text:
\section{Introduction}
\noindent Throughout this paper, without other mention, $k$ will denote a field of characteristic zero. A
$k$-scheme $X$ of finite type is called {\it rationally chain connected} if for an uncountable
algebraically closed field $K$ containing $k$, any two $K$-points of $X$ can be connected by a chain
of rational curves defined over $K$. A $k$-scheme $X$ is called
{\it rationally connected} if any two $K$-points of $X$ can be
connected by an irreducible rational curve defined over $K$. Thus
a rationally connected variety is always geometrically
irreducible. See \cite{kollarrc} for general background on
rational connectedness and rational chain connectedness. The
notions of rational connectedness and rational chain connectedness
are known to coincide when $X/k$ is smooth and proper. A result of
Campana and Koll\'ar-Miyaoka-Mori says that all smooth Fano
varieties are rationally connected (see\cite{Ca}, \cite{KMM} or \cite{kollarrc}). More generally, log ${\mathbb Q}$-Fano
varieties are rationally connected (\cite{zhang} or \cite{hm05}).
A degeneration of a proper rationally chain
connected variety is rationally chain connected, but
a degeneration of a proper rationally connected variety may not be
rationally connected. For example a cone over an elliptic curve is
not rationally connected but it is a degeneration of smooth cubic
surfaces which are rationally connected. This paper has been
motivated by the following question raised in \cite{kollarax}.

\begin{q}{\rm (Koll\'ar)}\label{mainq'}
Suppose we have a family of smooth proper rationally connected varieties
over a field $k$. Does the degeneration $X_0$ contain an rationally connected (in particular geometrically irreducible) subvariety ?
\end{q}

\noindent In \cite{kollarax} it was proved that
a degeneration of Fano varieties over a smooth curve always contains a geometrically irreducible subvariety.
A recent paper by Jason Starr \cite{starr}
shows that if the field $k$ (of any characteristic) contains an algebraically closed field, then
a degeneration of smooth proper seperably rationally connected varieties contains
a geometrically irreducible subvariety. In fact, A.J.de Jong proved, under the same assumption as in \cite{starr}, if ${\rm char}(k)=0$, then the degeneration contains a rationally connected variety. As observed in \cite{kollarax}, the
existence of a geometrically irreducible subvariety can fail even for a degeneration of elliptic curves. \\

\noindent  The following result, which is the main result of this paper, gives an affirmative answer to (\ref{mainq'}).

\begin{thm}\label{deg-ran}
Let $\pi:X \to C$ be a dominant proper morphism of $k$ varieties such that
\begin{enumerate}
\item $C$ is $klt$, and
\item The generic fibre of $\pi$ is a rationally connected variety.
\end{enumerate}
\noindent Then for any point $0 \in C$, $X_0=\pi^{-1}(0)$ contains a subvariety (defined over $k(0)$) which is geometrically irreducible and rationally connected.
\end{thm}

\noindent Our approach to (\ref{mainq'}) uses the Minimal Model Program, particularly as developed in \cite{mmpgen}. After running the minimal model program for a suitable birational model for X we obtain a Fano fibration. This is then used to reduce (\ref{deg-ran}) to the case when the generic fibre of $\pi$ is
a log Fano variety (\ref{mainthm}). We first prove (\ref{mainthm}) in the special case when $C$ is a smooth curve and $0\in C$ is a $k$-point. The general case of arbitrary $klt$ base $C$ is then reduced to this case using the following result.

\begin{thm}\label{exc.thm}
Let $(X,\Delta)$ be a $klt$ pair with
$\Delta$ effective. Let $\pi:X'\to X$ be a proper birational morphism. Then
for any point $p\in X$, there exists a rationally connected subvariety
of $\pi^{-1}(p)$ (defined over $k(p)$).
\end{thm}

\noindent The following definition was suggested by J. Koll\'ar.
\begin{defn}
A field $k$ is called {\it geometrically-$C_1$} if every smooth proper rationally connected
variety over $k$ has a $k$-point.
\end{defn}

\noindent Following are some of the consequences of (\ref{deg-ran}).
\begin{cor}\label{fanocor}
Every geometrically-$C_1$ field (of characteristic 0) is $C_1$.
\end{cor}
\begin{proof}
To show  that $k$ is $C_1$  we have to show that every hypersurface $H_d$ of degree $d$ in $\P^n_k$
has a $k$-point if $n \geq d$. If $H_d$ is smooth, it is Fano and hence
rationally connected and has a $k$-point by the given hypothesis. Even if $H_d$ is not smooth it
can be expressed as a degeneration of smooth Fano hypersurfaces and has a a rationally
connected subvariety $Z$ by (\ref{deg-ran}). Then $Z$ has a $k$-point since
its resolution $Z'$ is smooth proper and rationally connected and has a $k$-point
by hypothesis.
\end{proof}

\begin{cor}\label{surfacecor}
Let $k$ be a $C_1$ field. Then every degeneration
of a proper rationally connected surface defined over $k$ has a $k$-point.
\end{cor}

\begin{rem} (In this remark, $k$ is not necessary of characteristic 0).
The converse of (\ref{fanocor}) is a well known problem.
It is known that a smooth proper rationally connected variety $X/k$ has a $k$-point if
\begin{enumerate}
\item $k$ is any $C_1$ field and $\dim(X)\leq 2$ (see for example (\cite{kollarrc}, {\rm IV.6.8})),
\item $k$ is a field of transcendence degree one over an algebraically closed field
of characteristic zero \cite{ghs} (in the positive characteristic case one needs to
assume $X$ is separably rationally connected \cite{ds03}), or
\item $k$ is a finite field \cite{He}.
\end{enumerate}

\noindent However, the converse of (\ref{fanocor}) is unknown even for specific $C_1$ fields like the
maximal unramified extension of local fields.

\noindent In view of (\ref{deg-ran}), if a field $k$ is geometrically-$C_1$,
then not only do smooth proper rationally connected varieties over $k$ have
$k$-points, but one also gets $k$-points on the degenerations of such varieties.
In fact, besides (\ref{surfacecor}), the following was already known.
\begin{enumerate}
\item A degeneration of
smooth rationally connected varieties over a finite field has
a rational point (\cite{nr}, \cite{ex07}).
\item A degeneration of seperably rationally connected variety over a field of
transcendence degree one over an algebraically closed field has a rational point
(this result has been mentioned in \cite{starr} and can be derived by using
the properness of Kontsevich's moduli space of stable maps).
\end{enumerate}
\end{rem}

\noindent A stronger version of (\ref{mainq'}) was asked in \cite{kollarax}.

\begin{q}{\rm (Koll\'ar)}\label{mainq}
For a flat family of proper varieties over a smooth $k$-variety whose general fiber is
smooth and rationally connected, does every simple normal crossing ($snc$)
fiber $X_0$ have an irreducible component (defined over $k$) which is rationally connected (and
thus geometrically irreducible)?
\end{q}

\begin{rem}
Unlike (\ref{mainq'}), the answer to this question is unknown even if $k$ is algebraically closed.
Over an algebraically closed field ${\overline{k}}$ one can ask the following
more general question: Does a $snc$ rationally chain connected variety $X_0/{\overline{k}}$
always contain an irreducible component which is rationally connected? The answer to
this question is indeed true if $\dim(X_0)\leq 2$ but is false in higher dimensions (see (\ref{counterex})).
\end{rem}

\noindent We show that (\ref{mainq}) has an affirmative answer in the case that the generic fiber satisfies $\dim(X_t) \le 3$.
\begin{thm}\label{surface}
Let $\pi:X/k \to C/k$ be a dominant proper morphism
such that
\begin{enumerate}
\item $C$ is a smooth curve with a $k$-point $0$,
\item $\pi$ is smooth outside $0$,
\item The generic fibre of $\pi$ is geometrically rationally connected and $\dim(X_t)\le 3$, and
\item $X$ is smooth, and $X_0= \pi^{-1}(0)$ is $snc$ (over $k$).
\end{enumerate}
\noindent Then there exists an irreducible component of $X_0/k$ which is rationally connected.
\end{thm}

\noindent In Section \ref{section:fano} we prove Theorem (\ref{deg-ran}) in the case of degeneration of log Fano varieties, assuming Theorem (\ref{exc.thm}). In Section \ref{section:mmp} we prove Theorem (\ref{deg-ran}) using the results of the minimal model program as proved in (\cite{mmpgen}). The proof uses results of Section \ref{section:fano} and some results on Fano fibrations which are proved in Section \ref{section:surface}. The stronger result, Theorem (\ref{surface}), for degeneration of rationally connected threefolds (or lower dimension) is proved in Section \ref{section:surface}. Finally, in Section \ref{section:exc} we prove Theorem (\ref{exc.thm}).  \\

\noindent{\bf Acknowledgement}: We are indebted to our advisor, J\'anos Koll\'ar, for many
useful discussions and crucial suggestions. The relevance of \cite{hm05} and
minimal model program was pointed out to us by him. We would also like to thank
Tommaso de Fernex and H\'el$\grave{\rm e}$ne Esnault for useful suggestions and Yuri Prokhorov for informing us his example (\ref{counter}).
The second author would like to thank Jason Starr for
useful discussions. We are also grateful to the anonymous referee for enormous suggestions and corrections.

\section{Summary of known results}

\noindent In this section we first recall the related results from \cite{hm05} and \cite{kollarax}. In our proof, their ideas are heavily used. Then we state another main ingredient which is from the minimal model program theory (cf. \cite{kollarmori}) and proved in \cite{mmpgen}. \\ \\ \noindent The following is the main theorem of \cite{hm05}, stated in a slightly less generality.

\begin{thm}[\cite{hm05}]
  Let $(X,\Delta)$ be a $klt$ pair. Let $g:Y\to X$ be a birational morphism such that $-K_X$ is relatively big and ${\mathcal O}_X(-m(K_X+\Delta))$ is relatively generated for some $m>0$. Then every fiber of the the composite morphism $\pi:Y\to S$ is rationally chain connected.
\end{thm}

\noindent Specialising to the case when $g$ is the identity morphism, $(X,\Delta)$ is rationally chain connected and $Y\to X$ is a resolution of singularities, one gets the following important corollary.

\begin{cor}[\cite{hm05}]
Let $(X,\Delta)$ be a $klt$ pair which is rationally chain connected. Then $X$ is rationally connected.
\end{cor}
\noindent Recall that there exist singular varieties (for example the cone over an elliptic curve) which are rationally chain connected but not rationally connected. However, the above corollary says that such examples necessarily have singularities worse than $klt$. \\

\noindent In Section \ref{section:fano}, we consider the degeneration of Fano varieties (see (\ref{mainthm})), which has also been considered in \cite{hm05}. Part of (\cite{hm05}, 5.1) guarantees the existence of an irreducible component which is rationally chain connected modulo the intersection of some other components.
But this component may be neither geometrically irreducible nor rationally connected.
Although the statement of (\cite{hm05}, 5.1) does not imply (\ref{mainthm}),
a modification of methods used in (\cite{hm05}) will be enough to prove it. \\

\noindent In their paper, They prove the following technical result (\cite{hm05}, 4.1), which we only state in a weaker form we need.

\begin{prop}[\cite{hm05}, 4.1]\label{criterion}
Let $(F,\Delta)$ be a projective log pair, and let $t : F\dashrightarrow Z$ be a dominant rational map, respectively, where $F$ and $Z$ are projective,
with the following properties:
\begin{enumerate}
\item the locus of log canonical singularities of $(K_F+\Delta)$ does not dominate $Z$;
\item $F+\Delta$ has Kodaira dimension at least zero on the general fibre of $F\dashrightarrow Z$ (i.e.,
if $g : F'\to F$ resolves the indeterminacy of $F\dashrightarrow Z$, then $g^*(K_F + \Delta)$ has
Kodaira dimension at least zero on the general fibre of the induced morphism $F' \to Z$;
\item $K_F+\Delta$ has Kodaira dimension at most zero; and
\item there is an ample divisor $A$ on $F$ such that $A\le\Delta$.
\end{enumerate}
Then either $Z$ is a point, or it is uniruled.
\end{prop}

\noindent The main application of this proposition is when $t$ is the rational map from $F$ to its maximal rationally connected fiberation (cf. \cite{kollarrc}, IV Theorem 5.4). In fact, with the help of the following lemma, we obtain a quite useful criterion to show the rational connectedness of varieties.

\begin{lemma}[\cite{hm05}, 4.2(i)]\label{MRC}
Let $F$ be a normal variety. $F$ is rationally connected if and only if, for every nonconstant dominant
rational map $t: F \dashrightarrow Z$, $Z$ is uniruled.
\end{lemma}
\noindent Next we recall the main result of \cite{kollarax}.

\begin{thm}[\cite{kollarax}] Let $k$ be a field of characteristic zero and $C$ be a smooth curve, $Z$ an irreducible projective variety and $g:Z\to C$ a morphism. Assume that the general fibers $F_{gen}$ are
\begin{enumerate}
\item smooth,
\item geometrically connected, and
\item Fano.
\end{enumerate}
Then for every point $c$ of $C$, the fiber $g^{-1}(c)$ contains a $k(c)$-subvariety which is geometrically irreducible.
\end{thm}

\noindent In our proof, as a special case, and also as a middle step, we improve Koll\'ar's result by showing that the subvariety which he looks at is not only geometrically irreducible but also rationally connected (see (\ref{specialcase})). Then Theorem(\ref{deg-ran}) generalizes the above result to the case where the general fibers are rationally connected and $C$ is any $klt$ base. Moreover (\ref{deg-ran}) guarantees the existence of a subvariety which is in fact rationally connected.\\

\noindent Finally, in the very important paper \cite{mmpgen}. A large part of the minimal model program is proved. In particular, we state the following theorem which is a special case of their results. This will be the main tool in the proof of (\ref{deg-ran}).

\begin{thm}[\cite{mmpgen}]\label{mmp}
Let $(X,\Delta)$ be a $klt$ pair, projective over a base $S$. Assume $K_X+\Delta$ is not pseudoeffective. Then one can run a relative minimal model program for $(X,\Delta)$ to obtain a Fano fibration, i.e. there exists a sequence of maps
$$ X=X_0\stackrel{\phi_1}{\dashrightarrow}X_1 \stackrel{\phi_2}{\dashrightarrow} \cdots \stackrel{\phi_{r}}{\dashrightarrow}X_r  \stackrel{h}{\to} Y $$
where each $\phi_i$ is either a divisorial contraction or a flip and $h$ is a Fano fibration of relative Picard number one.
\end{thm}

\noindent The important case in which the above hypothesis is satisfied is when $X$ is a smooth rationally connected variety and $\Delta$ is empty. Indeed, in this case, the canonical divisor $K_X$ has negative intersection with any free rational curve and hence it cannot be pseudoeffective.

\section{Degenerations of Fano Varieties}\label{section:fano}

\noindent In this section we prove (\ref{deg-ran}) in the case when the generic fibre of $\pi:X\to C$ is a log Fano variety. Recall that a proper variety $X/k$ is called {\it log Fano} (or {\it log ${\mathbb Q}$-Fano}) if there exists an effective divisor $D$ on $X$ such that $(X,D)$ is $klt$ and $-(K_X+D)$ is nef and big. It is known that a log Fano variety is always rationally connected (cf. \cite{zhang}, \cite{hm05} ).

\begin{thm}\label{mainthm} Let $\pi:X \to C$ be a dominant proper morphism of $k$ varieties such that
\begin{enumerate}
\item $C$ is $klt$, and
\item The generic fibre of $\pi$ is a log Fano variety.
\end{enumerate}
\noindent Then for any point $0 \in C$, $X_0=\pi^{-1}(0)$ contains a subvariety (defined over $k(0)$) which is geometrically irreducible and rationally connected (over $k(0)$). In another words, Theorem (\ref{deg-ran}) is true in the case when the generic fibre of $\pi$ is a log Fano variety.
\end{thm}

\noindent The extension theorem (\cite{hm06}, 3.17) lies at the heart of the proof. We have to first introduce the terminology there.

\begin{defn}{\rm (\cite{hm06}, $3.7$)} Let $\pi:(X,\Delta)\to S$ be a relative log pair.
A Cartier divisor $D$ on $X$ will be called {\it $\pi$-transverse} to $(X,\Delta)$
if the natural map
$$\pi^*\pi_*({\mathcal O}_X(D))\to {\mathcal O}_X(D)$$ is sujective at the generic point of every log canonical centre
of $(X,\Delta)$. Obviously, the notion only depends on the divisor class of $D$. Similarly, a ${\mathbb Q}$-divisor $D$ will be called {\it $\pi$-${\mathbb Q}$-transverse} to
$(X,\Delta)$ if $mD$ is $\pi$-transverse
for some $m>0$.
\end{defn}

\noindent The proof of (\ref{mainthm}) relies on the following extension theorem
proved in \cite{hm06}. The statement given below is less general
than the one in \cite{hm06}.

\begin{thm}{\rm (\cite{hm06}, 3.17)}\label{extensionthm}
Let $\pi:X\to S$ be a projective morphism where $X/k$ is a smooth
variety of dimension $n$.
Let $F\subset X$ be a smooth divisor.
Let $B$ be effective ${\mathbb Q}$-divisor on $X$ such that
\begin{enumerate}
\item $B+F$ is $snc$,
\item the support of $B$ and $F$ do not have common component,
\item $\rdown{B}=0$,
\item $K_F+ B_{|F}$ is effective, and
\item $K_X +B+F$ is $\pi$-${\mathbb Q}$-transverse to $(X,\rup{B}+F)$.
\end{enumerate}
For every sufficiently $\pi$-ample divisor $H$ on $X$ and for every sufficiently
divisible positive integer $m$, the image of the morphism
$$ \pi_*({\mathcal O}_X( m(K_X+B+F)+(n+2)H))\to \pi_*({\mathcal O}_F(m(K_F+B_{|F})+(n+2)H_{|F}))$$
contains the image of
$$ \pi_*({\mathcal O}_F(m(K_F+B_{|F})+H_{|F}))\stackrel{(n+1)H_{|F}}{\longrightarrow} \pi_*({\mathcal O}_F(m(K_F+B_{|F})+(n+2)H_{|F})).$$
\end{thm}

\noindent In the situation of our interest, condition (4) above
will be checked as follows. There will be an effective ${\mathbb Q}$-divisor $A$ such that
\begin{enumerate}
\item[(i)]  $K_X+F+B\sim A$, and
\item[(ii)] $A$ and $B+F$ have no common components and $A+B+F$ is $snc$.
\end{enumerate}
Since $\rup{B}+F$ is a $snc$ divisor where each irreducible
component occurs with coefficient one, the log canonical centres of $(X,\rup{B}+F)$
are exactly all possible intersections of the components of $\rup{B}+F$. But since
$A+B+F$ is $snc$ and $A$ and $B+F$ have no common components,
${\mathcal O}_X(A)$ is generated by its global sections at the generic point of every log
canonical center of $(X,\rup{B}+F)$.

\noindent As we mentioned before, the proof is given first when $C$ is a smooth curve.

\begin{prop} \label{specialcase}
When $C$ is a smooth curve, Theorem (\ref{mainthm}) is true.
\end{prop}

\begin{proof} By extending the base field, we can assume $0\in C$ is a $k$-point. By assumption, the generic fiber $X_g$ of $\pi:X\to C$ is a log Fano variety. Thus there exists an effective divisor $D_g$ on $X_g$ such that $(X_g,D_g)$ is $klt$ and $-(K_{X_g}+D_g)$ is nef and big. Thus there exists an ample divisor $H_g$ and an effective divisor $E_g$ on $X_g$ such that $-(K_{X_g}+D_g) \sim H_g+E_g$. Furthermore, since $-(K_{X_g}+D_g)$ is nef and big, $E_g$ and $H_g$ can be chosen such that $(X_g,D_g+E_g)$ is $klt$ (cf. \cite{kollarmori} (2.61)). By replacing $D_g$ by $D_g+E_g$ we may assume $-(K_{X_g}+D_{g})\sim H_g$ is ample. Let $D$ be an effective divisor on $X$ such that $D_{|X_g}=D_g$. By shrinking $C$ we may assume that $C$ is affine and $D$ does not contain any vertical divisor outside $0$.

\noindent After successively blowing up $X$ with suitable centers and shrinking $C$ one can find a birational morphism $h: Y \to X$ such that

\begin{enumerate}
\item $Y$ is smooth, and the natural projection $\pi'(=\pi\circ h):Y\to C$ is smooth outside $0$, and
\item ${\rm Supp} (h^{-1}D + Y_0) +{\rm Ex} (h)$ is $snc$ (over $k$) in $Y$. Here $Y_0=h^{-1}(0)$.
\end{enumerate}
\noindent Then there exist effective ${\mathbb Q}$-Cartier divisors $G$ and $R$ on $Y$ having no common components
in their support and such that $K_{Y/X} +  G- R= h^*D$. Moreover, $G$ and $R$ satisfy
\begin{enumerate}
\item Any irreducible component of $G$ which is not supported in $Y_0$ appears in $G$ with coefficient strictly less than one (this is a consequence of the fact that $(X_g,D_g)$ is $klt$), and
\item $R$ is $h$-exceptional.
\end{enumerate}

\noindent  Note that the restriction of  $-(K_Y+G-R)$ to the generic fiber $Y_g$ of $Y$ over $C$ is the pull back of an ample divisor on $X_g$. Thus for some sufficiently small $h$-exceptional effective divisor $E$ on $Y$, $-(K_Y+G-R+E)$ is ample on the generic fibre. If necessary blow up $Y$ along smooth centers in the fiber $Y_0$ so that there exists a vertical ${\mathbb Q}$-Cartier divisor $\tilde\Delta$, supported in $Y_0$, such that $-(K_Y+G-R+E+\tilde\Delta)$ is relatively ample over $C$ (see \cite{kollarax}, page $5$).

\noindent Let $\tilde{\Delta} = \sum d_i \tilde{\Delta}_i$
where the divisors $\tilde{\Delta}_i$ are the irreducible components of the support $\tilde{\Delta}$. The only irreducible components of $G-R+E+\tilde{\Delta}$ which appear with coefficient $\geq 1$ are supported in $Y_0$. Thus by perturbing the coefficients $d_i$ slightly (by small rational numbers) and by changing $\tilde{\Delta}\to \tilde{\Delta}+rY_0$ for a suitable $r\in {\mathbb Q}$ we may assume
\begin{enumerate}
\item $-(K_Y+G-R+E+\tilde{\Delta})$ is $\pi$-ample, ${\rm Supp}(\tilde{\Delta})\subset X_0$,
\item There exists an irreducible divisor $F$ (over $k$) in $Y$, an effective ${\mathbb Q}$-Cartier divisors $A$, and a boundary $\tilde{B}$ in $Y$ such that $G-R+E+\tilde{\Delta}=F+\tilde{B}-A$ and such that $F$ and the supports of $A$ and $\tilde{B}$ have no pairwise-common components, and
\item There exist effective ${\mathbb Q}$-Cartier divisors $A_0$ and $R_0$ in $Y$ such that
$A=A_0+R_0$, ${\rm Supp}(A_0) \subset Y_0$ and ${\rm Supp}(R_0)$ is in the
exceptional set of $h$ and contains no component of $Y_0$.
\end{enumerate}
The necessary perturbations as well as the proof of the existence of $F$,  $\tilde{B}$ and $A$ can be found
in \cite{kollarax}. Notice that $A+\tilde{B}+F$ is $snc$ since it is contained in ${\rm Supp} (h^{-1}D + Y_0 +{\rm Ex} (h))$. In (\cite{kollarax}, Remark 8) it was proved that $F$ is geometrically irreducible. If $A$ were equal to $0$ then (\cite{hm05}, 5.1) would imply that $F$ is rationally connected. But in general $A$ can be nonzero. Fortunately, as we will show below, we are saved in this situation by the fact (3).\\

\noindent  Let $m$ be a sufficiently large and divisible integer and let $L_m$ be the Cartier divisor of a general section of ${\mathcal O}_X(m(-K_Y-F-\tilde{B}+A))$. Let $L= \frac{1}{m}L_m$. Let $B= \tilde{B} + L$ and $\Delta = F+B-A$. Thus we are finally in the following situation
\begin{enumerate}
\item[(i)] $-(K_Y+\Delta)\sim_{\pi'} 0$,
\item[(ii)] $\Delta = F+B-A$ where  $A+B+F$ is $snc$,
\item [(iii)] $F$ is geometrically irreducible ${\rm Supp}( F) \subset {\rm Supp} (Y_0)$,
\item [(iv)] $A$, $B$ are effective ${\mathbb Q}$-divisors and $\rdown{B}=0$,
\item[(v)] $A=A_0+R_0$ such that ${\rm Supp} (A_0)\subset {\rm Supp}(Y_0)$, $R_0$ is $h$-exceptional, and
\item[(vi)] There exists a $\pi'$-ample divisor $L$ such that $L \leq B$.
\end{enumerate}

\noindent To apply (\ref{MRC}), we now make the following two claims.
\begin{enumerate}
\item[]\hspace{-6mm}Claim(1): Let $p:F'\to F$ be a proper birational morphism
such that the induced rational map $f':F'\to Z$ is a morphism. Then $p^*(K_{F}+B_{|F})$
has Kodaira dimension at least zero on the general fibre of $f'$.
\item[]\hspace{-6mm}Claim(2): $\kappa(K_F+B_{|F})=0$.
\end{enumerate}
Notice that $(F,B_{|F})$ is $klt$ since $B+F$ is $snc$, $B$ and $F$ do not have common components
and $\rdown{B}=0$. Also the support of $B$ contains the support of an
ample divisor $L$. Hence once we prove the above two claims it will
follow that the hypotheses of (\ref{criterion}) are implied by our hypotheses. In fact, the first hypothesis of (\ref{criterion}) follows from the conclusion that $(F,B|_F )$ is klt, Claim 1 implies the second hypothesis,
Claim 2 implies the third hypothesis and the fourth hypothesis follows easily from the fact that $B+F$ is $snc$. So we can conclude that $Z$ is uniruled, thus proving the rational connectedness of $F$. \\

\noindent By restricting $K_Y + F + B \sim_{\pi '} A$ on $F$, we have $K_F+ B_{|F} \sim A_F $. But
$A$ and $F$ have no common components and hence $A_F$ is effective. Thus $p^*(K_F + B_{|F})$
is effective. Hence it is also effective when restricted to the general fibre of $f'$. This
proves Claim(1).
\noindent To prove Claim(2) we use (\ref{extensionthm}). Let $M$
be a sufficiently $\pi '$-ample divisor. To prove
$\kappa(K_F+B_{|F})=0$, it is enough to show that for $m$ large
and divisible, $h^0(F,m(K_F+B_{|F})+M)= length \left(\pi
'_*({\mathcal O}_F(m(K_F+B_{|F})+M_{|F})) \right)$ is bounded independent
of $m$. By (\ref{extensionthm}) it is enough to bound the rank of
the torsion free sheaf $\pi'_*({\mathcal O}_Y( m(K_Y+B+F)+(n+2)M))$.
$K_X+B+F\sim_{\pi '}A$ and hence we have to bound the rank of $\pi
'_*({\mathcal O}_Y(mA+(n+2)M))$. Since $A=A_0+R_0$ where $R_0$ is effective and $h$-exceptional,
this sheaf is isomorphic to a subsheaf of $\pi_*({\mathcal O}_X(mh_*(A_0) + h_*((n+2)M)))$. But since
${\rm Supp}(h_*(A_0))\subset {\rm Supp}(X_0)$, $\pi_*({\mathcal O}_X(mh_*(A_0) + h_*((n+2)M)))$ is isomorphic
to $\pi_*({\mathcal O}_X( h_*((n+2)M)))$ on $C-\{0\}$ and hence it has
constant rank.
\end{proof}

\begin{proof} [ Proof of (\ref{mainthm}) in General Case] To prove Theorem (\ref{mainthm}) for any $klt$ variety $C$ and an arbitrary point $0\in C$, we need Theorem (\ref{exc.thm}) in the following way.
Let $h:C'\to C$ be a resolution of $C$, then by (\ref{exc.thm}), there exists a rationally connected subvariety $Z \subset h^{-1}(0)$. Let $L$ be the function field of $Z$. Since $Z$ is itself rationally connected, to find a rationally connected subvariety of $X_0$, it suffices to find a rationally connected subvariety of $X_0\times_kL$ by \cite{ghs}.
Thus by base extending to $L$ we may assume there is a $k$-point $0' \in C'$ such that $h(0')=0$. Now let $D\subset C'$ be a general smooth curve passing through $0'$. Note that it is possible to find such a curve since $C'$ is smooth. Let $X_D= X\times_CD$. Since $D$ is general, the generic fibre of $X_D\to D$ is a log Fano variety. Thus (\ref{mainthm}) now follows from the above case.
\end{proof}

\section{Degenerations of Rationally Connected Varieties and the
Minimal Model Program}\label{section:mmp}

\noindent In this section we prove Theorem(\ref{deg-ran}). It suffices to prove Theorem(\ref{deg-ran}) after replacing $X$ by any other higher birational model $X'\to X$. Thus by resolution of singularities, we may assume $X$ is smooth, and $X_0=\pi^{-1}(0)$ is $snc$ (over $k$).
The proof of Theorem(\ref{deg-ran}) will proceed by studying the relative minimal
model program for the pair $(X,\Delta)$ where $\Delta$ denotes the sum of all
irreducible components of $X_0$, each appearing with coefficient one. In this situation
$X$ is smooth and $\Delta$ is $snc$.

\begin{lem}\label{dltklt}
Let $\phi:(X,\Delta)\dashrightarrow (X',\Delta')$
be a birational morphism obtained by running the relative minimal model program
for $(X,\Delta)$ (i.e. $\phi$ is a composition of extremal divisorial contractions and
flips). Then
\begin{enumerate}
\item[(i)] Every irreducible component $\Delta'_j$ of $\Delta'(=\sum_{j=1}^m\Delta'_j)$ is of the
form $\phi_*\Delta_i$ for some $i$, and
\item[(ii)] Every $k$-irreducible component of $\Delta'$ is $klt$.
In particular, after running the relative minimal model program,
the ${\rm Gal}(\bar{k}/k)$-conjugate components of $\Delta\times_k\overline{k}$ do not intersect.
\end{enumerate}
\end{lem}

\begin{ex}
Let $Y=(x^2+y^2+tz^2=0)\subset \P^2_{{\mathbb R}} \times {\mathbb A}^1_{{\mathbb R}}$ where $t$ is the
coordinate on the affine line $C:={\mathbb A}^1_{{\mathbb R}}$. Let $X$ be the blow up of
$Y$ at $([0,0,1],0)$, and denote the exceptional divisor to be $E$.
Then the projection $f: X \to C$ gives us a family which is
defined over ${\mathbb R}$, whose generic fibres are smooth conics. Look at
the point $t=0$ on $C$, the fibre $X_0$ consists of $E$ and the
birational transform of $Y_0=(x^2+y^2=0)$, which has 2
components $E_1, E_2$ over ${\mathbb C}$, and are conjugate to each other
under the Galois action of ${\rm Gal}({\mathbb C}/{\mathbb R})$. So $NE(X/C)({\mathbb R})$ is
generated by $E$ and $E_1+E_2$. We have $(K_X+E+E_1+E_2)\cdot E=0$
and $(K_X+E+E_1+E_2)\cdot (E_1+E_2)=-2$, so to run the relative
program for $f:(X,E+E_1+E_2) \to C$ over ${\mathbb R}$ , we can only
contract $E_1+E_2$.
\end{ex}

\begin{proof}[Proof of (\ref{dltklt})]
$(i)$ For arbitrary  $\phi$
as in Lemma 3.1, it is a morphism away from codimension 2 subsets of both the
domain and target, hence (i) is obvious. \\
\noindent $(ii)$ We devide the argument into several steps.\\
\noindent$Step(1)$ Let $\Delta'= \sum_{i=1}^r\Delta_i'$ where $\Delta'_i= \phi_*\Delta_i$ and
$\phi_*\Delta_i = 0 \ \forall \   r+1 \leq i \leq m$.
By Inversion of Adjunction, to prove $\Delta_1'$ is $klt$ it is
enough to prove $(X',\Delta'_1)$ is $plt$ (\cite{kollarmori}, $5.4$). For $I\subset \{1,...,m\}$ let $\Delta_I= \bigcap_{i\in I}\Delta_{i}$.
Since $X$ is smooth and $\Delta \subset X$ is $snc$,
$(X,\Delta)$ is log canonical and the only log canonical centers of $(X,\Delta)$ are
connected components of closed subsets $\Delta_I$.
Because during each step of the minimal model program discrepancy never decreases
(\cite{kollarmori}, $3.38$), $(X',\Delta')$ is also log canonical. \\
\noindent $Step(2)$ Let $E$ be an exceptional divisor of $X'$
with $a(E,X',\Delta')=-1$. We need to show $a(E,X',\Delta_1')>-1$.
We claim that $center_{X}E = \Delta_I$ for some $I$. Applying the nondecreasing of the discrepancy again,  we conclude $a(E,X,\Delta)=-1$.
But as remarked earlier the only log canonical centers of $(X,\Delta)$ are $\Delta_I$.
Hence $center_XE=\Delta_I$ for some $I$. \\
\noindent $Step(3)$ We claim that for every $i\in I$, $\phi_*\Delta_i\neq 0$, i.e.,
none of the components of $\Delta$ which contain the generic point of $center_XE$ can be
contracted to a lower dimension subvariety by $\phi$. This is a consequence of
(\cite{kollarmori}, $3.38$) since if $\phi$ is not an isomorphism at the generic point
of $center_XE$ then $$a(E,X',\Delta')>a(E,X,\Delta)\geq -1$$ which is contradictory to
our assumption. Since we have assumed $E$ is an exceptional divisor of $X'$,
this also shows that $center_XE$ is not a divisor. Thus $I$ contains at least two elements.
Without loss of generality we assume $2 \in I$.\\
\noindent $Step(4)$ Now we have $a(E,X',\Delta_1'+\Delta_2')\geq a(E,X',\Delta') = -1$.
But both $\Delta_1'$ and $\Delta_2'$ contain $center_{X'}E$ hence
$$ a(E,X',\Delta_1') > a(E,X',\Delta_1'+\Delta_2') \geq -1$$
\end{proof}

\noindent The following lemma is useful in proving Theorem(\ref{deg-ran}) because it allows us to replace $X$ by any other $klt$ birational model.

\begin{lemma}\label{birational}
Let $f:X'\dasharrow X$ be a birational map of varieties over $C$ such that both $X'$ and $X$ are $klt$. Then for $0\in C$, $X'_0$ has a rationally connected subvariety if and only if the same is true for $X_0$.
\end{lemma}
\begin{proof}Suppose $X_0$ contains a rationally connected subvariety $Z$. It is enough to prove $X_0'$ contains a rationally connected subvariety after replacing $X'$ by a higher birational model. Thus we may assume $f$ is in fact a morphism. Let $k(Z)$ be the generic point of $Z$. By Theorem(\ref{exc.thm}), there exists a rationally connected subvariety in the fiber of $X'$ over $k(Z)$. Take its closure $Z'$ in $X'$. By \cite{ghs} $Z'$ is rationally connected and is clearly contained in $X'_0$.
\end{proof}

\begin{proof}[Proof of (\ref{deg-ran})]
 Let $\{\Delta_i\}$ be the set of irreducible components of $X_0$. Let $\Delta = \sum_i \Delta_i$. We now want to
run a relative minimal model program for the pair $(X,\Delta)$ over $C$ and apply the main theorem (see (\ref{mmp})) of \cite{mmpgen}. However, $(X,\Delta)$ is not $klt$ but only log canonical.
Nevertheless, for sufficiently small and positive $\epsilon$, $(X,\Delta-\epsilon\pi^*(0))$
is $klt$. And for any curve class $\eta$ in $N_1(X/C)$, $\pi^*(0)\cdot \eta=0$, running the relative minimal model program for $(X,\Delta-\epsilon\pi^*(0))$, it is equivalent to running the relative minimal model program for $(X,\Delta)$. Thus in order to run the minimal model program resulting in a Mori fibration we only need to check that $K_X+\Delta$ is not $\pi$-pseudo-effective. But in our case the generic fibre, $X_g$, of $X\to C$ is a smooth rationally connected variety. The intersection number of $K_{X_g}$ with any free rational curve in $X_g$ is negative. Thus $K_{X_g}$ is not pseudo-effective. Since $\Delta$ is vertical, this implies that $K_X+\Delta$ restricted to the generic fiber is not pseudo-effective. In particular $K_X+\Delta$ itself cannot be pseudo-effective. After running the program one arrives at the following situation.

$$\xymatrix{
(X,\Delta) \ar@{-->}[r]^{\phi}\ar[d]_{\pi} & (X',\Delta')\ar[d]^{\pi'}\ar[r]^h & Y \ar[dl]^f \\
C \ar@{=}[r]                       & C                                 &
}$$ where
\begin{enumerate}
\item[(i)] Relative Picard number of $h$ is one,
\item[(ii)] $X$ and $X'$ are birational, and
\item[(iii)] $h: X'\to Y$ is a Fano contraction, i.e. the generic fiber of $h$ is a log Fano variety.
\end{enumerate}

\noindent The generic fiber of $Y\to C$ is also a rationally connected since it is dominated by the generic fiber of $X' \to C$. Thus by induction on dimension, $Y_0=f^{-1}(0)$ has a rationally connected subvariety $Z/k$. Let $p$ be the generic point of $Z$. By (\ref{fibrationsing}), $Y$ is $klt$. The generic fiber of $X'\to Y$ is a log Fano variety. Thus by (\ref{mainthm}) there exists a rationally connected subvariety of $h^{-1}(p)$ defined over $k(p)$. Let $Z'$ be its closure in $X'$. By \cite{ghs}, $Z'$ is a rationally connected subvariety of $X'_0$. Since both $X'$ and $X$ are $klt$, an application of lemma(\ref{birational}) completes the proof.

\end{proof}

\section{Fano Fibrations}\label{section:surface}
\noindent In this section we prove some results (of independent interests) on Fano fibrations obtained by contraction of an extremal ray. These results will then be used to prove (\ref{surface}).

\noindent Let $\pi:(X,\Delta)\to S$ be a relative log canonical pair and with
$X$ ${\mathbb Q}$-factorial with $\Delta$ effective.
Let $h:X\to Y$ be a Fano fibration obtained by
contracting an extremal ray. $Y$ is automatically
${\mathbb Q}$-factorial by (\cite{kollarmori}, $3.36$). Let $\Delta=\sum_ia_i\Delta_i$  where $\Delta_i$ are irreducible.
Let $n=dim(X)$ and $r=dim(Y)$.

\begin{lem}\label{equicodim}
Let $D$ be any prime Weil divisor on $X$. Then either $h(D)= Y$ or $h(D)$ is a divisor.
\end{lem}
\begin{proof}
Assume that $\dim(h(D))=k\leq r-2$.
After cutting $Y$ and then cutting $X$ by general very ample hypersurfaces,
we may assume $h^*(H_Y)^k\cdot H_{X}^{n-k-2}$ is a surface whose
image in $Y$ is still 2 dimensional (because $r-k\geq 2$), and
$D\cdot h^*(H_Y)^{k}\cdot H_{X}^{n-k-2}$ is a curve which is
contracted to a point on $Y$, then we have $D^2\cdot
h^*(H_Y)^k\cdot H_{X}^{n-k-2} < 0$. On the other hand, let $E$
be a curve on the fibre over a general point. Since $E$ and $D$ do
not intersect $E\cdot D=0$, so $D\cdot h^*(H_Y)^{k}\cdot
H_{X}^{n-k-2}$ and $E$ cannot be numerically proportional. But this
contradicts the fact that the relative Picard number of $h$ is
one.
\end{proof}
\begin{lem}\label{oneone} Let $D$ be a prime Weil divisor on $X$ which does not
dominate $Y$.
Then ${\rm Supp}(D)={\rm Supp}(h^{-1}h(D))$.
In particular if $\Delta_i$ and $\Delta_j$ do not dominate $Y$,
then $h(\Delta_i)$ and $h(\Delta_j)$ are distinct divisors on $Y$ unless $i=j$.
\end{lem}
\begin{proof} Assume there exists an irreducible component $D'$ of $h^{-1}h(D)$
different from $D$. Then $h(D')\subset h(D)$. But by the previous lemma(\ref{equicodim}),
both are irreducible divisors in $Y$. Hence $h(D')=h(D)$.
Cut $Y$ and $X$ by general very ample
hypersurfaces so that $h^*(H_Y)^{r-1}\cdot H_{X}^{n-r-1}$ is a
surface whose image is a curve. $D'\cdot h^*(H_Y)^{r-1}\cdot H_{X}^{n-r-1}$ and
$D\cdot h^*(H_Y)^{r-1}\cdot H_{X}^{n-r-1}$ are curves on
$h^*(H_Y)^{r-1}\cdot H_{X}^{n-r-1}$, which are mapped to the same
point on $Y$. Then by Zariski's lemma (\cite{bpv}, {\rm III}$.8.2$) they cannot be numerical
proportional which is a contradiction since $h$ has relative Picard number one.
\end{proof}

\noindent The hypothesis of (\ref{deg-ran}) says that $X$ is smooth and the divisor $\Delta$
whose support is $X_0$ is $snc$ (each irreducible component of $\Delta$ appearing
with coefficient one).
However, we know $(X,\Delta)$ is $dlt$ (cf. \cite{kollarmori}), which is a property preserved under Log Minimal Model Program. Now we pinpoint the property we need for the $dlt$ condition as follows,
\begin{point}\label{hypommp}(Hypothesis) Let $\pi:(X,\Delta)\to S$ be
projective log canonical pair where $\Delta$
is a vertical divisor. Assume $X$ is ${\mathbb Q}$-factorial.
Let $\{\Delta_i\}_{1 \leq i \leq m}$ be the set of all irreducible components of
$\Delta$ and assume each $\Delta$ appears with coefficient one in $\Delta$.
Assume that the only log canonical centers of $(X,\Delta)$ are given by $\Delta_I$
where $I\subset \{1,..,m\}$ and $\Delta_I=\bigcap_{i\in I}\Delta_i$.
\end{point}

\begin{rem}
(\ref{hypommp}) in particular implies that for each $i\in \{1,..,m\}$, $(X,\Delta_i)$ is $plt$.
Hence each irreducible component $\Delta_i$ is $klt$ by inversion of adjunction.
We will show (see (\ref{dltklt})) that even after performing divisorial contractions and
flips, the resulting divisor has $klt$ components. The following proposition proves
the same for Fano fibrations.
\end{rem}

\begin{prop}\label{fibrationsing}
Let $(X,\Delta)$ satisfy hypothesis(\ref{hypommp}). Let $X\to Y$ be a Fano fibration
obtained by contracting an extremal ray. Let $D_i$ be the image of $\Delta_i$ in $Y$ and
let $D= \sum_{i=1}^mD_i$.
Then $(Y,D)$ also satisfies (\ref{hypommp}).
\end{prop}

\begin{rem}To prove (\ref{fibrationsing}) we need Kawamata's canonical bundle formula
as proved in \cite{Kol05}. A similar method
has been used before in \cite{ambro} to study singularities of Fano contractions.
\end{rem}

\begin{defn}\cite{Kol05}(Standard normal crossing assumptions)\label{nc}
We say that a projective morphism
$f:X\to Y$ together with ${\mathbb Q}$-divisors $R$, $B$ satisfy the standard normal crossing assumptions
if the following hold
\begin{enumerate}
\item $X$,$Y$ are smooth,
\item $R+f^*B$ and $B$ are $snc$ divisors,
\item $f$ is smooth over $Y\setminus B$, and
\item $R$ is a relative $snc$ divisor over $Y\setminus B$.
\end{enumerate}
\end{defn}

\begin{thm}[Kawamata's canonical bundle formula, see \cite{Kol05}]\label{KK}
Let $f:X\to Y$ and $R$,$B$ satisfy the standard normal crossing assumptions.
Let $F$ be the generic fibre of $f$. Assume that $K_X+R\sim_{{\mathbb Q}} f^*H_R$ for some
${\mathbb Q}$-divisor $H_R$ on $Y$. Let $R=R_h+R_v$ be the horizontal and vertical parts of $R$ and assume that when we write
$R_h=R_{h(\geq 0)}-R_{h(\leq 0)}$ as the difference of its positive and negative parts, we have
$h^0(F,\rdown{R_{h(\leq 0)}|_F})=1$. Then we can write
$$K_{X/Y}+R \sim_{\mathbb Q} f^*(J(X/Y,R)+B_R)$$
where
\begin{enumerate}
\item The moduli part $J(X/Y,R)$ is nef and depends only on $(F,R_h|_F)$ and $Y$,
\item The boundary part $B_R$ depends only on $f:X \to Y$ and $R_v$. More precisely, $B_R$ is the unique smallest ${\mathbb Q}$-divisor supported on $B$ such that
$$red(f^*B)\geq R_v+f^*(B-B_R).$$
Moreover,
\item The pair $(Y,B_R)$ is $lc$ iff The pair $(X,R)$ is $lc$.
\item If $\rdown{R_h}\leq 0$, then the pair $(Y,B_R)$ is $klt$ iff the pair $(X,R)$ is $klt$.
\item $B_i$ appears in $B_R$ with nonnegative coefficient if $f_*{\mathcal O}_X(\rdown{-R_v})\subset {\mathcal O}_{B_i,Y}$
\end{enumerate}
\end{thm}
\begin{rem}
In the above formula, $J(X/Y,R)$ is only defined as a ${\mathbb Q}$-divisor class. However, $B_R$ is a ${\mathbb Q}$-divisor.
\end{rem}

\begin{proof}[Proof of (\ref{fibrationsing})]
Since $h$ is a contraction of a $-(K_X+\Delta)$-negative ray, $-(K_X+\Delta)$
is $h$-ample. Thus there exist ample ${\mathbb Q}$-divisors $H_X$ (resp. $H_Y$) on $X$ (resp. $Y$)
in general position such that
$$ K_X + \Delta + H_X \sim_{{\mathbb Q}} h^*(H_Y)$$
Choose log resolutions $g:\overline{X} \to (X,\Delta)$ and $d:\overline{Y}\to Y$ such
that we have the following commutative diagram
$$\xymatrix{
\overline{X} \ar@{->}[r]^{g}\ar[d]_{\overline{h}} & X\ar[d]^{h} \\
\overline{Y}\ar@{->}[r]^{d}                                   & Y
}$$
There exists a Cartier divisor $R$ on $\overline{X}$ such that
$$ K_{\overline{X}/X}+R= g^*(\Delta+H_X)$$
After further blowup we may assume $\overline{h}:\overline{X}\to \overline{Y}$, the
divisor $R$, and a reduced divisor $B$ on $\overline{Y}$ containing ${\rm Supp}(d^{-1}(D))$
satisfy the standard normal crossing assumptions (\ref{nc}).

\noindent We can choose above $H_X=\sum d_iH_i$, where $0<d_i \ll 1$ and $H_i$ are divisors which are in general position.   We now apply (\ref{KK}) to $\overline{h}:\overline{X}\to \overline{Y},R$ and $B$. Since the negative part of $R$ consists of
exceptional divisors,  $h^0(F,\rdown{R_{h(\leq 0)}})=1$. Thus by (\ref{KK})
we have
$$K_{\overline{X}}+R \sim_{{\mathbb Q}}
\overline{h}^*(K_{\overline{Y}}+J(\overline{X}/\overline{Y},R)+B_R)$$
\begin{lemma}
For every $i$ the coefficient in $B_R$ of the birational
transform of $D_i$ in $\overline{Y}$ is $1$. Furthermore, every log canonical center $W$ of
$(\overline{Y},B_R)$ satisfies that $d(W)=\cap_J D_j$ for
some $J\subset \{1,...,m\}$.
\end{lemma}
\begin{proof}
The first part of the lemma is because of (\ref{KK}(2)) and the fact that $R$ is a subboundary (i.e. the coefficients are less or equal to 1, not necessarily nonnegative) which contains a
component dominating $D_i$ for every $i$, whose coefficient in $R$ is 1.

\noindent The reason for the second part is similar to the
one just given. Let $D$ be an exceptional divisor of $\overline{Y}$
whose center in $\overline{Y}$ is a log canonical center $W$
of $(\overline{Y},D)$. To show $d(W)$ is of the form $\bigcap_JD_J$
in $Y$, we may blow up $\overline{Y}$ and $\overline{X}$ further
and assume $D=W$ and
the coefficient of $D$ in $B_R$ is $1$. But this implies that there is a
component $E$ of $R_v$ dominating $B$ such that the coefficient of
$E$ in $R_v$ is one. The image of $E$ in $X$
is a log canonical center of $(X,\Delta)$ and hence is of the form
$\bigcap_{i\in J}\Delta_i$ for some $J\in \{1,...,m\}$.
Thus $d(W)=\bigcap_{i\in J}D_J$.
\end{proof}

\noindent Now we only need to prove that $(Y,D)$ is log canonical. For this we first
claim that
\begin{lemma}
$d_*B_R$ is an effective divisor.
\end{lemma}
\begin{proof}
By (\ref{KK}($2$)) it follows
that if an irreducible component $B_1$ appears in $B_R$ with negative coefficient
then every component of $\overline{h}^*B_1$ appears in $R_v$ with negative coefficient.
But since all negative components of $R$ have to be $g$ exceptional, it follows
that $B_1$ is also $d$-exceptional. This proves the lemma.
\end{proof}
\noindent To check $(Y,D)$ is canonical, it is enough to check that for every divisor
$E$ on $\overline{Y}$ $a(E,Y,D)\geq -1$. Since $J(\overline{X}/\overline{Y})$
is nef, for every $\epsilon >0$ we can choose an effective divisor
$J_{\epsilon}\sim_{{\mathbb Q}} J(\overline{X}/\overline{Y})+d^*H_Y$ such that
for every divisor $E$ on $Y$ $a(E,\overline{Y},J_{\epsilon}+B_R)>-1-\epsilon$.
Because $K_{\overline{Y}}+J_{\epsilon}+B_R \sim_{{\mathbb Q}} d^*(A)$ for some
${\mathbb Q}$-divisor $A$ on $Y$, then we have
$$K_{\overline{Y}}+J_{\epsilon}+B_R \sim_{{\mathbb Q}} d^*(K_Y+d_*(J_{\epsilon}+B_R))$$
But $D\leq d_*(J_{\epsilon}+B_R)$. Hence for every divisor $E$ on $Y$ we have
$$ a(E,Y,D)\geq -1-\epsilon $$
Taking the limit as $\epsilon \to 0$ we see that $(Y,D)$ is log canonical.
\end{proof}

\begin{proof}[Proof of (\ref{surface})]
The idea of the proof is similar to that of (\ref{deg-ran}). Let us assume $\dim X_t=3$, for the case of smaller dimension, the argument is similar but eaiser. Let $\Delta$ be the sum of all irreducible components of $X_0$, each with coefficient one. Then after running the relative minimal model program for $(X,\Delta)$ we arrive at one of the following two cases.
\begin{enumerate}
\item[Case(1)]
$$\xymatrix{
(X,\Delta) \ar@{-->}[r]^{\phi}\ar[d]_{\pi} & (X',\Delta')\ar[d]^{\pi'} \\
C \ar@{=}[r]                       & C
}$$ where $\pi'$ has relative Picard number one.
\item[Case(2)]
$$\xymatrix{
(X,\Delta) \ar@{-->}[r]^{\phi}\ar[d]_{\pi} & (X',\Delta')\ar[d]^{\pi'}\ar[r]^h & Y \ar[dl]^f \\
C \ar@{=}[r]                       & C                                 &
}$$ where
\begin{enumerate}
\item[(i)] Relative Picard number of $h$ is one,
\item[(ii)] $X$ and $X'$ are birational, and $h: X'\to Y$ is a Fano contraction of relative dimension two, and
\item[(iii)] $Y/C$ is of relative dimension one with generic fiber isomorphic to $\P^1_{k(C)}$.
\end{enumerate}
\item[Case(3)]
$$\xymatrix{
(X,\Delta) \ar@{-->}[r]^{\phi}\ar[d]_{\pi} & (X',\Delta')\ar[d]^{\pi'}\ar[r]^h & Y \ar[dl]^f \\
C \ar@{=}[r]                       & C                                 &
}$$ where
\begin{enumerate}
\item[(i)] Relative Picard number of $h$ is one,
\item[(ii)] $X$ and $X'$ are birational, and $h: X'\to Y$ is a Fano contraction of relative dimension one, and
\item[(iii)] $Y/C$ is of relative dimension two.
\end{enumerate}
\end{enumerate}

\noindent {\it Case ($1$)}: Every irreducible component of $\Delta'$ contributes to the relative Picard number of $\pi'$. Since the relative Picard number is one $\Delta'$ must be irreducible. It is also geometrically connected since fibres of $\pi'$ are geometrically connected. By lemma(\ref{dltklt}) $\Delta'$ is $klt$, in particular normal. A normal geometrically connected variety over $k$ is geometrically irreducible. Hence $\Delta'$ is geometrically irreducible.
\noindent  Since $\Delta'$ is a degeneration of a proper rationally chain connected scheme, it is rationally chain connected.  But $\Delta'$ is $klt$, hence by \cite{hm05} it is in fact rationally connected. This proves that $\Delta'$ is geometrically irreducible as well as rationally connected. By (\ref{dltklt}($i$)), $\Delta' = \phi_*\Delta_i$ for some irreducible component $\Delta_i$ of $\Delta$. Hence $\Delta_i$ is also rationally connected and geometrically irreducible.\\

\noindent {\it Case ($2$)}:
Let $\{\Delta_i'\}_{1\leq i \leq s}$ be the irreducible components of $\Delta'$ and let $\Delta_i'=\phi_*\Delta_i$.
Let $D=\sum_{i=1}^sD_i$. By (\ref{fibrationsing}) $(Y,D)\to C$ satisfies hypothesis (\ref{hypommp}).

\noindent We first claim that there exists an irreducible components $D_{j_0}$ of $D$ which is rationally connected and in particular geometrically irreducible. The generic fibre of $Y/C$ is isomorphic to $\P^1$ and hence $-(K_Y+D)$ is not pseudo-effective. Thus after running the minimal model program for $(Y,D)/C$ we arrive at the following situation.
$$\xymatrix{
(Y,D) \ar@{-->}[r]^{\phi}\ar[d]_{\pi_Y} & (Y',D')\ar[d]^{\pi'_Y} \\
C \ar@{=}[r]                       & C
}$$ where $\pi'_Y$ has relative Picard number one. The claim now follows by an argument similar to that in Case($1$).

\noindent Now look at $\Delta'_{j_0}$. It is geometrically irreducible by (\ref{oneone}). Let $\Delta''\to \Delta_{j_0}$ be a resolution of singularities. Then by \cite{hm05}(1.2), every fibre of $h':\Delta''\to D_{j_0}$ is rationally chain connected. In particular the generic fiber is rationally connected since it is also smooth. This together with \cite{ghs} and the fact that $D_{j_0}$ is rationally connected proves that $\Delta''$ and hence $\Delta'_{j_0}$ is rationally connected. Since $\Delta_{j_0}$ and $\Delta_{j_0}'$ are birational, $\Delta_{j_0}$ is also rationally connected.\\

\noindent{\it Case ($3$)} The proof is similar as in Case (2). We need to use induction to prove there exists a rationally connected subvariety in $Y_0$. Hence it suffices to prove that when we run the minimal model program for $(Y,D)$, it terminates with a Fano contraction. In another words, it suffices to show the non-pseudo-effectivity of $K_Y+D$. Now $\phi|_{X_t}: X_t\to Y_t$ is a Fano contraction of an extremal ray of $K_{X_t}$, and $X_t$ is a terminal three-fold. Apply the Theorem 1.2.7 of \cite{mp}, we conclude that $Y_t$ only contains Du Val singularity of type A. In particular, since $X$ is rationally connected, $Y_t$ is a rational surface with canonical singularity. Take the minimal resolution $d:\overline{Y}_t \to Y_t$, then
$$H^0(Y_t,mK_{Y_t})=H^0(\overline{Y}_t,d^*(mK_{Y_t}))=H^0(\overline{Y}_t,mK_{\overline{Y}_t})=0.$$
Since $K_Y+D|_{Y_t}=K_{Y_t}$, we conclude that $K_Y+D$ is non-pseudo-effective.
\end{proof}

\begin{rem}\label{hard}

\noindent One can also try to answer (\ref{mainq'}) or (\ref{mainq}) in higher dimensions by the same method as above. In this case one would not have to deal with the case of degeneration of Fano varieties separately. The missing property here is the non-pseudo-effectivity of $K_Y+D$ where $(Y,D)$ is a result of a Fano contraction. This non-pseudo-effectivity is by no means automatic except in the case when relative dimension of $Y/C$ is one. In fact, in dimension $\geq 2$, Koll\'ar has constructed a family of rational varieties with only quotient singularities, but whose canonical divisor is ${\mathbb Q}$-ample (cf. \cite{kollar06}).
As a stronger evidence to show the restriction of our approach toward (\ref{mainq}), Prokohorv even constructs an example of $h:X\to Y$ a Fano contraction, where $X$ is a rationally connected variety of terminal singularity, but $Y$ has a ${\mathbb Q}$-effective canonical bundle.

\end{rem}
\begin{ex}[Prokhorov]\label{counter}
\noindent Let $S$ be the surface $x^n+y^n+z^n+w^n=0$ in $\P^3$, the group $G={\mathbb Z}/n$ acts on $\P^3$ and $S$ with weights $(0,0,1,1)$.
Let $Y:=S/G$, and $\pi:S \to Y$ is the quotient morphism. Since $\pi$ is \'etale outside finite points, thus $\pi^*(K_Y)=K_S$. So when $n\ge 4$, $K_Y$ is ${\mathbb Q}$-effective. We claim $Y$ is rational. In fact, the projection map $p:S\to \P^1$ by sending $(x,y,z,w)$ to $(x,y)$ will factor through $Y$. The generic fiber of the map $p$ is birational to the affine curve $(z^n+w^n+\lambda=0)\subset {\mathbb A}^2$, so the generic fiber of the map from $Y$ to $\P^1$ is birational to the curve $(z^n+w^n+\lambda=0)/ G$, which is also rational. We conclude that $Y$ is a rational surface.

\noindent Now consider the action of $G(={\mathbb Z}/n)$ on $\P^{n-1}$ with the weights $(0,1,...,n-1)$. We define $X:= (\P^{n-1}\times S) /G$, where $G$ acts diagonally. For a fixed point $x$ of a $g\in G$ on $(\P^{n-1}\times S) /G$, if we write the eigenvalue of $g$ on $T_x$ as $e(r_1),...,e(r_{n+1})$, where $e(x):=e^{2\pi ix}$ and $0\le r_i<1$. Then to prove $X$ contains only terminal singularity, it suffices to prove that for any element nontrivial element $g$ and any $x$ stablized by $g$, the $age$ satisfies
$${\rm age}_x(g):=r_1+\cdots+r_{n+1}>1.$$
(cf. \cite{reid}, Theorem 4.6). Now when $n\ge 4$, we can see that the above condition holds for our case.

\noindent We define the map $h:X\to Y$ by descending the second projection of $p_2: \P^{n+1}\times S\to S$, which is equivariant. It is easy to see that $h$ is a Mori contraction of an extremal ray. Since any terminal variety is an output of a minimal model program which starts with a smooth variety (simply reverse the desingularization process), we in fact have an example, where we start with a smooth rationally connected variety, but running the minimal model program terminates with a contraction to a variety of ample canonical bundle.
\end{ex}

\noindent Finally we give an example of a simple normal crossing variety of dimension $3$ which is rationally chain connected but none of its components are rationally connected.
\begin{ex}\label{counterex}
\noindent Let $C$ be any smooth curve in $\P^2$ of genus $g \geq 2$. Let
$i:C \hookrightarrow \P^2$ be the closed embedding of $C$ in $\P^2$.
Let
$$
\xymatrix{
            &  \P^2\times_k C = X_1 \\
C\times_k C\ar[ru]^{i_1}\ar[rd]^{i_2} &   \\
             &  \P^2\times_k C = X_2
}
$$
be two maps defined by
$i_1 = (i,id_{C})$ and $i_2 = (id_{C}, i)$. Glue $X_1$ and $X_2$ along the maps $i_1,i_2$. The
resulting three dimensional variety $X=X_1\bigcup X_2$ is simple normal crossing as well as
rationally chain connected. But neither $X_1$ nor $X_2$ is rationally connected.
\end{ex}

\section{Rationally Connected Subvarieties of the Exceptional Locus}\label{section:exc}

\noindent In this section we prove Theorem(\ref{exc.thm}).

\begin{lem}
Let $X/k$ be a normal variety. Let $\pi:X'\to X$ be a proper birational morphism.
Then after further blowing up $X'$ if necessary we can find an effective exceptional
divisor $A$ on $X'$ such that $-A$ is $\pi$-ample.
\end{lem}
\begin{proof}
It is enough to prove the lemma for one particular $\pi:X'\to X$ with $X'$ smooth.
For any quasiprojective variety $S$, if $f:S_{{\mathcal I}} \to S$ is the blow up of an ideal sheaf ${\mathcal I}$
on $S$ then $E({\mathcal I})= \pi^{-1}{\mathcal I}$ is an effective divisor on $S_{{\mathcal I}}$ which is anti $f$-ample.
Now let $\pi:X'\to X$ be the resolution of singularities obtained by successively
blowing up centers disjoint from the smooth locus of $X$, in particular $\pi$ is projective. Suitable positive
linear combination of (strict transforms of) divisors of the form $E({\mathcal I})$ gives
us an effective divisor $A$ on $X'$ such that $-A$ is $\pi$-ample. Since $X$
is normal this divisor $E$ is $\pi$-exceptional.
\end{proof}

\begin{proof}[Proof of (\ref{exc.thm})]
For getting hold of a geometrically irreducible subvariety of $\pi^{-1}(p)$
we follow the general idea of the proof of main theorem in \cite{kollarax}.
First by base extending to $k(p)$ we may assume $p$ is a $k$-point.
By replacing $X$ by a suitable open neighbourhood of $p$ we may assume $X$ is
affine.
We claim that there exists an effective ${\mathbb Q}$-divisor
$\Delta_1\sim_{\mathbb Q} 0$ passing through $p$ such that
$(X,\Delta+\Delta_1)$ is $klt$ outside $p$ and log canonical at $p$.
Since $X$ is affine, taking a $\Delta_1$ to be a suitable rational multiple of
a sufficiently general section of ${\mathcal O}_X$ passing through $p$ with high multiplicity
does the job. \\

\noindent By further blowing up $X'$ we may assume
\begin{enumerate}
\item[(i)]$\pi:X'\to X$ is a log resolution of $(X,\Delta+\Delta_1)$,
\item[(ii)]$\pi^{-1}(p)$ is a divisor, and
\item[(iii)]There is an effective exceptional divisor $A$ on $X'$ such that
$-A$ is $\pi$-ample.
\end{enumerate}
Let $\{E_i\}_{1\leq i \leq r}$ be the exceptional divisors lying over $p$ and let
$\{F_j\}_{1\leq j \leq s}$ be the remaining exceptional divisors of $\pi$.\\

\noindent Henceforth we will use the notation "$\sim_{\mathbb Q} $" to denote
${\mathbb Q}$-numerical equivalence of ${\mathbb Q}$-divisors relative to $\pi$. In another words, for two ${\mathbb Q}$-Cartier divisors $A$ and $B$, we write $A\sim_{{\mathbb Q}}B$ iff
$$A\cdot \eta=B\cdot \eta,$$
for any $\eta \in N_1(X'/X)$, then we have
$$ K_{X'} \sim_{{\mathbb Q}} \sum_{i=1}^r a_iE_i + \sum_{j=1}^sb_jF_j - \Delta'$$
where
\begin{enumerate}
\item[(i)] $a_i\geq -1  \ \forall \ i$ with equality holding for at least one $i$,
\item[(ii)] $b_j >-1\ \forall j$, and
\item[(iii)] $\Delta'$ is the birational transform of $\Delta+\Delta_1$ hence $\rdown{\Delta'}=0$.
\end{enumerate}
Let
$$ A = \sum_{i=1}^rc_i E_i + \sum_{j=1}^sd_jF_j$$
By replacing $A$ by $\frac{1}{n}A$ for $n$ large enough we may assume the
coefficients $c_i$ and $d_j$ are sufficiently small as compared to one.
For small rational numbers $\epsilon>0$ and $\{\delta_i\}_{1\leq i \leq r}$
consider the following divisor
$$K_{X'}-\pi^*(K_X+\Delta+(1-\epsilon)\Delta_1)-(A+\sum_i\delta_iE_i)  \sim_{\mathbb Q}  K_{X'}-(A+\sum_i\delta_iE_i) -\Delta'(\epsilon)$$
$$\sim_{{\mathbb Q}}\sum_{i=1}^r a_i(\epsilon,\delta)E_i + \sum_{j=1}^sb_j(\epsilon)F_j - \Delta'(\epsilon) $$
where $\Delta'(\epsilon)$ is the birational transform of $\Delta+(1-\epsilon)\Delta_1$.
Notice that $b_j(\epsilon)>-1$ for all $\epsilon >0$ and all $j$.
Choose $\epsilon$ and $\delta_i's$ such that
\begin{enumerate}
\item[(i)] $A+\sum_i\delta_iE_i=A'$ is effective and such that $-A'$ is $\pi$-ample
(this is true for all $\delta_i$ small enough because ampleness is an open condition), and
\item[(ii)]$a_i(\epsilon,\delta)\geq -1$ with equality holding for exactly one $i$ (say $i=1$).
\end{enumerate}
Thus we can write
$$ K_{X'}-A' \sim_{{\mathbb Q}} -E_1 + E - M$$
where
$$ E = \sum_{i\geq 2}\rup{a_i(\epsilon,\delta)}E_i+\sum_j\rup{b_j(\epsilon)}F_j$$
is an effective exceptional divisor and $M$ is an effective $snc$ divisor (having no
common components with $E_1$) and such that $\rdown{M}=0$.

\noindent Thus by relative Kawamata-Viehweg vanishing theorem we get
$$ R^1\pi_*{\mathcal O}_{X'}(-E_1+E)=0.$$
This gives us a surjection
$${\mathcal O}_X \cong \pi_*{\mathcal O}_{X'}(E)\to \pi_*{\mathcal O}_{E_1}\to 0$$
which proves $E_1$ is geometrically irreducible. \\

\noindent We claim that $E_1$ is also rationally connected.
Since $-A'$ is $\pi$-ample and since $X$ is affine it is actually ample.
Choose a general divisor $W$ such that $-A'\sim_{{\mathbb Q}}W$ and such that $\rdown{W}=0$.
Since $W$ is general, it has simple normal crossings with all exceptional
divisors and also with $M$. Let $M'=M+W$. Then $M'$ supports an ample divisor and
we have
$$K_{X'}+E_1+M' \sim_{{\mathbb Q}} E$$
But $E$ is an effective divisor having no common components with $E_1$.
Thus $K_{E_1}+M'_{|E_1}$ is effective and as in proof of (\ref{mainthm}) rational
connectivity of $E_1$ will follow by (\cite{hm05}, $4.1$)
once we show $\kappa(K_{E_1}+M'_{|E_1})=0$. As in the proof of (\ref{specialcase})
this follows from (\ref{extensionthm}) and using the fact that $E$ is effective
exceptional.
\end{proof}

\begin{rem}
Theorem(\ref{exc.thm}) can fail for varieties having log canonical singularities.
See (\cite{kollarax}, $9$) for an example.
\end{rem}

redpajama_set_name: RedPajamaArXiv


text:
Каза́нська вулиця — вулиця в Оболонському районі міста Києва, місцевість Пріорка. Пролягає від Вишгородської до Автозаводської вулиці.

Історія
Вулиця виникла в 1952 році під сучасною назвою внаслідок злиття Кошицького провулку (виник у XIX столітті, інша назва — Кожищев провулок, від прізвища поселенця) і 77-ї Нової вулиці (виникла на початку 30-х років XX століття). Сучасна забудова — з 1970-х років.

Примітки

Джерела

Вулиці Оболонського району Києва

redpajama_set_name: RedPajamaWikipedia


I am studying performance and job satisfaction of family member and non-family member employees in a family-run business. The purpose of this study is to see how employees are affected when they work for a family business and to explore whether employees are more or less motivated to work for the family firm. This study shows how working for a family business has an impact on an employee's performance based on the care and consideration received from the family that owns the business, and whether it leads to more or less organizational commitment. To gather data for my research project, I conducted interviews with 15 employees of a local family restaurant. The interviewees included both family-member and non-family-member employees to compare perceptions of management as well as general morale and satisfaction working for the company. The study found that nonfamily employees were more motivated to serve customers and were more committed to their work than family members. My research project helps to identify the advantages and disadvantages of working for a family firm. It also creates a deeper understanding of how different business operating structures impact employees and overall business performance.
Echevarria, Sloane N. How Working for a Small Family Business Impacts Employee Satisfaction and Performance. Honors Thesis. East Carolina University, May 2017. The Scholarship. http://hdl.handle.net/10342/6278. April 23, 2019.
Echevarria, Sloane N, "How Working for a Small Family Business Impacts Employee Satisfaction and Performance" (Honors Thesis., East Carolina University, May 2017).
Echevarria, Sloane N. How Working for a Small Family Business Impacts Employee Satisfaction and Performance [Honors Thesis]. Greenville, NC: East Carolina University; May 2017.


Dear Sir, I like words.
A letter from copywriter Robert Pirosh to potential employers in 1934. As proof of the power of a well-crafted letter, Pirosh went on to write for the Marx Brothers and earned an Academy Award for his script Battleground.
Read the full letter at Letters of Note, a site paying homage to correspondence. Shaun Usher is the editor, and he offers equally entertaining and enlightening companion sites: Lists of Note and Letterheady.
The Letters of Note book will be available in November 2012.
Newer PostThankful Thursday: Is that you, gratitude?


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text:
Simonds lecture to compare property rights in U.S., the world
CARBONDALE, Ill. — Concerns over government use of eminent domain to seize private property for public use is the focus of a lecture next week at the SIU School of Law. The issue has recently been in the news as landowners along the southern U.S. border object to President Donald Trump building a border wall on their land.
Shruti Rajagopalan will present the law school's 2019 Gene and Katy Simonds Lectureship in Democracy on March 27. Rajagopalan's lecture, "Property Rights and Property Wrongs: A Comparative Perspective," begins at 5 p.m. in the Hiram H. Lesar Law Building Auditorium at SIU Carbondale.
The lecture is free, and the public is invited. A Q-and-A and reception will follow. While on campus Rajagopalan will also meet with law students and faculty.
Reporters, photographers and camera crews are welcome to attend the Simonds Lectureship in Democracy with Shruti Rajagopalan at 5 p.m., March 27, in the SIU School of Law Auditorium. For more information or to arrange interviews, contact Michele Mekel, the law school's director of external relations, at mmekel@siu.edu or 618/453-8768.
A 2005 U.S. Supreme Court decision is an example of eminent domain issues
Rajagopalan is an assistant professor of economics at State University of New York, Purchase College, and a Fellow at the Classical Liberal Institute at NYU School of Law. George Mocsary, assistant professor in the School of Law, said Rajagopalan's lecture is timely and relevant to the ongoing conversation about eminent domain "that was kicked to the fore" by a 2005 U.S. Supreme Court decision involving a proposed economic development project in New London, Connecticut.
In that case, the city proposed to seize 15 commercial and residential private properties and transfer the land to sell to private developers for the purpose of economic development and job creation. The plaintiffs argued the use of eminent domain to sell private property to private developers was not part of the public use clause under the Fifth Amendment.
In a 5-4 decision, the Supreme Court ruled in favor of the city taking the private property and selling it for private development in hopes of economic development.
Those attending the lecture will get a sense of "what this country should avoid in terms of eminent domain power and a sense of what is going on in other places in the world," Mocsary said. He noted that the private developer was unable to continue with the project and that the land remains vacant.
The property owner's fight was developed into a 2017 movie "Little Pink House," referencing a home that belonged to the lead plaintiff, Susette Kelo.
Speech includes economic analysis of comparative legal and political systems
Rajagopalan's research interests include law and economics, public choice theory, and constitutional economics and her research is in peer-reviewed journals, law reviews and books.
Rajagopalan holds a doctorate in economics from George Mason University, with a bachelor's degree and bachelor of laws degree from University of Delhi, and a master of laws degree from the European masters in law and economics program at the University of Hamburg, Ghent University and the University of Bologna.
Rajagopalan has published opinion editorials on Indian political economy in publications including The New York Times and The Wall Street Journal, and she has a fortnightly column "The Impartial Spectator" in Mint.
Gene and Katy Simonds Lecture
Shruti Rajagopalan will present "Property Rights and Property Wrongs: A Comparative Perspective"
5 p.m. March 27
SIU School of Law Auditorium
This is the fourth lecture in the series established in 2015 through a gift from Emma K. (Katy) Simonds. The funds support a lecture, symposium or debate on a topic of current or recurring public interest, which explore commonsense solutions to issues related to the common good by applying principles of limited government, popular sovereignty, personal liberty, personal responsibility and federalism.
Media Contact: Michele Mekel, the law school's director of external relations, at mmekel@siu.edu or 618/453-8768.
Transfer New Student Orientation
Location: Student Center Corker Lounge
7:45 AM to 2:00 PM - Student Center Corker Lounge
National Engineering Competition
Location: SIU Arena SIU Arena
8:00 AM to 10:00 PM - SIU Arena SIU Arena
SI Music Festival Presents: Gianni Schicchi
Location: Shryock Auditorium Auditorium
7:30 PM to 9:30 PM - Shryock Auditorium Auditorium Admission Adutls $20 / Students $8

redpajama_set_name: RedPajamaCommonCrawl


If you are in the need of a Tow Truck in Roy or maybe you need a tow truck in one of the many surrounding areas you can rest assured that Roy Towing will send a Tow Truck your way quickly to get you safely back on the road in no time. Roy Towing has the Best Tow truck drivers. Roy Towing is known for punting our costumers first and having the best costumer service is key to making sure you share your great towing experience with your friends and family. We understand that Towing needs are not always needed in the best of circumstances and Roy Towing will send our tow truck driver out with a smile, to so anything he can with in the realm of possibilities to make your day just a little bit better.
If you have been in a car accident the most important thing you can do is make sure you are not hurt. Even if you were in a car accident and you feel like you are fine you may still need to be checked out by a Doctor or a chiropractor to insure that you are in fact not injured in a car accident.


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text:
Constative
News, Celebrity Lifestyle, Facts And References
Facts File
Football Sports
Roberto Mancini is the new Inter coach
Chinwe Esther
Roberto Mancini has been confirmed as the new coach of Inter Milan. The news means that the former Manchester City boss will embark on a second spell in charge of the Nerazzurri, having first coached the club between 2004 and 2008.
The former Italian international was most recently in charge of Turkish club Galatasaray, who he left in the summer. He replaces Walter Mazzarri, who was sacked on Friday with the club ninth in the Italian Serie A table.
A statement from Inter owner Erick Thohir read: "Today I made the difficult decision to dismiss Walter Mazzarri as head coach. It was a decision that I reached in conjunction with my management team. It was a difficult decision because he has been continuously supportive of the direction that we have wanted to take the club, and he has worked tirelessly and selflessly for Inter, with both belief and passion.
"I would like to thank him sincerely for his efforts. However, our goal is to regain our rightful position as one of the very top clubs in Europe and that's why I am so happy to welcome back Roberto Mancini. His track record both at Inter and elsewhere speaks for itself, and his experience and hunger for success will drive the team to the next level."
Inter won three consecutive Serie A titles during Mancini's first spell at the San Siro, and he also led Manchester City to a dramatic English Premier League title success in 2012 when Sergio Aguero's late goal snatched the title away from local rivals United.
Mazzarri's dismissal came after a 2-2 home draw with Verona in their last outing. Inter are currently 12 points behind leaders Juventus after just four wins from their 11 league games to date, although they top their Europa League group with an unbeaten record from four matches. Inter said Mancini would be presented to the media at a press conference on Saturday afternoon at the club's Appiano Gentile training ground.
Boko Haram captures Chibok
EFCC arraigns prison accountant for N83m fraud
2 thoughts on "Roberto Mancini is the new Inter coach"
Ejinwa Anthony says:
do what you knw how to do best. …win trophies
Copyright © 2019 Constative


IOServer is a piece of industrial control software that runs on Windows. It contains a built-in web server to support the "XML Server" feature. This web server can be abused to download any file on the file system without authentication, if the "Root Directory" setting does not contain a trailing backslash. Note that a sample configuration provided with the product exhibits a vulnerable configuration.
If the "XML Server" feature is turned on (apparently by configuring a valid "Root Directory" on the "XML Server" tab), the integrated web server listens on the configured port (81 by default). If the "Root Directory" configuration value does not contain a trailing backslash, then these issues exist.
A directory traversal vulnerability exists such that the web server can be tricked to serve up any file on the server, outside of the configured "Root Directory". On Windows, one common thing to do with an issue like this is to download the backup copy of the SAM, in order to retrieve password hashes and mount an offline attack on them. Any other potentially sensitive file on the server can be accessed this way as well, if the attacker knows the path to it.
Note that directory traversal is only needed if you want to break out of the "Root Directory". The web server will serve up files inside the "Root Directory" just by asking it for them. This may or may not be considered a vulnerability, depending on the expectations of the administrator.
Also, the web server allows directory listings in some cases, whether inside or outside the "Root Directory". Again, this may or may not be considered a vulnerability depending on expectations.
Note also that the "XML Server Settings" tab contains fields for a "Read Password" and "Write Password", which are apparently intended to only restrict access to the devices via the XML web interface (they are not intended to restrict access to other content served by the web server). By default, they are set to blank. But due to their intended use of resticting device access, they do nothing to stop access to the file system as described here even if they are set to some value.
Install the software and use the sample Modbus Web Server configuration from http://www.ioserver.com/mbtcp.io to get it working. This sample configuration uses the "Root Directory" value of "C:Program FilesIOServerDocs" (note the lack of a trailing backslash). This directory does not exist unless you install the documentation package (isodoc.exe, available from the vendor web site). You can get to a vulnerable configuration if you leave the "Root Directory" alone and install the documentation package, or just modify the "Root "Directory" value to something valid (but lacking a trailing backslash) like "C:Program FilesIOServer". Note that the software or server may need to be restarted, then you will see a web server listening on the configured port (81 by default).
Note that more recent versions of the software default to "C:IOServer" rather than "C:Program FilesIOServer". This does not affect the issues discussed here.
Note that it is only this directory traversal issue that was fixed in version 1.0.19.0. Accessing files and directory listings inside the "Root Directory" is apparently intended operation.
Unexpected arbitrary access to the file system can lead to the disclosure of sensitive information. Worst case, disclosure of the system's password hashes can lead to compromise of the passwords, and therefore, of the server.
Note that the CVSS calculation was based only on the directory traversal issue, since it is the most obvious and serious vulnerability addressed here.
This was downloaded from http://www.ioserver.com/driver18.exe, which was the current version as of 23 April 2012. Older versions were not available, but are assumed vulnerable.
Ensure that the "Root Directory" value has a trailing backslash. This eliminates most of the issues, although you can still get a directory listing of the "Root Directory" itself (but not subdirs) even with this in place.
Of course, limit access to the web server port to only trusted hosts/networks. Even a fixed version should probably not be exposed to the Internet or other untrusted networks.
The directory traversal issue was fixed in version 1.0.19.0 (http://www.ioserver.com/driver19.exe), released on 2012-06-20. However, arbitrary file access and directory listings inside the "Root Directory" and its subdirs are still possible in this version, unless the trailing backslash is in place. Note that even with the trailing backslash, directory listing of the "Root Directory" itself is still possible.
It is recommended that all three of the above actions (use the trailing backslash, limit access to trusted networks/hosts only, and upgrade to 1.0.19.0 or later) be taken in order to secure this web server as much as possible.
Well, actually, this vulnerability was found by Nessus, plugin ID 10297 (which looks for generic web server directory traversal issues, not specific to IOServer of course). All I did was see it in the scan results, verify it, reproduce it for testing, report it to the vendor and write this advisory.
Thanks to my friends at foofus.net, and to IOServer for being very responsive and open.


Send this football Favorites cutting board gift baskets anytime of the year to any football fan you know. Always a favorite among football fans. This sports football gift basket is in a league of its own!


Prior to becoming a member of Infor, Beth led the provision chain enterprise intelligence staff at Fairview Health System (MN), worked as a perioperative nurse in the U.S. Military, and held positions in surgical procedure and scientific administration with Baldwin Area Medical Center (WI). Beth holds a B.S.N. and a grasp's in technology management; she is pursuing her PhD in healthcare informatics with the College of Minnesota, the place she can also be a guest lecturer.
On Friday, in a joint letter, the insurance coverage business lobby and the affiliation that represents Blue Cross Blue Defend plans came out strongly in opposition to one of the improvements within the latest draft. They joined client teams, patient advocates and organizations representing medical doctors, hospitals, drug abuse treatment centers and religious leaders who've expressed opposition to the invoice.
Frailty and getting old in clinics and public health. This journal's part is devoted at presenting studies on scientific problems with frailty and age-associated conditions. This multidisciplinary section notably welcomes experiences from clinicians coming from different backgrounds and specialties coping with the heterogeneous scientific manifestations of superior age. Moreover, this part of the journal additionally comprises reports on frailty- and age-related social and public health points.


There exists in the Archives much material which will be of interest to historians, including some items which may shed light into the lives of the descendants of conversos, or "Crypto Jews," who settled in the region.
This page provides a look at the Crypto-Jewish experience in the U. S. Southwest.
Freund, Michael. "Marranos No More," The Jerusalem Post, October 30, 2002. Freund's column concerns the descendants of Jews forcibly converted in Spain and Portugal during the Inquisition who are now seeking to return to the Jewish people.
Golden, Gloria. 2004. Remnants of Crypto-Jews Among Hispanic Americans. Mountain View, CA: Floricanto Press.
Saint Louis University, Madrid, Spain. Converso and Morisco Studies. Converso and morisco are the terms applied to those Jews and Muslims who converted to Christianity in large numbers and usually under duress in late Medieval Spain The Converso and Morisco Studies conferences and publications examine the implications of these mass conversions, for the converts themselves, for their heirs (also referred to as conversos and moriscos ) and for Medieval and Modern Spanish culture.
Herrera, M. Miriam. Crypto-Jews/Anusim Links Ms. Herrera is a converso poet descended from Natives of the Rio Grande Valley of South Texas. She has compiled a list of resources on these categories: History & Definitions, The Spanish & Mexican Inquisitions, Tools for Researching Crypto-Jewish Heritage, Writers & Artists, Personal Stories, Culture & Folklore, and Book Reviews.
Valerio, Max. "The Atlantic Monthly Exposé," HALAPID 8:1 (Winter 2001) p. 4-5, 8-10.
Kulanu Website: "All Of Us", is an organization which reflects the community of interests of individuals of varied backgrounds and religious practices dedicated to finding and assisting lost and dispersed remnants of the Jewish people. It includes links to past newsletters, articles on a variety of related topics, and contains information on upcoming events sponsored by the Society for Crypto-Judaic Studies.
Interested in more? Join the Kulanu_Mexico Yahoo group. "Kulanu activities in Mexico in locating, assisting and interacting with Mexican Jewish groups and emerging Anousim communities."
Visit The Society for Crypto-Judaic Studies website. You will find many informative articles in its publication, HaLAPID.


text:
Home › World › Asia › Gunmen kill 15 in southern Thailand's worst attack in years
Gunmen kill 15 in southern Thailand's worst attack in years
Suspected separatist insurgents stormed a security checkpoint in Thailand's Muslim-majority south and killed at least 15 people, including a police officer and many village defence volunteers, security officials said on Wednesday.
It was the worst single attack in years in a region where a Muslim separatist insurgency has killed thousands.
The attackers, in the province of Yala, also used explosives and scattered nails on roads to delay pursuers late on Tuesday night.
"This is likely the work of the insurgents," Colonel Pramote Prom-in, a regional security spokesman, told Reuters.
"This is one of the biggest attack in recent times."
There was no immediate claim of responsibility, however, as is common with such attacks.
A decade-old separatist insurgency in predominantly Buddhist Thailand's largely ethnic Malay-Muslim provinces of Yala, Pattani, and Narathiwat has killed nearly 7,000 people since 2004, says Deep South Watch, a group that monitors the violence.
The population of the provinces, which belonged to an independent Malay Muslim sultanate before Thailand annexed them in 1909, is 80 percent Muslim, while the rest of the country is overwhelmingly Buddhist.
Some rebel groups in the south have said they are fighting to establish an independent state.
Authorities arrested several suspects from the region in August over a series of small bombs detonated in Bangkok, the capital, although they have not directly blamed any insurgent group.
The main insurgency group, the Barisan Revolusi Nasional (BRN), denied responsibility for the Bangkok bombings, which wounded four people.
In August, the group told Reuters it had held a secret preliminary meeting https://www.reuters.com/article/us-thailand-south-exclusive/exclusive-muslim-insurgent-group-says-it-met-with-thai-government-idUSKCN1V7039 with the government, but any step towards a peace process appeared to wither after the deputy prime minister rejected a key demand for the release of prisoners.
MuslimsSeparatistsTerror attackThailand
Thai Officials Meet with Major Southern Insurgent Group
Officials from Thailand have held their first formal meeting in recent years with Muslim separatists from southern Thailand belonging to…
Thailand: Thousands protest against government with run
A large protest against Thailand's military-backed government on Sunday saw thousands join the "Run Against Dictatorship" in Bangkok, with some…
Ukraine, pro-Russia separatists carry out landmark prisoner exchange
Pro-Russia separatists and the Kyiv government carried out a landmark prisoner swap of Ukrainian soldiers in exchange for Ukraine-held separatists on…
Thai Opposition Leader Asks Supporters to Mobilise in Bangkok on Saturday
Thai opposition party leader Thanathorn Juangroongruangkit called on supporters to mobilize in Bangkok on Saturday, days after the national election…
Aung San Suu Kyi: Accusations of Genocide Against Myanmar 'Misleading'
Myanmar's de facto leader Aung San Suu Kyi is leading her country's defense at the International Court of Justice in…
Russian court convicts 11 people over 2017 St Petersburg metro bombing
A Russian court has convicted 11 people for organising a deadly bomb attack in Saint Petersburg in 2017, Euronews reports….


Flashback Sunday, and this fascinating clip of the health care debate under Richard Milhous.
class-tiered privilege would be a rather vulgar anecdote, if it were not for the collateral damage.
And learn some history so we can finally stop this deja vu all over again.
This entry was posted on Sunday, August 30th, 2009 at 9:11 am and is filed under Uncategorized.	You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.


text:
Il campionato italiano di calcio di Eccellenza regionale 2001-2002 è stato l'undicesimo organizzato in Italia. Rappresenta il sesto livello del calcio italiano. Il campionato si è concluso con la vittoria del Cisco Collatino per il girone A e dell'Anagni per il girone B, entrambi al loro primo titolo.

Tutte le squadre iscritte al campionato di Eccellenza, assieme a quelle del campionato di Promozione, hanno diritto a partecipare alla Coppa Italia Dilettanti Lazio 2001-2002.

Stagione

Aggiornamenti
 La Cisco Tor Sapienza acquisisce il Collatino e forma il Cisco Collatino, il Tor Sapienza si iscrive comunque al campionato.
 Il , retrocesso nella stagione precedente, si fonde con il Villanova e si iscrive al campionato con il nome di Guidonia Villanova.
 La Velletri Castelli Romani torna alla vecchia deniminazione di .
 Il Maccarese cambia denominazione in Giada Maccarese.
 L'Atletico Formia cambia denominazione in Società Sportiva Formia Calcio.
 A completamento di organici viene ripescato dalla Promozione il Nettuno.

Di seguito la composizione dei gironi fornita dal Comitato Regionale Lazio.

Girone A

Squadre partecipanti

Classifica finale
Fonte:

Legenda:
      Promossa in Serie D 2002-2003.
      Ammessa ai Play-off nazionali.
      Retrocesse in Promozione Lazio 2002-2003.

Regolamento:
Tre punti a vittoria, uno a pareggio, zero a sconfitta.

Note:
Il Guidonia Villanova ha scontato 10 punti di penalizzazione.
Il Guidonia Villanova promosso in Serie D 2002-2003 in quanto vincitore dei play-off nazionali.

Risultati

Tabellone

Girone B

Squadre partecipanti

Classifica finale
Fonte:

Legenda:
      Promossa in Serie D 2002-2003.
      Ammessa ai Play-off nazionali.
      Retrocesse in Promozione Lazio 2002-2003.

Regolamento:
Tre punti a vittoria, uno a pareggio, zero a sconfitta.

Risultati

Tabellone

Note

Bibliografia
L'Annuario F.I.G.C. 2001-02, Roma (2002) conservato presso:
 tutti i Comitati Regionali F.I.G.C.-L.N.D.;
 la Lega Nazionale Professionisti a Milano;
 la Biblioteca Nazionale Centrale di Firenze;
 la Biblioteca Nazionale Centrale di Roma.
 ANNOGOL 2003 di Carlo Fontanelli - GEO Edizioni

Collegamenti esterni

Lazio


text:
Q: setting environment variables in ansible permanently i am using ansible to add permanent environment variables in ubuntu bashrc .
i have these settings defined in prod_vars file:
enviornment_variables:
  PRODUCTION:
    MONGO_IP: 0.0.0.0
    MONGO_PORT: 27017
    ELASTIC_IP: localhost
    ELASTIC_PORT: 9200

how can i export it using a task? i kniow about lineinfile module but i do not want to repeat for every env var
- name: set env in the bashrc files
  lineinfile: dest=/home/user/.bashrc line='export MONGO_IP=enviornment_variables[PRODUCTION][MONGO_IP]'

also above command gives synatx error?

A: Instead of using lineinfile module, use the blockinfile module.
So something like this should work:

- name: Adding to environment variables for user
    blockinfile:
        path: /home/user/.bashrc
        insertafter: EOF
        block: |
          export {{ item.key }}={{ item.val }}
        marker: "# {mark} {{ item.key }}"
    with_dict:
        "{{ enviornment_variables['PRODUCTION'] }}"

ps: The spelling error in "environment" literally took 20+ minutes for me to identify!

redpajama_set_name: RedPajamaStackExchange


text:
CEIBS Hosts Paris Webinar on New Opportunities for China-France Economic Cooperation
February 3, 2021. Paris, France – CEIBS, the Paris Region Chamber of Commerce and Industry and Bpifrance jointly organised a webinar today entitled An Historic Opportunity for French Enterprises to Enter the Chinese Market under the China-EU Comprehensive Investment Agreement. Chinese Ambassador to France and Monaco Lu Shaye, EU Ambassador to China Nicolas Chapuis, former WTO Director-General and CEIBS Distinguished Professor Pascal Lamy, CEIBS Vice President and Dean Ding Yuan, Paris Region Chamber of Commerce and Industry Chairman Didier Kling and Bpifrance President Nicolas Dufourcq participated in the event.
During the webinar, the panel reviewed the China-EU Comprehensive Investment Agreement reached by China and the EU on December 30 last year, and spoke highly of the unprecedented opportunities that the agreement provides for European companies – especially French companies – to enter the Chinese market. According to the agreement, China will open up sectors including new energy vehicles, telecommunications manufacturing, cloud computing, financial services, healthcare, corporate services and service industries related to sea and air transportation, environmental protection and construction to EU companies.
Ambassador Lu said that the agreement is the result of the joint efforts of both China and the EU. Moreover, he added that it reflects China's determination in participating in the international economy and further opening up to the outside world. It is a balanced, high-level, mutually beneficial and win-win agreement, he continued, that provides for companies from both China and Europe. In particular, it will result in greater market access, an improved business environment, stronger institutional guarantees and brighter prospects for cooperation.
Many areas of the agreement are open to European companies for the first time and will create conditions to better share in the development dividends of the Chinese market in the post-COVID-19 era. Ambassador Lu said that the China-EU Comprehensive Investment Agreement also serves to refute the so-called 'economic decoupling' fallacy. In the context of globalisation, economic decoupling is harmful to any country and will not work. The Chinese government, he noted, has put forward the concept of economic 'dual circulation,' which places emphasis on the integration of the Chinese and world economies while developing the Chinese market. Ambassador Lu concluded by noting that the completion of the agreement is not the end, but rather a new starting point. Finally, he said he hopes that China and the EU will work together to promote the ratification process and make the agreement effective as soon as possible.
Next, Ambassador Chapuis noted that the agreement covers a wide range of fields and in many respects provides unprecedented opportunities for European companies to enter the Chinese market. He cited the position of the EU-China Chamber of Commerce on this issue, saying that the agreement is of great significance to European companies in China. Ambassador Chapuis also emphasised that the signing of the agreement shows that in today's world, the best way to resolve any disagreement and conflict of interest is to negotiate, adding that, while the agreement promotes economic cooperation, it can also promote world peace. Ambassador Chapuis said that in the context of today's global environment, 'market protectionism' and 'economic decoupling' run counter to the times. Moreover, he continued, the China-EU Comprehensive Investment Agreement will help maintain and consolidate the world's multilateral economic system. Finally, Ambassador Chapuis called on European companies to pay attention to the agreement to ensure that it is successfully passed and signed by the EU in 2022.
Professor Lamy stated that he believes that the China-EU Comprehensive Investment Agreement represents a brand new '21st century agreement,' which goes beyond the usual minor tinkering of agreements between countries. However, he cautioned, it is too early to evaluate the impact of the agreement and it is still necessary to see it implemented. Nevertheless, he said that the agreement allows European companies to make up for the shortcomings left by the signing of the first phase of the economic and trade agreement between China and the United States.
European companies are now ahead of American companies in the Chinese market and Professor Lamy said that the agreement sends a clear signal that 'economic decoupling' is not feasible and that the EU and China are willing to enter into deeper economic exchange and integration. He added that there are still different opinions on the agreement within the EU, but that the webinar was very helpful for all parties to understand the agreement and will help the agreement be successfully passed within the EU in the future.
Next, Professor Ding said that it is gratifying that the China-EU Comprehensive Investment Agreement has finally reached a principled agreement after seven years, and it represents an unprecedented opportunity for European companies to enter the Chinese market. However, he added, the Chinese market is not what it used to be, and it is now full of competition. For example, he said, with regards to the medical field, France's technology is indeed very good, but Japan and South Korea are also quite strong and have cultivated the Chinese market for many years. Therefore, he advised, French companies need to identify partners and understand market characteristics while studying the content of the agreement. Professor Ding further proposed that with the rapid growth of the Chinese middle class and rising demand for high-end consumer goods, French companies are looking at huge opportunities to enter the Chinese market.
Chairman Kling then said that the Chamber of Commerce, as a home for enterprises, is particularly concerned about what new business opportunities the agreement will present for SMEs. The Paris Region Chamber of Commerce and Industry has maintained an office in China for a long time, and often holds various activities at its headquarters in Paris to help French companies understand China. Compared with the past, he said, China has made significant technological progress and French companies have little advantage. As such, he added, they must re-understand the Chinese market and keep pace with the times. Mr. Kling explained that he believes the agreement brings the cooperation between Chinese and French companies closer, but it requires the unremitting efforts of French entrepreneurs to turn potential opportunities into real opportunities. Furthermore, he said he believes that CEIBS has been helpful for entrepreneurs on both sides in understanding each other's markets.
Next, President Dufourcq said that the agreement is undoubtedly good news for French companies. As for positioning the agreement as a 'major breakthrough' or a 'general breakthrough,' he said it does not make much sense in itself. What is really important, he noted, is to implement the content of the agreement and to find new business opportunities for French companies. Mr. Dufourcq admitted that certain manufacturing sectors that were opened in the agreement, such as electric vehicles, represent no comparative advantage. As French automakers have no comparative advantage, he explained, such dividends may instead be enjoyed by German automakers. Nevertheless, he pointed out that the in-vehicle IoT technology used in German cars – and even in Tesla cars – is largely provided by French companies. French company Valeo, for example, has more than 30 projects in China. Bpifrance has invested in a large number of innovative projects in China. In fact, Mr. Dufourcq said, France is a country full of entrepreneurial spirit and innovation. The agreement, he added, provides new opportunities for French companies to enter the Chinese innovation market. Finally, he said that what French companies were most worried about before was that China would fall into the trap of 'economic decoupling,' and the agreement reached in principle shows the firm attitude of China and the EU against this trend.
More than 200 participants, including member companies of the Paris Region Chamber of Commerce and Industry, member companies of Bpifrance, CEIBS European alumni and students from the school's two major teaching and research bases in Europe and Africa participated in the webinar. Participants commented that they believe that the agreement is of great significance to the promotion of French companies' direct investment in China. The French business community also welcomed the agreement and said it hoped that China and France can further strengthen exchange and take practical steps to promote global economic cooperation in the post-epidemic era.
The webinar was hosted by Dr. Xu Bo, Special Advisor for International Affairs, Office of the Dean, CEIBS.
Dr. Xu Bo


How many kids get excited to visit the dentist? Mine do. Eye exam? Yep. Well, except maybe Joey, because he hates getting drops put in his eyes (gets that from Rob).
Joey's vision is 20/15, but he can't resist a prop. I love the Napolean Dynamite specs. If he grows his hair out again, it'll be the perfect Halloween costume.
Ever since the boys quit soccer, our Saturdays have been blissfully open. This week was an exception. The boys left at 7:45 a.m. for a merit badge day. I joined them for a couple of leader classes from 10:00-12:00. Robbie had most of the merit badges they offered, so he came home with me after lunch, and Joey stayed until 4:00 and worked on rank advancement. Here they are in the fingerprinting class, which they both enjoyed. Their other favorite was the electronics badge.
Robbie and I hurried home, changed and cut brownies to take to Grandpa Backman's 90th birthday open house. We feel so lucky to have him in our lives.
It was great to see Rob's brother Rich, from Atlanta. We missed Matt & his family, and of course Joe wasn't there.
Judy is the oldest of seven daughters. We love all of Rob's aunts.
Here's a shot of all the family there—two of the Champneys came from California and Texas, and the Prices came from California, Texas and Germany! It was so fun to see everyone.
And the great-grandkids—this should have been a video, not a picture. It was pretty comical.
We picked Joey up just in time for a photo before everyone went home.
In case anyone else is doing a kindergarten learning center on St. Patrick's Day. :) I couldn't find a graph that was free and high resolution with all eight charms, so I made one. Feel free to use it. Just give each kid a baggy with Lucky Charms in it, have them sort their charms into piles, then color in the graph with a matching crayon. When they're done, they can eat the cereal.
My "theme" this month is color.
I love the way the girl's rooms turned out—that's another post to come, but here's a sneak peek.
I'm also so excited to see a few flowers and green leaves sprouting up and can't wait for Spring!
I stocked up on lots of colorful fruits and veggies at Target, and will be making yummy Spring Rolls this week.
Ainsley spent most of the morning coloring—right now she's into Bratz Girls' coloring pages (I can't stand them, but I guess it's better than watching TV all morning).
I painted the girls' fingernails green for St. Patrick's Day.
Rob very grudgingly agreed to let me take a photo of him wearing the new shirt I picked up at Costco—I love the color!
Robbie had a water polo game tonight. Water polo is seriously fun to watch, even though it makes me tired just watching. :) They won.
Finally, I found Joey reading in his room, instead of putting on his pj's. No collage of photos representing this day would be complete without piles of dirty clothes. Just keeping it real.
It seems like every haircut Joey gets takes an inordinate amount of coaxing and/or bribery, and is usually a picture taking event, because it happens so rarely. I guess if long hair is the worst form of rebellion we get, we'll count ourselves lucky.
Joey's class is studying debate, and he did a good job arguing against vending machines in elementary schools.
The Kindy 500 happened to fall on the same day as Dr. Seuss day, thus the crazy hair and socks.