbox{$\mu$m})
are detected in the central part of the silicate emission between
$\sim$9 and 11~\mbox{$\mu$m}\ in the data sets obtained at the shortest baselines
(\#1 and \#3), while the differential phases observed at the longer baselines
(\#2 and \#4) are close to zero.
As shown in Fig.~\ref{obsvis_bmgem}c, the differential phases derived from
the calibrators for the data sets \#1 and \#3 are close to zero.
The phases of two calibrators for the data set \#3
are in agreement within $\pm$5\degr, which we take as the error of
the observed differential phases
(the phase above
12~\mbox{$\mu$m}\ is more uncertain due to the increasing effect of the
atmospheric water vapor).
This confirms that the non-zero differential phases observed toward BM~Gem
are real, revealing a photocenter shift in the silicate feature caused by
an asymmetric circumstellar environment.
The non-zero values are observed only
in the central part of the silicate feature, although one might expect the
differential phase to deviate from zero over the whole feature.
However, as explained above, the silicate emission is completely
resolved out at $\la$9~\mbox{$\mu$m}\ and $\ga$11~\mbox{$\mu$m}\ even at the shortest
baselines.  The visibilities observed at these wavelengths correspond to
the unresolved, thus symmetric star.
This is why differential phases are nearly zero
at these wavelengths, as well as in the data taken with the longer
baselines.

The wavelength dependence of the $N$-band visibilities observed for
BM~Gem is in marked contrast to that previously observed toward
IRAS08002-3803 and IRAS18006-3213 with MIDI by Ohnaka et al. (\cite{ohnaka06})
and Deroo et al. (\cite{deroo07}), respectively.
The $N$-band visibilities of these two
objects show an increase from 8 to $\sim$10~\mbox{$\mu$m}\ and remain
nearly constant or slightly decrease longward of 10~\mbox{$\mu$m}.
We also note that Deroo et al. (\cite{deroo07}) observed non-zero
differential phases toward IRAS18006-3213, which are characterized by
a significant phase jump at $\sim$8.3~\mbox{$\mu$m},
completely different from that observed toward BM~Gem.
Given that the visibility levels observed toward IRAS18006-3213 range from
$\sim$0.1 to $\sim$0.7, the difference in the $N$-band visibilities and
differential phases cannot be solely attributed to a difference in the
object's angular size and/or baselines used in the observations.
As outlined above, the dusty environment of BM~Gem is very different
in mineralogy and optical thickness from that of IRAS08002-3803 and
IRAS18006-3213.
Therefore, the distinct $N$-band visibilities and differential phases
observed toward BM~Gem most likely reflect the difference in dust chemistry
and the structure of the circumstellar environment
between BM~Gem and IRAS08002-3803/IRAS18006-3213.

\begin{figure*}[!hbt]
\resizebox{\hsize}{!}{\includegraphics[width=12cm]{10229F3.ps}}
\caption{
{\bf a:} $N$-band visibilities of BM~Gem obtained with MIDI (symbols)
and those predicted by the best-fit asymmetric-disk model discussed in
Sect.~\ref{sect_model} (lines).
The error bars are shown only in the inset, which shows the enlarged view of
the visibilities observed between 9.4 and 11.4~\mbox{$\mu$m}.  The data set numbers
are defined in Table~\ref{table_obs}.
The model visibilities are shown by the solid, dashed-dotted, dashed, and
dotted lines for the data sets \#1, \#2, \#3, and \#4, respectively,
with the same colors as the corresponding data sets.
{\bf b:} Calibrated differential phases obtained for BM~Gem (symbols) and
those predicted by the asymmetric-disk model (lines) shown in the same
manner as in the panel {\bf a}.
The dashed-dotted and dotted lines (red and orange lines) are almost entirely
overlapping.
{\bf c--e:} 8, 10, and 13~\mbox{$\mu$m}\ images predicted by the best-fit model.
The radius of the ring is 20~mas with a Gaussian radial cross
section with a FWHM of 27~mas at 10~\mbox{$\mu$m}.
The azimuthal intensity modulation is
characterized by $a$ = 0.2 and $\Phi$ = 280\degr.
The image at each wavelength is normalized so that the integrated flux is
1 (the color on the central star is saturated).
A color version of this figure is available in the electronic
edition.
}
\label{model_bmgem}
\end{figure*}

\section{Geometrical model}
\label{sect_model}

To characterize the asymmetric silicate emitting region,
we adopted an ``asymmetric-disk'' model, in which
an unresolved star is surrounded by a ring with a Gaussian radial cross
section, and the azimuthal brightness distribution of the ring is sinusoidally
modulated.
This model was used to interpret the interferometric data on
Herbig Ae/Be stars by Monnier et al. (\cite{monnier06}).
The intensity at the radial distance $r$ from the star and the position angle
$\phi$ is expressed as
\[
I_{\lambda}(r, \phi) \propto e^{-\left(\frac{r-R_{\lambda}}{\sigma_{\lambda}}\right)^2}
\times \{ a \cos (\phi - \Phi) + 1\},
\]
where $R_{\lambda}$ and $\sigma_{\lambda}$ denote the
radius and the Gaussian width of the ring, respectively.
The intensity contrast between
the brightest and faintest points of the ring is determined by the parameter
$a$, and $\Phi$ is the position angle of the brightest point.
$R_{\lambda}$ and $\sigma_{\lambda}$ are dependent on wavelength,
but they cannot be well constrained by the current MIDI data at $\la$9~\mbox{$\mu$m}\
and $\ga$11~\mbox{$\mu$m}, because the extended component is resolved out at these
wavelengths.  To take into account that the emission at the longer
wavelengths originates in the cooler, outer region, we assumed the radius to be
expressed as
\[
R_{\lambda} = R_{10\mu{\rm m}} \times \left(\frac{\lambda}{10}\right)^p,
\]
where $R_{10\mu{\rm m}}$ denotes the ring radius at 10~\mbox{$\mu$m}.
The ratio between $R_{\lambda}$ and $\sigma_{\lambda}$ was assumed to be
independent of wavelength.
Therefore, the free parameters of the model are $R_{10\mu{\rm m}}$, $p$,
the ratio $\sigma_{\lambda}/R_{\lambda}$, $a$ and $\Phi$.

We first derived the ratio of the stellar flux to the total flux at each
wavelength as follows.  If we approximate
the stellar radiation with a blackbody of effective temperature \mbox{$T_{\rm eff}$}\
(3000~K for BM~Gem, Ohnaka \& Tsuji \cite{ohnaka99}),
the observed flux $F_{\lambda}^{\rm obs}$ (star + extended dust emission) is
given by
\[
F_{\lambda}^{\rm obs} = \theta_{\star}^2 B_{\lambda}(\mbox{$T_{\rm eff}$})/f_{\lambda}^{\star},
\]
where $\theta_{\star}$ and $f_{\lambda}^{\star}$ are the stellar angular radius
and the ratio of the stellar flux to the total flux, respectively.
As mentioned above, the visibilities at $\la$9~\mbox{$\mu$m}\ and $\ga$11~\mbox{$\mu$m}\
represent the fractional flux contribution of the star.
Using the observed 8~\mbox{$\mu$m}\ visibility for $f_{\lambda}^{\star}$
and the observed 8~\mbox{$\mu$m}\ flux, $\theta_{\star}$ was
derived to be 1.1~mas.  Using this $\theta_{\star}$ and the MIDI
spectrum, we determined $f_{\lambda}^{\star}$ for all wavelengths in the
$N$ band.
We note here that as Fig.~\ref{obsspec_bmgem}c shows, the SED of BM~Gem
shortward of $\sim$5~\mbox{$\mu$m}\ is well fitted by the blackbody of 3000~K,
suggesting the absence of hot dust, which would have manifested itself as an
additional (unresolved) component in our MIDI observations.
This verifies our assumption that the observed visibility at 8~\mbox{$\mu$m}\
represents the fractional flux of the star.

The intensities of the point-source-like star and the extended
component at each wavelength were set so that the flux ratio
is consistent with the derived $f_{\lambda}^{\star}$.
The visibility amplitude and phase were calculated from
the two-dimensional complex visibility computed at each wavelength.
The differential phase was calculated in the same manner as
the EWS software derives it from MIDI data:
a least-square-fit with a straight line (i.e., a linear function with
respect to wavenumber) was performed on the model phase as a function
of wavenumber.  The differential phase was then derived by subtracting the
fitted linear component from the model phase.
The sign of differential phase measured with MIDI is defined as described
in Ratzka et al. (\cite{ratzka08}).  In the model fitting, we excluded
the observed differential phase data above 12~\mbox{$\mu$m}\ because of the
larger systematic uncertainties.

Figures~\ref{model_bmgem}a and \ref{model_bmgem}b show a comparison between
the best-fit model and the MIDI data.
This model is characterized by a broad ring with a radius of 20~mas
and a Gaussian radial cross section with a FWHM of 27~mas at 10~\mbox{$\mu$m}\
($\sigma_{\lambda}/R_{\lambda}$ = 0.8).  The wavelength dependence of the ring
radius is characterized by $p = 2$, which results in overall sizes of
$\sim$45, 70, and 120~mas at 8, 10, and 13~\mbox{$\mu$m}, respectively,
as shown in Figs.~\ref{model_bmgem}c--\ref{model_bmgem}e.
The intensity of the ring has a mild azimuthal modulation with $a$ = 0.2,
and the brightest point is offset by 20~mas (at 10~\mbox{$\mu$m}) at a position
angle of 280\degr\ with respect to the star.
Figure~\ref{model_bmgem}a reveals that the observed $N$-band visibilities
are reasonably reproduced by the model.
As shown in Fig.~\ref{model_bmgem}b, the predicted differential phases also
agree fairly with the observed data.
The model predicts the non-zero differential phase values to be
somewhat too large at $\la$9~\mbox{$\mu$m}\ and too small at $\ga$9~\mbox{$\mu$m}\
compared to the observed data.  Also, the model visibility for the data set
\#3 is lower than observed.
However, given the simple nature of the model, the fit can be regarded as
fair.
The $\chi^2$ value is 3.8 for this best-fit model.
The formal errors of the ring radius at 10~\mbox{$\mu$m}\ ($R_{10\mu {\rm m}}$),
$p$, the ratio $\sigma_{\lambda}/R_{\lambda}$, azimuthal brightness
modulation ($a$), and the position angle of the brightest point ($\Phi$) are
$\pm 2$~mas, $\pm 0.5$, $\pm 0.05$, $\pm 0.1$, and $\pm 40$\degr, respectively.
However, it should be kept in mind that
the extended silicate emission is nearly resolved out by our
MIDI observations, which makes it difficult to glean information on
the overall geometry of the silicate emitting region.
Therefore, the above model parameters would better be regarded as
preliminary despite the rather small formal errors,
until they are confirmed by future MIDI observations with
a better $(u,v)$ coverage.

\section{Discussion}
\label{sect_discuss}

Given the spectroscopic signatures of a circum-companion accretion disk
discovered toward BM~Gem by Izumiura (\cite{izumiura03}) and
Izumiura et al. (\cite{izumiura08}), it is tempting to interpret the above
model as a circum-companion disk.
In the scenario of Yamamura et al. (\cite{yamamura00}),
the radiation pressure from the primary star drives an outflow from the
circum-companion disk.
Depending on the orbital period of the binary system and the outflow velocity,
the material blown away from the disk may trail like a cometary tail and show a
spiral structure surrounding the whole binary system.
The material surrounding the whole system (corresponding to the
``cometary tail'') can be represented by the broad ring of the asymmetric-disk
model, while the circum-companion disk (the ``coma'' of the cometary
trail) appears as the offset bright region.
In our model, the offset of the bright feature is dependent on
wavelength as the ring's radius, although the position of the companion
is independent of wavelength.  However, since the mid-infrared emission
reflects the distribution of dust, not exactly the companion itself, such
wavelength dependence of the offset may also be expected.

If the above model represents the circum-companion disk,
the offset of the bright feature from the central star, 13--34~mas,
corresponds to the angular separation between the primary star
and the companion.
This favors a main-sequence star companion rather than a white dwarf
companion, because the luminosity from an accretion disk around a white
dwarf at separations of 13--34~mas (16--41~AU at a distance of 1.2~kpc)
is estimated to be $\sim$5--18~\mbox{$L_{\sun}$}\ in
the Bondi-Hoyle formulation (Eq.~5 in Izumiura et al. \cite{izumiura08}).
This is significantly higher than the observationally derived value of
$\sim$0.2~\mbox{$L_{\sun}$}\ (0.03--0.6~\mbox{$L_{\sun}$}, Izumiura et al. \cite{izumiura08}).
However, it should be noted that the Bondi-Hoyle-type accretion could
overestimate the accretion rate as much as by an order of magnitude
(e.g., Nagae \cite{nagae04} and references therein).
In this case, the predicted accretion luminosity is $\sim$0.5--1.8~\mbox{$L_{\sun}$},
which is marginally compatible with the observed range.
Therefore, a main-sequence star companion seems more likely, although
a white dwarf companion cannot be entirely excluded.

\section{Conclusion}
\label{sect_concl}

Our VLTI/MIDI observations of the silicate carbon star BM~Gem have revealed
that the $N$-band visibilities are characterized by a steep
decrease from 8 to $\sim$10~\mbox{$\mu$m}\ and a gradual increase longward of
10~\mbox{$\mu$m}, reflecting the optically thin 10~\mbox{$\mu$m}\ emission feature
emanating from sub-micron-sized amorphous silicate grains.
Furthermore, we have detected significant non-zero differential phases
($\sim \! -70$\degr) between $\sim$9 and 11~\mbox{$\mu$m}, which reveals the
asymmetric nature of the silicate emitting region.
The wavelength dependence of the observed $N$-band visibilities and
differential phases is remarkably different from that previously observed
toward two silicate carbon stars, IRAS08002-3803 and IRAS18006-3213.
This is most likely to reflect the differences in mineralogy, optical
thickness, and geometry (circumbinary or circum-companion) of the disks.

Our simple geometrical modeling of the observed visibilities and differential
phases suggests the presence of a bright region which is offset from the
central star by 13--34~mas (16--41~AU at 1.2~kpc) at a position angle of
$280 \pm 40$\degr.
This bright region can be interpreted as a circum-companion disk.
The derived offset from the central star makes a white dwarf companion
unlikely (thus leaving a main-sequence star companion more favorable),
because a white dwarf companion at these separations would produce
too much accretion luminosity compared to that observed.

However, the lack of detailed information on the geometry of the
silicate dust distribution makes this circum-companion
disk interpretation not yet definitive.  For example, we cannot yet
entirely exclude the possibility that the asymmetric-disk model
represents a circumbinary disk with an asymmetric structure.
MIDI observations with shorter baselines are indispensable for putting tighter
constraints on the size and shape of the silicate emitting region.
Also, high-resolution imaging at violet/blue wavelengths
is important for direct detection of the
circum-companion accretion disk.  Combination of the violet/blue
high-resolution imaging and MIDI observations will provide definitive
confirmation that silicate dust is stored in a circum-companion disk.

\begin{acknowledgement}
We thank the ESO VLTI team on Paranal and in Garching and the MIDI
team for carrying out the observations and making the data reduction
software publicly available.
We also thank the referee, H.~van~Winckel, for his constructive comments.

\end{acknowledgement}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

The combination of particle production from multiple parton interactions (MPI) (excluding parton-parton scattering with the highest momentum transfer) and beam-beam remnant (BBR) interactions is commonly called the underlying event (UE). The UE usually produces particles at low transverse momenta $p_{T}$ that cannot be experimentally distinguished from particles produced from initial (ISR) and final state radiation (FSR). These processes cannot be completely described by perturbative quantum chromodynamics (QCD) calculations, and require phenomenological models, whose parameters are tuned by means of fits to data. The properties of the UE are measured as a function of conventional observables related to the impact parameter of the pp collision, such as the average number of charged particles and the scalar sum of their $p_{T}.$ The data are corrected for detector effects using the iterative D'Agostini method~\cite{unfold} and compared to Monte Carlo (MC) event generators, as well as with earlier results at $\sqrt{s}$~=~1.96~\cite{uecdf} and 7 TeV~\cite{uedycms}. In this paper, we present recent results on measurements of UE activity using events with a leading track and leading jet at a centre-of-mass energy of 13 TeV~\cite{CMS-PAS-FSQ-15-007}, and measurements of the UE activity using events with a  Z boson (with muonic decay) at a centre-of-mass energy of 13 TeV~\cite{CMS-PAS-FSQ-16-008}. The UE measurement with Z boson events is complementary to measurements with a leading jet and leading track, which corroborate the universality of the UE.

\section{UE measurement using leading charged particle tracks and charged jets}

The UE measuerement is performed using leading charged particles as well as leading charged particles jets as reference objects at the centre-of-mass energy of 13 TeV~\cite{CMS-PAS-FSQ-15-007}. A leading charged particle or charged particle jet is required to be produced in the central pseudorapidity region ($|\eta| <$ 2) and with transverse momentum $p_{T} >$ 0.5 ($p_{T}^{jet} >$ 1) GeV for the leading charged particle (charged particle
jet).

The data used in this analysis are selected from an unbiased sample of events whenever there
is a beam crossing in the CMS detector. This corresponds to an integrated luminosity of 281 nb$^{-1}$.

The UE activity is quantified in terms of the average number of charged particles in an event per unit $\Delta\eta\Delta\phi$ area (particle density) and the scalar sum of their $p_{T}$ in an event per unit $\Delta\eta\Delta\phi$ area ($\Sigma p_{T}$ density), with $|\eta|<$ 2 and $p_{T} >$ 0.5 GeV  in the region orthogonal to the azimuthal direction of the leading charged particle or jet, referred to as the transverse region ($60^{\circ} <|\Delta\phi|<$ 120$^{\circ}$). The transverse region can be split into 2 halves depending on the sign of
$|\Delta\phi|$.  The transMAX (transMIN) densities are then defined as the densities in the transverse half with a higher (lower)
activity.  The transDIF density is then defined as the difference of transMAX and transMIN densities.
Adding all sources of uncertainty in quadrature results in a total systematic uncertainty of about 8--9\%

\begin{figure}[htbp]
\begin{minipage}[t]{.45\textwidth}
\centering
\includegraphics[width=1\textwidth]{ntracks_leadingjet_transmax.pdf}
\caption {{Comparisons of corrected transMAX average particle densities with the various simulations as a function of $p_{T}^{jet}$~\cite{CMS-PAS-FSQ-15-007}.}} \label{fig:TransMAX}
\end{minipage}
\hfill
\begin{minipage}[t]{.45\textwidth}
\centering
\includegraphics[width=1\textwidth]{ntracks_energy_leadingjet.pdf}
\caption {{Comparisons of corrected transAVE particle densities with various simulations at $\sqrt{s} =$ 0.9, 2.76, 7, and 13 TeV as a function of $p_{T}^{jet}$~\cite{CMS-PAS-FSQ-15-007}.}} \label{fig:allEnergyComp}
\end{minipage}
\end{figure}

The transMAX particle densities are shown in figure~\ref{fig:TransMAX}, as a function of $p_{T}^{jet}$.  The measurements are better described by the Monash tune of PYTHIA8. The PYTHIA8 CUETP8M1 tune describes the measurements within 10$-$20\%. The predictions by the CUETHS1 tune of HERWIG fails in the low $p_{T}$ region. EPOS describes the low $p_{T}$ rising region well but fails to describe the plateau region by 20\%. Distributions for the average
$\Sigma p_{T}$ densities (not shown) also reveal similar behaviour. In all plots, the densities increase sharply up to 5 (12$-$15) GeV and then rise slowly with increasing $p_{T}$ ($p_{T}^{jet}$). The level of agreement between simulations and the measurements falls within 10$-$20\%
in  the  plateau  region  but  differs  in  the  low $p_{T}$ region.   The  sharp  rise is  interpreted in the MC models as due to an increase in the MPI contribution which reaches a plateau at high $p_{T}$.  Comparisons between various MC simulated samples and data across centre-of-mass energies of 0.9, 2.76, 7, and 13 TeV are made for transAVE as a function of  $p_{T}^{jet}$ as shown in figure~\ref{fig:allEnergyComp}. There is a strong rise in the UE activity as a function of the centre-of-mass energy, as predicted by the MC tunes. The  transMIN  (not  shown)  densities  exhibit  a  stronger $\sqrt{s}$ dependence  than  the transDIF (not shown) density, indicating that the activity coming from MPI grows more with $\sqrt{s}$ than that from ISR and FSR.

\section{UE measurement using the Z-boson  process}

A measurement of the UE activity using events with a  Z-boson (with muonic decay) at a
centre-of-mass energy of 13 TeV~\cite{CMS-PAS-FSQ-16-008} is presented. The Z-boson production is experimentally clean and theoretically well understood, allowing a clear identification of the UE activity.

The analysis is performed with a data sample of pp collisions corresponding to an integrated luminosity of 2.1 fb$^{\rm -1}$
at a centre-of-mass energy of 13 TeV, collected using the CMS detector at the LHC~\cite{Chatrchyan:2008aa}. Events are triggered with the
requirement of at least two isolated muon candidates with $p_{T} >$ 17 GeV and 8 GeV for leading and subleading muons
respectively.

The offline selection criteria require each event to have at least one well reconstructed primary vertex.
Both muons are required to lie within a range of $|\eta| <$ 2.4.  The events with two oppositely charged muons are further required to have an invariant mass
(M$_{\mu\mu}$) in the window of 81--101 GeV. After all the selections, there are about
1.3 million Z candidate events in the data, which is in agreement with simulated samples within 1--2\%. Selected events
have background contributions, mainly from top-quark and diboson processes, of about 0.3\%.

In the selected Z-boson events, all the tracks with $p_{T}>~$0.5 GeV and $|\eta|<$ 2 are considered for the UE measurements.
The track selection efficiencies for data and simulated samples agree within 4--5\%.

The UE activity is again quantified in terms of the particle density and their $\Sigma p_{T}$ density. These observables are calculated in different phase-space regions defined with respect to the resultant azimuthal direction of the two selected muons, classified as {\it towards}
region ($|\Delta\phi|<$ 60$^{\circ}$), {\it transverse} region ($60^{\circ} <|\Delta\phi|<$ 120$^{\circ}$), and {\it away} region
($|\Delta\phi|>$ 120$^{\circ}$).

For the comparison with predictions from different simulations and tuning of model parameters, the UE
distributions are corrected to the stable charged particle level using the iterative D'Agostini method~\cite{unfold}, which properly considers the bin-to-bin migrations. The unfolded measured distributions can get biased due to the selection criteria and simulated samples used for the
unfolding. The total systematic uncertainty in the particle and $\Sigma p_{T}$ densities is about 4.8--7.8\%.

The unfolded distributions of the UE activity as a function of $p_{T}^{\mu\mu}$ are compared with predictions from various
simulations. In order to understand the MPI evolution with centre-of-mass energies, measurements are also compared
with previous results from Tevatron and LHC.

Figure~\ref{fig:allEnergyComp2} shows the UE activity as a function of $p_{T}^{\mu\mu}$ at centre-of-mass energies of
1.96, 7, and 13 TeV. The predictions of the \textsc{powheg} event generator with \textsc{pythia}8, as well as with
\textsc{herwig}++, are also shown. The ratios of simulations to the measurements are shown in the bottom panel of each plot.
The \textsc{powheg} + \textsc{pythia}8 predictions describe the measurements within 10\% at centre-of mass energies of
1.96 TeV and 7 TeV, and within 5\% at 13 TeV. The combination of \textsc{powheg} and \textsc{herwig}++ describes the
measurements within 10--15\%, 10--20\%, and 20--40\% at a centre-of-mass energy of 13 TeV, 7 TeV, and 1.96 TeV respectively.

\begin{figure}[htbp]
\begin{minipage}[t]{.45\textwidth}
\centering
\includegraphics[width=1\textwidth, height=7.7cm]{4comp_nchg_ptmm_trans_AllEnergies.pdf}
\caption {{Comparison of UE activity measured at $\sqrt{s}=$13 TeV with the measurements at 7 TeV and 1.96 TeV, by the CMS and CDF
experiments, for particle density in the {\it transverse} region as a function of $p_{T}^{\mu\mu}$~\cite{CMS-PAS-FSQ-16-008}.}} \label{fig:allEnergyComp2}
\end{minipage}
\hfill
\begin{minipage}[t]{.45\textwidth}
\centering
\includegraphics[width=1\textwidth]{5ptsum_ptmm_UE.pdf}
\caption {{$\Sigma p_{T}$ density, with $p_{T}^{\mu\mu}$~$<$~5~GeV as a function of centre-of-mass
energy for data, and predictions from simulations by \textsc{powheg} + \textsc{pythia}8, and \textsc{powheg} + \textsc{herwig}++~\cite{CMS-PAS-FSQ-16-008}.}} \label{fig:dataMCpickPt}
\end{minipage}
\end{figure}

To quantify the energy dependence of the UE activity, events with a $p_{T}^{\mu\mu}$ smaller than 5 GeV are considered.
An upper cut on $p_{T}^{\mu\mu}$ reduces the radiation contribution and the resulting UE activity comes mainly from MPI. With a
requirement of $p_{T}^{\mu\mu}$~$<$~5~GeV, the UE activity is similar in the {\it towards} and {\it transverse} regions.
Therefore, the UE activity is combined in these two regions. Figure~\ref{fig:dataMCpickPt} shows the UE activity, after
an upper cut of 5 GeV on $p_{T}^{\mu\mu}$, as a function of the centre-of-mass energy for data and the simulated samples. The
predictions from \textsc{powheg} + \textsc{pythia}8, without MPI, are also shown.
It is clear from the comparison of the distributions, with and without MPI, that there is very small contribution from
radiation, which increases very slowly with centre-of-mass energy. The energy evolution is better described by \textsc{powheg} events hadronized with
\textsc{pythia}8, whereas hadronization with \textsc{herwig}++ overestimates the UE activity at all energies.

\section{Summary}

Measurements of underlying event activity at 13 TeV using events with inclusive Z bosons and leading jets/tracks at 13 TeV are presented.
There is scope of further improvements in the underlying event modeling, especially in the energy dependence. The present measurements, in combination with previous results, will be important for further optimization of the model parameters in various simulations.


text:
\section{Introduction}

Neutrino scattering experiments have contributed to our understanding of
electroweak physics for more than three decades. Early determinations of
$\sin^{2}\theta_{W}$ served as the critical ingredient to the Standard
Model's successful prediction of the W and Z boson masses. More precise
investigations in the late 1980's set the first useful limits on the top
quark mass. Just as early measurements contributed to the accurate
predictions of the W, Z, and top quark masses before their direct
observation, recent results from neutrino-scattering experiments
combined with $e^+e^-$ and $p\bar{p}$ collider data likewise constrain
the Higgs boson mass. \\

The measurement presented here represents the most precise determination
of the electroweak mixing angle from neutrino-nucleon scattering to date.
The result is a factor of two more precise than the previous most accurate
$\nu$N measurement \cite{ccfr}.

\section{Methodology}

In deep inelastic neutrino-nucleon scattering, the weak mixing angle can
be extracted from the ratio of neutral current (NC) to charged current (CC)
total cross sections. Previous measurements relied on the Llewellyn-Smith
formula, which relates these ratios to $\sin^{2}\theta_{W}$ for neutrino
scattering on isoscalar targets \cite{llewellyn-smith}:

\begin{equation}
R^{\nu} \equiv \frac{\sigma(\nu_{\mu}N\rightarrow\nu_{\mu}X)}
                 {\sigma(\nu_{\mu}N\rightarrow\mu^-X)}
= \frac{1}{2}-\sin^2\theta_W+\frac{5}{9}(1+r)\sin^4\theta_W,
\end{equation}

\begin{equation}
R^{\overline{\nu}} \equiv \frac{\sigma(\overline{\nu}_{\mu}N\rightarrow\overline{\nu}_{\mu}X)}
                 {\sigma(\overline{\nu}_{\mu}N\rightarrow\mu^+X)}
= \frac{1}{2}-\sin^2\theta_W+\frac{5}{9}(1+ \frac{1}{r})\sin^4\theta_W,
\end{equation}

where

\begin{equation}
r \equiv \frac{\sigma({\overline \nu}_{\mu}N\rightarrow\mu^+X)}
                {\sigma(\nu_{\mu}N\rightarrow\mu^-X)} \sim \frac{1}{2},
\end{equation}

\noindent
The above equations are exact only for tree level scattering off
an idealized isoscalar target composed of first generation light quarks.
Corrections must be made for the non-isoscalar target, the heavy quark content
of the nucleon, radiative effects, higher-twist processes, the longitudinal
structure function ($R_L$), and charm production. This last effect is
most important. Unfortunately, previous determinations of $\sin^2\theta_W$
measured in this way were subject to the same charm production uncertainties
(resulting from imprecise knowledge of the charm quark mass) that dominated
the CCFR error \cite{ccfr}. This ultimately limited the precision of neutrino
measurements of electroweak parameters.

An alternate method for determining $\sin^2\theta_W$ that is much less
dependent on the details of charm production and other sources of model
uncertainty is derived from the Paschos-Wolfenstein quantity, $R^{-}$
\cite{pw}:

\begin{equation}
R^{-} \equiv \frac{\sigma(\nu_{\mu}N\rightarrow\nu_{\mu}X)-
                   \sigma(\overline{\nu}_{\mu}N\rightarrow\overline{\nu}_{\mu}X)}
                  {\sigma(\nu_{\mu}N\rightarrow\mu^-X)-
                   \sigma(\overline{\nu}_{\mu}N\rightarrow\mu^+X)}
= \frac{R^{\nu}-rR^{\overline{\nu}}}{1-r}=\frac{1}{2}-\sin^2\theta_W
\end{equation}

\noindent
Because $R^-$ is formed from the difference of neutrino and antineutrino
cross sections, almost all sensitivity to the effects of sea quark scattering
cancels. This reduces the error associated with heavy quark production
by roughly a factor of eight relative to the previous analysis. The
substantially reduced uncertainties, however, come at a price.
Unlike $R^{\nu}$, the ratio $R^-$ is more difficult to measure experimentally
because neutral-current neutrino and antineutrino events have identical
observed final states. The two samples can only be separated via
\emph{a priori} knowledge of the incoming neutrino beam type.

\section{The Neutrino Beam and the NuTeV Detector}

High-purity neutrino and antineutrino beams were provided by the Sign Selected
Quadrupole Train (SSQT) at the Fermilab Tevatron during the 1996-1997
fixed target run. Neutrinos are produced from the decay of pions and kaons
resulting from interactions of 800 GeV protons in a BeO target. Dipole magnets
immediately downstream of the proton target bend pions and kaons of specified
charge in the direction of the NuTeV detector, while wrong-sign and neutral
mesons are stopped in beam dumps. The resulting beam is almost purely
neutrino or antineutrino depending on the selected sign of the parent mesons.
The measured $\overline{\nu}_{\mu}$ contamination in the $\nu_{\mu}$ beam is less than
1/1000 and the $\nu_{\mu}$ contamination in the $\overline{\nu}_{\mu}$ beam is less than
1/500. In addition, the beam is almost purely muon-neutrino with a small
contamination of electron neutrinos ($1.3\%$ in neutrino mode and
$1.1\%$ in antineutrino mode).

Neutrino interactions are then observed in the NuTeV detector, which is
located approximately 1.5 km downstream of the proton target.
The detector consists of an 18m long, 690 ton steel-scintillator target
followed by an instrumented iron-toroid spectrometer (Figure \ref{fig:labe}).
The target calorimeter is composed of 168 3 m x 3 m x 5.1 cm steel plates
interspersed with liquid scintillation counters (spaced every two plates)
and drift chambers (spaced every four plates). The scintillation counters
provide triggering information as well as a determination of the longitudinal
event vertex, event length and visible energy deposition. The mean position
of hits in the drift chambers help establish the transverse event vertex.
The toroidal spectrometer, which determines muon sign and momentum, is not
directly used for this analysis.

\begin{figure}[ht]
\centerline{\epsfxsize 3.5 truein \epsfbox{zeller0101fig1.eps}}
\caption[]{The NuTev detector.}
\label{fig:labe}
\end{figure}

\noindent
The detector was continuously calibrated through exposure to a wide energy
range of test beam hadrons, electrons and muons delivered in a separate
beamline during each accelerator cycle.

\section{Event Selection}

For inclusion in this analysis, events must deposit at least 20 GeV of visible
energy in the detector. This cut ensures full efficiency of the event
trigger, proper vertex determination, and reduction of cosmic ray background
events. Events must also have a vertex that lies within a 1.0 m box around
the center of the detector which is at least 0.4 m of steel from the upstream
end and 2.4 m of steel from the downstream end of the detector. This
``box cut'' ensures shower containment and minimizes the contribution
from $\nu_{e}$ interactions. The chosen longitudinal fiducial volume ensures
both that the event is neutrino-induced and that a meaningful event length can
be measured. After all cuts, the total analysis sample consists of 1.3 million
neutrino and 0.3 million antineutrino events with a mean energy of
approximately 125 GeV.

\section{Event Separation}

In order to measure $\sin^2\theta_W$, observed neutrino events must be
separated into charged current (CC) and neutral current (NC) categories.
Figure \ref{fig:events} shows typical CC and NC candidate events in the
NuTeV detector.

\begin{figure}[hbt]
\begin{center}

\mbox{\psfig{figure=zeller0101fig2.eps,width=2.5in,bbllx=185bp,bblly=375bp,bburx=1012bp,bbury=770bp,clip=0,silent=0}
\epsfxsize=1.0in\epsfbox[0 -30 148 148]{zeller0101fig3.eps}}

\mbox{\psfig{figure=zeller0101fig4.eps,width=2.5in,bbllx=185bp,bblly=375bp,bburx=1012bp,bbury=770bp,clip=0,silent=0}
\epsfxsize=1.0in\epsfbox[0 -30 148 148]{zeller0101fig5.eps}}
\caption[]{Event displays of a NuTeV charged current candidate event (top)
           and neutral current candidate (bottom).}
\label{fig:events}
\end{center}
\end{figure}

Both CC and NC neutrino interactions initiate a cascade of hadrons in the
target that is registered in both the scintillation counters and drift
chambers. Muon-neutrino CC events are distinguished by the presence of a final
state muon. The muon typically penetrates well beyond the hadronic shower and
deposits energy characteristic of a minimum-ionizing particle in a large
number of scintillation counters. The muon track is clearly visible in the CC
event at the top of Figure \ref{fig:events}. On the other hand, there is no
final state muon in muon-neutrino NC events; the final state neutrino is
invisible, so $\nu_{\mu}$ NC events can only be recognized by their hadronic
shower.

Given the differing event topologies, one can statistically separate CC and NC
interactions on the basis of event length (\emph{i.e.}, on the presence or
absence of a muon in an event). The length of an event is based on the
longitudinal energy deposition in the calorimeter and is simply the number of
scintillation counters spanned by the event. Events with a long length
(spanning $>$ 20 counters) are identified as CC candidates and events with a
short length (spanning $\leq$ 20 counters) are identified as NC candidates.
The experimental quantity that is measured in both modes is the ratio:

\begin{equation}
{\rm\textstyle R_{meas}} = \frac{\# \hspace{0.05in} \rm\textstyle SHORT \hspace{0.05in} events}
                       {\# \hspace{0.05in} \rm\textstyle LONG \hspace{0.05in} events}
              = \frac{\# \hspace{0.05in} \rm\textstyle NC  \hspace{0.05in} candidates}
                     {\# \hspace{0.05in} \rm\textstyle CC \hspace{0.05in} candidates}
\end{equation}

Neither sample is of course entirely pure. Long events are predominantly
$\nu_{\mu}$ CC interactions but contain small contaminations from both
$\nu_{\mu}$ NC events where the hadronic shower fluctuates to long lengths,
as well as muons which are produced in upstream neutrino interactions in the
shielding and later undergo a hard bremsstrahlung interaction in the detector.
Short events are primarily $\nu_{\mu}$ NC interactions but also contain
contributions from low energy $\nu_{\mu}$ CC events in which the muon ranges
out, wide angle $\nu_{\mu}$ CC events in which the muon exits the side of
the detector, $\nu_{e}$ CC events and cosmic rays. The ratios
({$\rm\textstyle R_{meas}$})
of short to long events measured from the NuTeV data, are 0.4198 $\pm$ 0.0008
in the neutrino beam and 0.4215 $\pm$ 0.0017 in the antineutrino beam.
A Standard Model value of $\sin^2\theta_W$ can be directly extracted from
these measured ratios by using a detailed Monte Carlo simulation of the
experiment. The Monte Carlo must include the integrated neutrino fluxes,
the neutrino cross section as well as a detailed description of the NuTeV
detector. The following sections will discuss these three components.

\section{The muon-neutrino and electron-neutrino Fluxes}
The Monte Carlo simulation requires neutrino flux information as input.
In particular, a precise determination of the electron neutrino contamination
in the beam is essential. The measured short to long ratios are directly
impacted by the presence of electron neutrinos in the data sample because
$\nu_{e}$ charged current interactions, which usually lack an energetic
muon in the final state, are almost always identified as neutral current
interactions in the detector.

The electron neutrino flux is estimated using a detailed beam Monte Carlo.
Most of the observed $\nu_{e}$'s result from
$K^{\pm}\rightarrow \pi^0e^{\pm} \nu_{e}$\hspace{-0.15in}$^{^{(-)}}$ decay.
The beam simulation can be tuned to describe $\nu_{e}$ and
$\overline{\nu}_{e}$ production from charged kaon decay with high accuracy
given that the $K^{\pm}$ decay contribution is well constrained from
measurements of the observed $\nu_\mu$ and $\overline{\nu}_\mu$ fluxes
(as shown in Figure \ref{fig:flux}). Because of the precise alignment of the
beamline elements and the low acceptance for neutral particle propagation
in the SSQT, the largest uncertainty in the calculated electron neutrino flux
results from the 1.5\% uncertainty in the
$K^{\pm}\rightarrow \pi^0e^{\pm} \nu_{e}$\hspace{-0.15in}$^{^{(-)}}$ branching
ratio. The result is a factor of three reduction in the electron-neutrino flux
uncertainty when compared to CCFR.

\hspace{0.1in}
\begin{figure}[ht]
\centerline{\epsfxsize 6.5 truein \epsfbox{zeller0101fig6.eps}}
\caption[]{$\nu_{\mu}$ and $\overline{\nu}_\mu$ energy spectra comparing
           data (points) and the tuned beam Monte Carlo (histogram).}
\label{fig:flux}
\end{figure}

\section{Cross Section Model}

Neutrino-nucleon deep-inelastic scattering processes are simulated using a
leading order cross section model. The cross section model incorporates
leading order parton momentum distributions measured using the same target
and cross section model as NuTeV \cite{ccfr_sf}. Small modifications adjust
the parton densities to produce the inherent up-down quark asymmetry
consistent with muon scattering \cite{muon} and Drell-Yan \cite{drell} data.
A leading order analysis of dimuon events in CCFR \cite{dimuon} provides
the shape and magnitude of the strange sea. Mass suppression from
charged-current charm production is modeled using a leading order
slow-rescaling formalism whose parameters are measured from the same
high-statistics dimuon sample. Electroweak and QED radiative corrections
to the scattering cross sections are applied using radiative correction
routines supplied by D. Yu. Bardin \cite{bardin}. Higher twist contributions
are quantified using a Vector-Dominance-Model as implemented by Pumplin
\cite{pumplin} and constrained by lepto-production data \cite{lepto}.
A global analysis performed by L. Whitlow \cite{whitlow} provides a
parameterization of the longitudinal structure function, $R_{L}$.

\section{Detector Response}

The Monte Carlo must also accurately simulate the response of the detector to
the product of neutrino interactions in the target. The critical detector
parameters that must be modeled are the calorimeter response to muons,
measurement of the neutrino interaction vertex, and the range of hadronic
showers in the calorimeter. Precise determination of the various detector
effects are made possible through extensive use of both neutrino data and
large samples of calibration beam data. The efficiency, noise, and effective
size of the scintillation counters are measured using neutrino data or
test beam muons. Vertex finding resolutions and biases are studied using
neutrino data combined with information from a detailed GEANT-based simulation
of the detector. Test beam pion data provides information on hadronic shower
lengths.

\section {Event Length Distributions}

An important test of the Monte Carlo is its ability to predict the length
distribution of events in the detector. Figure \ref{fig:length} shows
event length distributions of the final data sample compared to the Monte
Carlo prediction assuming the best-fit $\sin^2\theta_W$ (see next section).
Events reaching the toroid, which comprise about 80\% of the CC sample, have
been left out for clarity but are included in the normalization of the Monte
Carlo to the data. Data and MC agree well in each beam mode.

\hspace{0.1in}
\begin{figure}[ht]
\centerline{\epsfxsize 4.5 truein \epsfbox{zeller0101fig7.eps}}
\vspace{0.3in}
\caption[]{Event length comparisons for neutrino and antineutrino events.
           The solid curve is the Monte Carlo prediction. The neutral current
           to charged current event separation (indicated by the arrow) is
           made at a length of 20 counters, approximately 2m of steel.}
\label{fig:length}
\end{figure}
\hspace{0.1in}

\section{Extraction of $\mathbf{\sin^2\theta_W}$}

Given separate high-purity neutrino and antineutrino data sets, NuTeV
measures the following linear combination of $R^{\nu}$ and $R^{\overline{\nu}}$:

\begin{equation}
     R^{-} = R^{\nu}-x R^{\overline{\nu}},
\end{equation}

\noindent
where x is chosen to be 0.5316. This value is obtained using Monte Carlo to
minimize the dependence of $R^{-}$ on the charm quark mass. This approach
explicitly minimizes uncertainties related to the suppression of charm quark
production, largely eliminates uncertainties related to sea quark scattering,
and reduces many of the theoretical and detector uncertainties common to both
the neutrino and antineutrino samples. The single remaining free parameter in
the Monte Carlo, $\sin^2\theta_W$, is then varied until the model calculation
of $R^{-}$ agrees with what is measured in the data. The preliminary result
from the NuTeV data sample for $M_{\rm\scriptstyle top}$=175 GeV and
$M_{\rm\scriptstyle Higgs}$=150 GeV is:

\begin{equation}
    \sin^2\theta_W^{({\rm\scriptstyle on-shell)}}=0.2253\pm0.0019({\rm\textstyle stat.})\pm0.0010({\rm\textstyle syst.})
\end{equation}

\noindent
Leading terms in the one-loop electroweak radiative corrections \cite{bardin}
to the W and Z self-energies produce a small residual dependence of our
result on $M_{\rm\scriptstyle top}$ and $M_{\rm\scriptstyle Higgs}$. We explicitly quote the
dependence of our measurement on the top and Higgs masses. The effect
is small given the existing uncertainty on the top quark mass and the
logarithmic dependence on the Higgs mass. For example, the total change in
$\sin^2\theta_W$ from varying the Higgs mass from 100 GeV to 1 TeV is only
0.0005 (as is indicated by the shaded gray bands outlining the NuTeV result
in Figure \ref{fig:mt-mw}).

\begin{equation}
  \delta \sin^2\theta_W=
         -0.00435\left[\left(\frac{M_{\rm\scriptstyle top}}{175 GeV}\right)^{2}-1\right]
         +0.00048\log \frac{M_{\rm\scriptstyle Higgs}}{150 GeV}. \nonumber
\end{equation}

\noindent
Having chosen the convention,
$\sin^2\theta_W^{({\rm\scriptstyle on-shell)}}\equiv 1- \frac{M_{W}^2}{M_{Z}^2}$,
where $M_{W}$ and $M_{Z}$ are the physical gauge boson masses,
our result implies:

\begin{equation}
    M_{W}=80.26\pm0.10({\rm\textstyle stat.})\pm0.05(\rm\textstyle syst.)
         =80.26\pm0.11 GeV
\end{equation}

\noindent
A comparison of this result with direct measurements of $M_W$ is shown in
Figure \ref{fig:mw}. Our measurement is in good agreement with Standard Model
expectations and is consistent with the most recent measurements from W and Z
production as well as from other neutrino experiments
(Figure \ref{fig:mt-mw}).

\begin{figure}[ht]
\mbox{
\begin{minipage}{0.4\textwidth}
\centerline{\epsfxsize 2.5 truein \epsfbox{zeller0101fig8.eps}}
\caption[]{Direct W boson mass measurements compared with this result
           (at the time of DPF99).}
\label{fig:mw}
\end{minipage}\hspace*{0.02\textwidth}
\begin{minipage}{0.4\textwidth}
\centerline{\epsfxsize 2.5 truein \epsfbox{zeller0101fig9.eps}}
\caption[]{Experimental constraints presented on the $M_{W}$-$M_{\rm\scriptstyle top}$
           plane (at the time of DPF99). The two narrow lines indicate
           the Standard Model predictions for $M_{\rm\scriptstyle Higgs}$=100 and
           1000 GeV.}
\label{fig:mt-mw}
\end{minipage}
}
\end{figure}

\noindent

If these plots are now updated to include new measurements from
CDF \cite{cdf}, D\O \cite{dzero} and LEP \cite{lep} that were presented at
Electroweak Moriond in March of this year, it is interesting to note that
tighter constraints on the Higgs mass are beginning to emerge
(Figure \ref{fig:mt-mw99}).

\begin{figure}[ht]
\mbox{
\begin{minipage}{0.4\textwidth}
\centerline{\epsfxsize 2.5 truein \epsfbox{zeller0101fig10.eps}}
\caption[]{Direct W boson mass measurements compared with this result (updated
           to include new measurements presented at EW Moriond99).}
\label{fig:mw99}
\end{minipage}\hspace*{0.02\textwidth}
\begin{minipage}{0.4\textwidth}
\centerline{\epsfxsize 2.5 truein \epsfbox{zeller0101fig11.eps}}
\caption[]{Experimental constraints presented on the $M_{W}$-$M_{\rm\scriptstyle top}$
          plane (updated to include new measurements presented at EW
          Moriond99).}
\label{fig:mt-mw99}
\end{minipage}
}
\end{figure}

\section{Conclusions}
NuTeV has successfully completed its data-taking and has extracted a
preliminary value of $\sin^2\theta_W$. The precision of this result is
a factor of two improvement over previous measurements in $\nu N$
scattering because of reduced systematics associated with measuring the
Paschos Wolfenstein ratio, $R^{-}$. Interpreted within the framework of
the Standard Model, this result is equivalent to a determination of the
W mass and is consistent with direct measurements of $M_{W}$.