text:
\section*{Introduction}

If  $X = V(f) \subset \P^N$ is  a  hypersurface,
the hessian determinant of $f$ (from now on simply  called   the hessian of $f$ or, by abusing language, the hessian  of $X$)  is the determinant of the
hessian matrix of $f$.

Hypersurfaces with vanishing hessian
were studied systematically  for the first time in  the fundamental paper \cite{GN}, where  P. Gordan and M. Noether  analysed
O. Hesse's claims in \cite{Hesse1, Hesse2}
according to which these hypersurfaces should be necessarily cones.  Clearly the claim is true if $\deg(f)=2$
so that the first relevant case for the problem is that of cubic hypersurfaces. The cubic hypersurface
$V(x_0x_3^2 +  x_1x_3x_4 +  x_2x_4^2)\subset {\mathbb P}^4$ has vanishing hessian but
it is not a cone, see \cite{Perazzo}. By adding the term $\sum_{i=5}^Nx_i^3$ we get examples of
 irreducible cubic hypersurfaces in ${\mathbb P}^N$ with $N\geq 4$ with vanishing hessian
 which are not cones.

Notwithstanding, the  question is quite subtle
because,  as it was firstly pointed out in \cite{GN},
Hesse's  claim is true for $N\leq 3$,  see also \cite{Lossen, GR} and \cite[Section 7]{Russo}.
Moreover,  hypersurfaces with vanishing hessian  and the Gordan-Noether Theory developed in \cite{GN} have a  wide range of applications
in different areas of mathematics such as Algebraic and  Differential Geometry (see \cite{AG, AG2, FP}), Commutative Algebra and the theory of EDP (see \cite{DeBondt1, Russo, Rodrigo}), Approximation Theory and Theoretical Physics (see \cite{FP,AG}) and
Combinatorics (see \cite{GoZa}).

Hypersurfaces with vanishing hessian have been forgotten by  algebraic geometers  for a long time and recently they were rediscovered in other contexts.
For example the cubic hypersurface in ${\mathbb P}^4$ recalled above is  celebrated nowadays in the modern differential geometry literature as the
{\it Bourgain-Sacksteder Hypersurface} (see \cite{AG2, AG, FP}).

Many classes of hypersurfaces with vanishing hessian, which are not cones, are ruled by a family of linear
spaces
along which the hypersurface is not developable. In particular, this ruling is different from the one  given by the fibers of the Gauss map. These examples and their generalizations are known in differential geometry as
{\it twisted planes}, see for example \cite{FP}. Despite the huge number of papers dedicated to this subject by differential geometers very few classification or structure results have been obtained.
Moreover, the global point of view provided by polarity, used systematically in this paper, has been completely overlooked in other areas.

The known  series of examples of hypersurfaces $X\subset {\mathbb P}^N$ with vanishing hessian, which are not cones,
have been  constructed by  Gordan and Noether, Perazzo, Franchetta, Permutti, see \cite{GN, Perazzo, Franchetta, Permutti1, Permutti2},
and later have been revisited and generalised in \cite[Section 2]{CRS}, see also \cite{CuRaSi, CuRaSi2}.

All these examples share several geometrical behaviours, see  in particular Subsection \ref{geopro}. For instance
there exists  a linear subspace $L \subset \operatorname{Sing} X$, dubbed the {\it core of $X$} in \cite{CRS}, such that,
letting  $L_{\alpha} = \P^{k+1}\supset L$ and letting
$L_{\alpha} \cap X = \mu_{\alpha} L \cup X_{\alpha}$ for some $\mu_{\alpha}\in\mathbb N$,
the variety  $X_{\alpha}$ is a cone with vertex $V_{\alpha} = \operatorname{Vert}(X_{\alpha}) \subset L$ tangent to $Z_X^* \subset L$, where $Z_X\subsetneq ({\mathbb P}^N)^*$ is the closure
of the image of the polar map of $X$. When  the cones $X_{\alpha}$ split into a union of linear spaces, the hypersurface is a twisted plane.
Furthermore,  the dual variety $X^*\subset({\mathbb P}^N)^*$ of most of the examples in these series
 tends to be a divisor in  $Z_X$. From the perspective of   Segre's Formula, recalled in Section \ref{prelHess},
 this  means
that the rank of the hessian matrix of a homogeneous polynomial with vanishing hessian determinant should be equal to the rank of the hessian matrix modulo the ideal generated by $f$, see Section
\ref{prelHess} for precise definitions. This  seemed to be the most natural and general behaviour, at least at a first glance.

On the other hand,  if
$Y\subset({\mathbb P}^N)^*$ is an arbitrary  non-degenerate irreducible  variety of  dimension $n\geq 1$, then, after identifying ${\mathbb P}^N$ with $({\mathbb P}^N)^{**}$,
the dual variety  $X=Y^*\subset{\mathbb P}^N$  is not a cone and, in general,  one expects that  $X$ is a hypersurface with non  vanishing hessian. If this is the case,  $Z_X=({\mathbb P}^N)^*$ and $\operatorname{codim}(X^*, Z_X)=\operatorname{codim}(Y,({\mathbb P}^N)^*)=N-n$
is arbitrary large.

These remarks motivate the search of hypersurfaces with vanishing hessian $X\subset{\mathbb P}^N$ such that $X^*$
has arbitrary codimension in $Z_X\subsetneq ({\mathbb P}^N)^*$.
Here we shall  present
a general construction of such hypersurfaces, starting from any irreducible non-degenerate
variety whose dual variety is a hypersurface and  based on the  Dual Cayley Trick.
The geometrical  properties of these hypersurfaces
are different from those described above (for example $V_\alpha$ is not contained in $L$) and their  dual varieties can have arbitrary codimension in the image
of the polar map. The ubiquity of the examples suggests that the classification of hypersurfaces with vanishing hessian for $N\geq 5$
might be very intricate, perhaps requiring a completely different approach not  based (only) on Gordan--Noether
Theory, which worked for $N\leq 4$, see \cite{GN, Franchetta, GR, Russo}. Other interesting series of examples of hypersurfaces with vanishing hessian such that $\operatorname{codim}(X^*,Z_X)$ is large
 has been recently constructed in \cite{CuRaSi, CuRaSi2} (see also Remark \ref{CuRS} for a possible geometrical description
of this series of examples).

The paper is organised as follow. In Section 1 we fix the notation and introduce the main definitions. Section 2 is devoted to the construction of the first
series of examples, leading to Theorem \ref{codimension} and ending with the description of the geometrical properties of the examples. In Section 3 we
briefly recall the definitions of resultant and discriminant and we apply them to calculate explicitly the dual of rational normal scroll surfaces in Theorem \ref{dualSab}.
Then we introduce the Cayley Trick and the Dual Cayley Trick and apply the Dual Cayley Trick to a generalisation of the series of examples constructed
in Section 2 (see Theorem \ref{codimensionk}), also providing a new conceptual proof of  the part (ii) of Theorem \ref{codimension}. In Section 4 we  prove that the duals of internal projections
from a point of Scorza Varieties have vanishing hessian and we describe the geometrical properties of these hypersuperfaces and of their polar maps.
\medskip

{\bf Acknowledgements}. We  wish to thank the referee for a very careful reading, for pointing out some inaccuracies  and for many useful suggestions leading to a significant improvement of the exposition.

\section[Preliminaries and definitions]{Preliminaries and definitions}\label{prelHess}

 Let  $f(x_0,\ldots,x_N)\in\mathbb  K[x_0,\ldots, x_N]_d$ be a homogeneous polynomial of degree $d\geq 1$  without multiple irreducible factors   and let $X = V(f) \subset \P^N$  be the associated projective  hypersurface. We shall always assume that   $\mathbb K$ is an algebraically closed field
of characteristic zero. Let
$$ \operatorname{H}(f)=\left[\frac{\partial^2 f}{\partial x_i \partial x_j}\right]_{0\leq i,j\leq N}$$
be the {\it hessian matrix of $f$}  (or of $X$).

Clearly $\operatorname{H}(f)=\mathbf 0_{(N+1)\times(N+1)}$ if and only if $d=1$. Thus, from now on,
{we shall suppose $d\geq 2$. Let
 $$\operatorname{hess}_f=\det(\operatorname{H}(f))$$
be the  {\it hessian (determinant) of $f$} (or of $X$, in which case
it will be denoted by  $\operatorname{hess}_X$, which
is defined modulo multiplication by a non zero element in~$\mathbb K$).
\medskip

There are two possibilities:

 \begin{enumerate}
\item either $\operatorname{hess}_f= 0$
\medskip
or
\item $\operatorname{hess}_f\in\mathbb K[x_0,\ldots, x_N]_{(N+1)(d-2)}$.
\end{enumerate}
\medskip

We shall be interested in case (1), that is in {\it hypersurfaces with vanishing hessian} ({\it determinant}).
\medskip

\subsection{The polar map}\label{ss:polarmap}
Let
$$
\nabla_f=\nabla_X:{\mathbb P}^N\dasharrow({\mathbb P}^N)^{*}$$
be {\it the polar (or gradient) map of $X=V(f)\subset{\mathbb P}^N$} which associates to $p\in {\mathbb P}^N$
the {\it polar hyperplane to $X$ with respect to $p$.} In coordinates it is defined by
$$\nabla_f(p)=\Big(\frac{\partial f}{\partial x_0}(p):\cdots :\frac{\partial f}{\partial x_N}(p)\Big).$$
\index{Polar map $\nabla_f$ o $\nabla_X$}\index{$\nabla_f$}\index{$\nabla_X$}

The base locus scheme of $\nabla_f$ is the scheme

$$\operatorname{Sing}(X):=V\Big(\frac{\partial f}{\partial x_0},\ldots, \frac{\partial f}{\partial x_N}\Big)\subset {\mathbb P}^N.$$

Let
   $$Z_X := \overline{\nabla_f(\P^N)}\subseteq({\mathbb P}^N)^{*}$$
   be  {\it the  polar image of ${\mathbb P}^N$}.\index{$Z$}\index{Polar image}

We can consider the rational map $\nabla_f$
as the quotient by the natural $\mathbb K^*$-action of the affine morphism
$$\nabla_f:\mathbb K^{N+1}\to\mathbb K^{N+1}$$
defined in the same way. Thus we have
the following key formula:
\begin{equation}\label{KeyFormula}
\operatorname{H}(f)=\operatorname{Jac}(\nabla_f:\mathbb K^{N+1}\to\mathbb K^{N+1}),
\end{equation}
that is, the hessian matrix of $f$ is the Jacobian matrix of the affine morphism $\nabla_f$.
Hence, $\operatorname{hess}_f=0$ if and only if  $Z_X\subsetneq {\mathbb P}^N$ so that $\operatorname{hess}_f=0$ if and only if $\frac{\partial f}{\partial x_0}, \frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_N}$
are algebraically dependent.

The restriction of $\nabla_f$ to $X$ is the Gauss map of $X$:
$$
\begin{array}{cccc}
\mathcal G_X=\nabla_{f|X}:& X&\dasharrow&\mathbb ({\mathbb P}^N)^{*}\\
&X_{\operatorname{reg}}\ni p  &\longmapsto& \mathcal G_X(p)=[T_pX]
\end{array}
$$
which to a non-singular point $p\in X_{\operatorname{reg}}$ associates the point $[T_pX]\in({\mathbb P}^N)^*$
representing the projective tangent hyperplane $T_pX$ to $X$ at the smooth point $p$.
Then, by definition,
$$X^*:=\overline{\mathcal G_X(X)}\subseteq Z_X$$
is the {\it dual variety of $X$}.
\medskip

If $A$ is a matrix with entries in $\mathbb K[x_0,\ldots, x_N]$ and if $f\in \mathbb K[x_0,\ldots,x_N]$,
then $\operatorname{rk}_{(f)}A$ denotes the {\it rank of $A$ modulo $(f)$}, that is the maximal order
of a minor not belonging to the ideal generated by $f$. With this notation, obviously $\operatorname{rk} A=\operatorname{rk}_{(0)}A$.
\medskip

\begin{lemma}\label{rkGauss}{\rm (\cite{Segre})}
Let $X=V(f)=X_1\cup\cdots\cup X_r\subset{\mathbb P}^N$ be a reduced hypersurface with $X_i=V(f_i)$,  $f=f_1\cdots f_r$
and $f_i$ irreducible. Then:
\medskip
\begin{enumerate}
\item[(i)] If $p_i\in X_i$ is general, then
\begin{equation}\label{Segreformula}
\operatorname{rk}(\operatorname{d}\mathcal G_X)_{p_i}=\operatorname{rk}_{(f_i)}H(f)-2.
\end{equation}
In particular,
\begin{equation}\label{Segreformulaeq}
\dim(X_i^*)=\operatorname{rk}(\operatorname{d}\mathcal G_X)_{p_i}=\operatorname{rk}_{(f_i)}H(f)-2\leq \operatorname{rk} H(f)-2= \dim(Z_X)-1.
\end{equation}
\medskip
\item[(ii)] If $X$ is irreducible, then $f^{N-\dim(X^*)-1}$ divides $\operatorname{hess}_f$.
\end{enumerate}
\end{lemma}
\medskip

We point out an immediate consequence for further reference.
\medskip

\begin{cor}\label{XdualinZ} Let $X=V(f)\subset{\mathbb P}^N$ be a reduced hypersurface with vanishing
hessian. Then
$$X^*\subsetneq Z_X\subsetneq ({\mathbb P}^N)^*.$$
\end{cor}
\medskip

We shall also need the following remark.
\medskip

\begin{lemma}\label{projectionHess}{\rm (\cite[Lemma
3.10]{CRS})}
Let  $X=V(f)\subset{\mathbb P}^N$ be a hypersurface. Let
$H={\mathbb P}^{N-1}$ be a hyperplane not contained in $X$, let
$h=H^*$ be the corresponding point in $({\mathbb P}^N)^{*}$ and let
$\pi_h$ denote the projection from the point $h$. Then we have a commutative diagram:
\begin{equation*}
\xymatrix{ H \ar@{-->}[rr]^{\nabla_{X\cap H}} \ar@{-->}[d]_{(\nabla_{X})|_H} & & H^{\ast}\\ ({\mathbb P}^N)^{\ast}\ar@{-->}[rru]_{\pi_h} }
\end{equation*}
In particular, $\overline{\nabla_{X\cap H}(H)}\subseteq \pi_h(Z_X).$
\end{lemma}
\medskip

\section{Hypersurfaces with vanishing hessian constructed from any non-degenerate variety}

\subsection{Hypersurfaces with vanishing hessian constructed from duals of arbitrary non-degenerate subvarieties}

Let us consider ${\mathbb P}^{2n+1}$ with homogeneous coordinates
$$(u:v:x_1:\dots:x_n:y_1:\cdots:y_n),$$
${\mathbb P}^1$ with homogeneous coordinates $(s:t)$ and ${\mathbb P}^{n-1}_{\mathbf z}$ with homogeneous
coordinates $(z_1:\cdots:z_n)$, where $\mathbf z=(z_1,\ldots,z_n)$.
Let  $\mathbf x=(x_1,\ldots, x_n)$, let $\mathbf y=(y_1,\ldots, y_n)$,
 let $$\phi_1:{\mathbb P}^{2n+1}\dasharrow{\mathbb P}^1$$ be the rational map defined by
$$\phi_1(u:v:\mathbf x:\mathbf y)=(u:v)$$
and let $\phi_2:{\mathbb P}^{2n+1}\dasharrow{\mathbb P}^{n-1}_{\mathbf z}$ be the rational map defined by
\begin{equation}\label{phi2}
\phi_2(u:v:\mathbf x:\mathbf y)=(ux_1-vy_1:\cdots:ux_n-vy_n).
\end{equation}

Let $g(z_1,\ldots, z_n)\in\mathbb K[z_1,\ldots,z_n]_d$ be a reduced irreducible polynomial such that
 the associated  irreducible hypersurface of degree $d$
 $$Y^*=V(g)\subset{\mathbb P}^{n-1}_{\mathbf z}$$
 is  not a cone. This is equivalent
 to
$$Y=Y^{**}=\overline{\nabla_g(Y^*)}\subset({\mathbb P}^{n-1}_{\mathbf z})^* $$
being non-degenerate.

Let
$$f(u,v,\mathbf x,\mathbf y)=g(ux_1-vy_1,\ldots, ux_n-vy_n)\in \mathbb K[u,v,\mathbf x,\mathbf y]_{2d}$$
and let $X=V(f)\subset{\mathbb P}^{2n+1}$. Clearly,
\begin{equation}\label{XfromY*}
V(f)=\overline{\phi_2^{-1}(V(g))}.
\end{equation}

The partial
derivatives of $f$ are  linearly independent over $\mathbb K$, due to the hypothesis on $g$, so that  $X=V(f)\subset{\mathbb P}^{2n+1}$ is not a cone.
One verifies that
$$\operatorname{Sing}(X)=V(u,v)\cup ({\mathbb P}^1\times{\mathbb P}^n)\cup \phi_2^{-1}(\operatorname{Sing}(V(g))),$$
where ${\mathbb P}^1\times{\mathbb P}^n\subset{\mathbb P}^{2n+1}$ is the Segre variety defined by the equations
$$
\operatorname{rk}
\left( \begin{array}{cccc}
v&x_1&\ldots&x_n\\
u&y_1&\ldots&y_n
\end{array}
\right)
=1.$$

From
\begin{equation}\label{dxi}
\frac{\partial f}{\partial x_i}=u\frac{\partial g}{\partial z_i}(u\mathbf x-v\mathbf y)
\end{equation}
and
\begin{equation}\label{dyj}
\frac{\partial f}{\partial y_j}=-v\frac{\partial g}{\partial z_j}(u\mathbf x-v\mathbf y),
\end{equation}
we deduce that, for every $i\neq j$,
\begin{equation}\label{ij-ji}
\frac{\partial f}{\partial x_i}\frac{\partial f}{\partial y_j}-\frac{\partial f}{\partial x_j}\frac{\partial f}{\partial y_i}=0.
\end{equation}

Thus $X\subset{\mathbb P}^{2n+1}$ has vanishing hessian since the partial derivates of $f$ are algebraically dependent.

\subsection{Polar image and dual variety of $X=V(g(u\mathbf x-v\mathbf y))\subset{\mathbb P}^{2n+1}$}

We need to introduce some more notation.
Let $$(u':v':x'_1:\dots:x'_n:y'_1:\cdots:y'_n)$$ be homogenous coordinates on $({\mathbb P}^{2n+1})^*,$ dual to the coordinates chosen on ${\mathbb P}^{2n+1}$.
Let $$L'=V(x_1',\ldots,x'_n,y'_1,\ldots, y'_n)={\mathbb P}^1_{u',v'}\subset ({\mathbb P}^{2n+1})^*$$ and let $$W={\mathbb P}^1\times{\mathbb P}^{n-1}\subset V(u',v')={\mathbb P}^{2n-1}_{\mathbf x',\mathbf y'}\subset({\mathbb P}^{2n+1})^*$$
be the Segre variety defined by the equations:
\begin{equation}\label{eqW}
\operatorname{rk}
\left( \begin{array}{ccc}
x_1'&\ldots&x'_n\\
y'_1&\ldots&y'_n
\end{array}
\right)
=1.
\end{equation}

Let  $S(L',W)\subset({\mathbb P}^{2n+1})^*$ be  the  cone with vertex $L'$ over the Segre variety $W={\mathbb P}^1\times{\mathbb P}^{n-1}$.
Thus $\dim(S(L',W))=n+2$ and $S(L',W)\subset({\mathbb P}^{2n+1})^*$ is defined by  the equations \eqref{eqW} of $W$.

Let ${\mathbb P}^{n-1}_{\mathbf z'}$ with homogeneous coordinates $(z'_1:\cdots:z'_n)$ be the dual of ${\mathbb P}^{n-1}_{\mathbf z}$ and consider ${\mathbb P}^n_{\mathbf z'}$ with homogeneous coordinates $(z'_0:z'_1:\cdots:z'_n)$. With this notation   ${\mathbb P}^{n-1}_{\mathbf z'}\subset{\mathbb P}^n_{\mathbf z'}$ is  the hyperplane of equation $z'_0=0$. Given $Y\subset{\mathbb P}^{n-1}_{\mathbf z'}$, let $\widetilde Y\subset{\mathbb P}^n_{\mathbf z'}$ be the cone
over $Y$ with vertex $(1:0:\cdots:0)$.

\begin{theorem}\label{codimension} Let the hypothesis and the notation be as above,  let $$Z_{Y^*}=\overline{\nabla_g({\mathbb P}^{n-1}_{\mathbf z})}\subseteq{\mathbb P}^{n-1}_{\mathbf z'},$$
let
$${\mathbb P}^1\times Z_{Y^*}\subset({\mathbb P}^{2n-1})^*$$ be the Segre embedding and let
$X=V(g(u\mathbf x-v\mathbf y))\subset{\mathbb P}^{2n+1}$. Then:
\medskip
\begin{enumerate}
\item[(i)] $Z_X=\overline{\nabla_f({\mathbb P}^{2n+1})}=S(L',{\mathbb P}^1\times Z_{Y^*})\subset({\mathbb P}^{2n+1})^*$;
\medskip
\item[(ii)] $X^*={\mathbb P}^1\times\widetilde Y\subset({\mathbb P}^{2n+1})^*$ Segre embedded;
\medskip
\item[(iii)] $\operatorname{codim}(X^*,Z_X)=\operatorname{codim}(Y, Z_{Y^*})+1.$
\end{enumerate}
\medskip

In particular, if $g(\mathbf z)$ has non-vanishing hessian determinant, then $Z_{Y^*}={\mathbb P}^{n-1}_{\mathbf z'}$
and $\operatorname{codim}(X^*,Z_X)=\operatorname{codim}(Y, {\mathbb P}^{n-1})+1.$
\end{theorem}
\begin{proof}
By definition  $\nabla_f:{\mathbb P}^{2n+1}\dasharrow({\mathbb P}^{2n+1})^*$ is given by
$$(\frac{\partial f}{\partial u}:\frac{\partial f}{\partial v}:\frac{\partial f}{\partial x_1}:\cdots:\frac{\partial f}{\partial x_n}:\frac{\partial f}{\partial y_1}:\cdots:\frac{\partial f}{\partial y_n}).$$

From \eqref{ij-ji} and from \eqref{eqW} we deduce that
\begin{equation}\label{in1}
\overline{\nabla_f({\mathbb P}^{2n+1})}\subseteq S(L',{\mathbb P}^1\times Z_{Y^*}).
\end{equation}

We also have
\begin{equation}\label{du}
\frac{\partial f}{\partial u}=\sum_{i=1}^nx_i\frac{\partial g}{\partial z_i}(u\mathbf x-v\mathbf y),
\end{equation}
\begin{equation}\label{dv}
\frac{\partial f}{\partial v}=-\sum_{j=1}^ny_j\frac{\partial g}{\partial z_j}(u\mathbf x-v\mathbf y).
\end{equation}

Let
$$p=(\tilde{u}':\tilde{v}': \tilde{\mathbf z}':\lambda\tilde{\mathbf z}')\in S(L',{\mathbb P}^1\times Z_{Y^*})$$
be a general point. In particular, we can suppose  $\tilde u'\neq\lambda \tilde v'$ and that  $\tilde{\mathbf z}'\neq\mathbf 0$. Then $[\tilde{\mathbf z}']\in Z_{Y^*}$ is general and by definition
there exists $\tilde{\mathbf z}\in{\mathbb P}^{n-1}_{\mathbf z}$ such that $\nabla_g(\tilde{\mathbf z})=\tilde{\mathbf z}'$.
Looking at \eqref{dxi} and \eqref{dyj}, we impose $v=-\lambda u$. If $u(\mathbf x+\lambda\mathbf y)=\tilde{\mathbf z}$ and if $v=-\lambda u$ hold,
then $\frac{\partial f}{\partial x_i}=u\tilde{\mathbf z}_i'$ and $\frac{\partial f}{\partial y_i}=u\lambda\tilde {\mathbf  z}_i'$. Hence, to find $q$ such that $\nabla_f(q)=p$, it is sufficient
to determine a solution of the system of equations
\begin{equation}\label{sistema1}
\left\{
\begin{array}{rl}
u(\mathbf x+\lambda\mathbf y)=&\tilde{\mathbf z}\\
\medskip

\sum_{i=1}^nx_i\frac{\partial g}{\partial z_i}(\tilde{\mathbf z})=&u\tilde{u}'\\
\medskip

-\sum_{i=1}^ny_i\frac{\partial g}{\partial z_i}(\tilde{\mathbf z})=&u\tilde{v}'\\
\end{array}
\right.
\end{equation}

From
$$u\tilde{u}'=\sum_{i=1}^n(\frac{\tilde{z_i}}{u}-\lambda y_i)\frac{\partial g}{\partial z_i}(\tilde{\mathbf z})=\frac{1}{u}d\cdot g(\tilde{\mathbf z})+\lambda u\tilde{v}',$$
we deduce
$$u^2=\frac{d\cdot g(\tilde{\mathbf z})}{\tilde{u}'-\lambda\tilde{v}'}.$$

If $\tilde u$ is a solution of this last equation and if
$\tilde{\mathbf a}=(\tilde{a}_1,\ldots, \tilde{a}_n)$ is a solution of the last linear equation in \eqref{sistema1},  then
$$p=\nabla_f(\tilde u:-\lambda \tilde u:\frac{\tilde{\mathbf z}}{\tilde u}-\lambda\tilde{\mathbf a}:\tilde{\mathbf a}),$$
yielding equality in \eqref{in1}.

\medskip

By restricting $\nabla_f$ to $X$, we deduce  that
$$X^*=\overline{\nabla_f(X)}\subseteq S(L', {\mathbb P}^1\times Z_{Y^*}).$$

Let $T={\mathbb P}^1\times {\mathbb P}^n\subset  ({\mathbb P}^{2n+1})^*$ be the Segre variety defined by the equations
$$
\operatorname{rk}
\left( \begin{array}{cccc}
v' & x'_1 & \ldots& x'_{n'}\\
u'&y'_1&\ldots&y'_n
\end{array}
\right)
=1.$$
Since $\deg(f)=2d$, Euler's Formula gives
\begin{equation}\label{Eulerf}
(2d)f=u\frac{\partial f}{\partial u}+v\frac{\partial f}{\partial v}+\sum_{i=1}^nx_i\frac{\partial f}{\partial x_i}+\sum_{i=1}^ny_i \frac{\partial f}{\partial y_i}=2(u\frac{\partial f}{\partial u}+v\frac{\partial f}{\partial v}).
\end{equation}

Thus,  for every $p\in X$, we have
\begin{equation}\label{extraeq}
u(p)\frac{\partial f}{\partial u}(p)+v(p)\frac{\partial f}{\partial v}(p)=0,
\end{equation}
yielding
$$\overline{\nabla_f(X)}=X^*\subseteq T\subset ({\mathbb P}^{2n+1})^*.$$

Indeed, for $i\neq j$ the equations $x'_iy'_j-x_jy'_i=0$ are satisfied by any point $\nabla_f(p)$ due to \eqref{ij-ji}. Due to \eqref{extraeq}, the equations
$v'y'_i-u'x'_i=0$ are satisfied by $\nabla_f(p)$ for every $p\in X$.
Furthermore, for every $(\mu:\nu) \in{\mathbb P}^1_{\mathbb K}$, the hypersurface $X\cap V(\mu u+\nu v)$ is singular so that  $(\mu:\nu:\mathbf 0:\mathbf 0)\in X^*\cap ((\mu:\nu)\times{\mathbb P}^n)$. By fixing $u,v$, by restricting to  $X\cap V(u\mathbf x-v\mathbf y)$ and by taking into account \eqref{dxi}, \eqref{dyj} and \eqref{extraeq} one deduces  that
$X^*={\mathbb P}^1\times \widetilde Y\subset T$  (see also the next sections for more details). The other conclusions are now clear.
\end{proof}
\medskip

\begin{rem}{\rm To prove equality in \eqref{in1} one could have argued also in this way. Letting $\rho=\operatorname{rk}(\operatorname{H}(g))=\dim(Z_{Y^*})+1$, it is sufficient to prove that $\operatorname{rk}(\operatorname{H}(f))=\dim(Z_X)+1$ is equal to
$\rho+3=\dim(S(L',{\mathbb P}^1\times Z_{Y^*}))+1$.

Clearly  $\operatorname{H}(f)$ is a $(2n+2)\times (2n+2)$ matrix,
whose rank can be computed in this way.  The $(2n)\times(2n)$ submatrix corresponding to the second partial derivatives with respect to the variables $x_i$ and $y_j$ has rank $\rho$ by \eqref{dxi} and \eqref{dyj}. The $2\times 2n$ submatrix of $\operatorname{H}(f)$ corresponding to the second partial derivatives with respect to the variables $(u,v)\times(x_i, y_j)$ increases the rank by $1$ by \eqref{du} and by \eqref{dv}. The $2\times 2$ submatrix corresponding to the second partial derivatives with respect to the variables $u,v$ increases the rank by $2$. In conclusion $\operatorname{rk}(\operatorname{H}(f))=\rho+1+2=\rho+3$ so that equality holds in \eqref{in1} (see also Remark \ref{rhohess}).}
\end{rem}
\medskip

\begin{rem}{\rm Obviously also other similar changes of variables, for example like $\mathbf z\to (u\mathbf x-v\mathbf y)^k$ with $k\geq 2$, will produce other interesting hypersurfaces
with vanishing hessian. Instead of pursuing further these generalizations, we prefer to focus on the geometrical properties of the previous examples
and on  the connections with the so called {\it Dual Cayley Trick}.
}
\end{rem}

\subsection{Geometrical properties of $X=V(g(u\mathbf x-v\mathbf y))\subset{\mathbb P}^{2n+1}$, of $Z_X^*$ and of the associated polar map}\label{geopro}

Let notation be as above and suppose that $V(g)\subset{\mathbb P}^{n-1}$ has non-vanishing hessian. Then $$Z_X=S(L',W)=S(L',{\mathbb P}^1\times{\mathbb P}^{n-1})\subset({\mathbb P}^{2n+1})^*$$ and
$$Z_X^*\simeq {\mathbb P}^1\times {\mathbb P}^{n-1}\subset V(u,v)=\langle Z_X^*\rangle\subset{\mathbb P}^{2n+1}$$
is the Segre variety defined in $V(u,v)$  by the equations:
$$
\operatorname{rk}
\left( \begin{array}{ccc}
x_1&\ldots&x_n\\
y_1&\ldots&y_n
\end{array}
\right)
=1.$$
In the terminology of \cite[Section 2.2]{CRS}, the linear space $$\Pi=V(u,v)={\mathbb P}^{2n-1}\subset \operatorname{Sing}(X)$$ is the {\it core of $X$}.
If $p=(0:0:\mathbf x:\mathbf y)\in V(u,v)$, then  we can take $(\mathbf x:\mathbf y)$ as coordinates on $V(u,v)$.
Let $L$ be the line of equations $\mathbf x=\mathbf 0=\mathbf y$,  let
$$\xi =(u:v:\mathbf 0:\mathbf 0)\in L,$$
and let
$$\Pi_\xi=\langle \Pi, \xi\rangle\subset{\mathbb P}^{2n+1}.$$
For a fixed $\xi$, the points of the hyperplane $\Pi_\xi$ can be parametrized by $(tu:tv:\mathbf x:\mathbf y)$ so that $(t:\mathbf x:\mathbf y)$ can
be taken as coordinates on $\Pi_\xi$.
Then $\Pi_\xi\cap X$ has the following equation in the hyperplane $\Pi_\xi$:
$$t^dg(u\mathbf x-v\mathbf y)=0$$
with $u,v$ fixed. Since $t=0$ is the equation of $\Pi\subset \Pi_\xi$, $\Pi_\xi\cap X$ contains $\Pi$ with multiplicity
$d$,
while $$V(g(u\mathbf x-v\mathbf y))\subset \Pi_\xi$$ is a cone with vertex a ${\mathbb P}^n$, which is
not contained in $\Pi$. The change of variable $u\mathbf x-v\mathbf y\mapsto \mathbf x$, $\mathbf y\mapsto \mathbf y$, $t\mapsto t$ shows that  the resulting equation
does not depend on the variables $\mathbf y$ and $t$, yielding that the vertex of the cone is the linear subspace of  $\Pi_\xi$ given by the $n$ linear equations $u\mathbf x-v\mathbf y=\mathbf 0$.

Varying the hyperplane in the pencil of hyperplanes through $V(u,v)$, the vertices  of the corresponding cones describe a Segre variety
${\mathbb P}^1\times {\mathbb P}^n$, which is the dual of $T\subset({\mathbb P}^{2n+1})^*$ and which cuts $V(u,v)$ along $Z^*_X$.

In particular, the series of examples constructed in this section is completely different from those known up to now,
which we shall simply call of {\it Gordan-Noether-Perazzo-Permutti-CRS type}. Indeed, in all these examples the intersection of the
linear spaces
$\Pi_\xi={\mathbb P}^{c+1}$ through  the core  $\Pi=<Z_X^*>={\mathbb P}^c$ with the hypersurface $X$ consists of the core with a suitable multiplicity and of a cone, whose vertex
is a linear space contained in $\Pi$.

From the algebraic point of view this new phenomenon means that there does not exist a suitable linear change of coordinates
such that we can  {\it separate the variables in the equation via the core}. We recall that Gordan-Noether-Perazzo-Permutti-CRS type hypersurfaces in  ${\mathbb P}^4$
exhaust the list of hypersurfaces with vanishing hessian that are not  cones.

\begin{example}{\rm In  $\P^7$  with homogeneous coordinates $(u:v:x_1:x_2:x_3:y_1:y_2:y_3)$, let
$$X = V((x_1u-y_1v)^2+(x_2u-y_2v)^2+(x_3u-y_3v)^2) \subset \P^7$$
and let $Y = V(z_1^2+z_2^2+z_3^2) \subset \P^2$ be the  self dual Fermat conic.
Letting the notation be as above, we have $X^* = \P^1 \times \widetilde Y\subset ({\mathbb P}^7)^*$. Let us remark that the construction of irreducible hypersurfaces
of this kind starts from  $\P^7$. Specialisations of above examples  have interesting applications, see \cite[Example 2.3]{DeBondt2} and \cite{DeBondt1}.}
\end{example}

\begin{example}{\rm In  $\P^5$ with homogeneous coordinates $(u:v:x_1:x_2:y_1:y_2)$, let
$$X = V((x_1u-y_1v)^2+(x_2u-y_2v)^2+u^4) \subset \P^5.$$
It is not difficult to see that $Z_X=V(\tilde{x}_1\tilde{y}_2-\tilde{x}_2\tilde{y}_1)\subset \P^5$ and that, taking into account the previous remarks, $X\subset{\mathbb P}^5$ is the first
example of a hypersurface with vanishing hessian that is not a cone and that it is not of Gordan-Noether-Perazzo-Permutti-CRS type.}

\end{example}

\section{Hypersurfaces with vanishing hessian constructed from cones with vertex a ${\mathbb P}^{k-1}$ via Dual Cayley Trick}

\subsection{Resultants and discriminants}
We  recall some well known facts on resultants and discriminants. A reference for most of
the properties listed below is \cite{GKZ}, see also \cite{Cox}.

Let $f_i(x_0,\ldots, x_N)$, $i\in\{0,\ldots,N\}$, be $N+1$ universal homogeneous polynomials
of degree $d_i\geq 1$. Then the {\it resultant of $f_0,\ldots, f_N$}, indicated by
$$\operatorname{Res} (f_0,\ldots, f_N),$$
is a polynomial in the coefficients of the $f_i$'s, which is homogeneous of degree
$d_0\cdots d_{j-1}d_{j+1}\cdots d_N$ in the variables corresponding to $f_j$
and which has degree
\begin{equation}\label{degRes}
d_0\cdots d_N\sum_{i=0}^N\frac{1}{d_i}.
\end{equation}

The polynomial $\operatorname{Res}(f_0,\ldots,f_N)$  has the following property:
given  homogeneous   polynomials $g_0,\ldots, g_N\in\mathbb K[x_0,\ldots, x_N]$ with $\deg(g_i)=d_i$,  the value of $\operatorname{Res} (f_0,\ldots, f_N)$ on the coefficients
of $g_0,\ldots, g_N$ is zero if and only if $g_0=\ldots=g_N=0$ has a non-zero solution in $\mathbb K^{N+1}$ (or equivalently
$V(g_0)\cap\ldots\cap V(g_N)\neq\emptyset$ where $V(g_i)\subset{\mathbb P}^N_\mathbb K$ is the projective hypersurface defined by $g_i$), see \cite{GKZ}.

For a universal  $f\in\mathbb K[x_0,\ldots,x_N]_d$, let
$$\Delta_{N,d}=\operatorname{Res}(\frac{\partial f}{\partial x_0},\ldots, \frac{\partial f}{\partial x_N}),$$
which is a homogeneous polynomial of degree $(N+1)(d-1)^N$ in the ${N+d}\choose d$ coefficients of
the universal $f \in\mathbb K[x_0,\ldots,x_N]_d$. By the previous geometrical} property of the resultant we deduce
that  the (geometrically irreducible) hypersurface
$V(\Delta_{N,d})\subset{\mathbb P}(\mathbb K[x_0,\ldots,x_N]_d)$, called the  {\it discriminant hypersurface}, is well defined and it describes the locus of  singular projective  hypersurfaces of degree $d$.
\medskip

 \subsection{Dual varieties of rational normal scrolls surfaces and some explicit examples of hypersurfaces with vanishing hessian}\label{dualscrolls}

 Let $1\leq a\leq b$ be integers and let $$S(a,b)\subset{\mathbb P}^{a+b+1}$$ be a rational normal scroll of degree $d=a+b$.
 The surface $S(a,b)\subset{\mathbb P}^{a+b+1}$ is
smooth and projectively generated by a rational normal curve $C_a=\nu_a({\mathbb P}^1)\subset{\mathbb P}^a$ and a rational normal curve $C_b=\nu_b({\mathbb P}^1)\subset{\mathbb P}^b$
with $\langle C_a\rangle\cap\langle C_b\rangle=\emptyset$, by taking the union of the lines $\langle \nu_a(p),\nu_b(p)\rangle$, $p\in{\mathbb P}^1$.

We shall choose coordinates $(x_0:\cdots:x_a:y_0:\cdots:y_b)$ on ${\mathbb P}^{a+b+1}$ such that $V(y_0,\ldots,y_b)=\langle C_a\rangle$ and such that
$V(x_0,\ldots, x_a)=\langle C_b\rangle$. Accordingly, $C_a\subset{\mathbb P}^{a+b+1}$ has parametrization $(s^a:s^{a-1}t:\cdots:st^{a-1}:t^a:0:0:\cdots:0)$
and $C_b\subset{\mathbb P}^{a+b+1}$ has parametrization $(0:0:\cdots:0:s^b:s^{b-1}t:\cdots:st^{b-1}:t^b)$.

The following result is well known, see for example \cite[Example 3.6]{GKZ}, and it shows the existence of a lot of significative examples of hypersurfaces with vanishing hessian, not cones.
Special projections of such examples in $\P^4$ produce examples of Gordan-Noether-Perazzo-Permutti-CRS hypersurfaces  (see \cite{CRS}).

\begin{theorem}\label{dualSab} Let notation be as above and let $(w_0:\cdots:w_a:z_0:\cdots:z_b)$ be dual coordinates
to $(x_0:\cdots:x_a:y_0:\cdots:y_b).$ Then
$$S(a,b)^*=V(\operatorname{Res}(f,g))\subset ({\mathbb P}^{a+b+1})^*$$ is a hypersurface of degree $a+b$,
where $f=w_0s^a+w_1s^{a-1}t+\cdots +w_at^a\in\mathbb K[s,t]_a$
is a general binary form of degree $a$ and where $g=z_0s^b+z_1s^{b-1}t+\cdots+z_bt^b
\in\mathbb K[s,t]_b$
is a general binary form of degree $b$.

In particular,
$$S(1,b)^*=V((-w_1)^bz_0+(-w_1)^{b-1}w_0z_1+\cdots-w_1w_0^{b-1}+w_0^bz_b)\subset({\mathbb P}^{b+2})^*$$
is a hypersurface of degree $b+1$, which is not a cone and which for $b\geq 2$ has vanishing hessian.
\end{theorem}
\begin{proof}
Let notation be as above. To calculate the parametric equations of
the tangent plane to $S(a,b)$ at a general point $q=\lambda\nu_a(s,t)+\mu\nu_b(s,t)$
with $(s:t)\in{\mathbb P}^1$ we shall suppose
$(s:t)=(1:t)$ and $(\lambda:\mu)=(1:\mu)$.
In particular we can suppose  $x_0=1$ and $y_0=\mu$.
Thus
the projective tangent space $T_qS(a,b)$ is spanned by the rows of the following matrix:
$$
\left( \begin{array}{cccccccc}
1&t&\ldots&t^a&\mu&\mu t&\ldots&\mu t^b\\
0&1&\ldots&at^{a-1}&0&\mu&\ldots&\mu b t^{b-1}\\
0&0&\ldots&0&1&t&\ldots&t^b
\end{array}
\right),
$$
and hence it is
also spanned by the rows of the matrix
\begin{equation}\label{parTq}
\left( \begin{array}{cccccccc}
1&t&\ldots&t^a&0&0&\ldots&0\\
0&1&\ldots&at^{a-1}&0&\mu&\ldots&\mu b t^{b-1}\\
0&0&\ldots&0&1&t&\ldots&t^b
\end{array}
\right).
\end{equation}
If $(w_0:\cdots:w_a:z_0:\cdots:z_b)$ are dual coordinates on $({\mathbb P}^{a+b+1})^*$, we get that
a point of $({\mathbb P}^{a+b+1})^*$ belongs to $S(a,b)^*$ if and only if
$$
\left\lbrace \begin{array}{lcl}
w_0+w_1t+\cdots + w_at^a&=&0\\
w_1+2w_2 t+\cdots+aw_{a}t^{a-1}+\cdots+\mu z_1+2\mu tz_2+\cdots+b\mu t^{b-1}z_b&=&0\\
z_0+tz_1+\cdots+t^bz_b&=&0
\end{array}\right.
$$
has a solution $(t,\mu)$. Since the second equation is linear in $\mu$, this happens if
and only if
$$
\left\lbrace \begin{array}{lcl}
w_0s^a+w_1s^{a-1}t+\cdots + w_at^a&=&0\\
z_0s^b+z_1s^{b-1}t+\cdots+z_bt^b&=&0
\end{array}\right.
$$
has a solution $(s,t)\neq (0,0)$. In conclusion, the equation of $S(a,b)^*$ is the resultant of two general homogeneous forms
of degree $a$ and degree $b$ in the variables $(s,t)$. Therefore, $S(a,b)^*$ is a hypersurface of degree $d=a+b$ by \eqref{degRes},
whose equation can be explicitly written (for example by using Sylvester Formula).

For $a=1$, the first equation is $sw_0+tw_1=0$, whose roots are $(-w_1,w_0)$. Thus the equation of $S(1,b)^*$ is obtained
by imposing that $(-w_1,w_0)$ is a solution of the second equation, that is
$$S(1,b)^*=V((-w_1)^bz_0+(-w_1)^{b-1}w_0z_1+\cdots+w_0^bz_b)\subset({\mathbb P}^{b+2})^*.$$

If $b>1$, then  the partial derivatives of the equation of $S(1,b)^*$ with respect to $z_i$ are algebraically dependent so that
the hypersurface $S(1,b)^*\subset({\mathbb P}^{b+2})^*$ has vanishing hessian and  is not a cone.
\end{proof}
\medskip

The previous analysis admits obvious generalizations we shall only mention without proofs. The surface $S(a,b)\subset{\mathbb P}^{a+b+1}$ can be seen as
the embedding of ${\mathbb P}(\O_{{\mathbb P}^1}(a)\oplus\O_{{\mathbb P}^1}(b))$ into ${\mathbb P}^{a+b+1}={\mathbb P}( \operatorname{H}^0(\O_{{\mathbb P}^1}(a))\oplus\operatorname{H}^0(\O_{{\mathbb P}^1}(b)))$
by the tautological line bundle $\O(1)$. Thus, letting $r\geq 1$ and letting
$${\mathbb P}^{N(a_0,\ldots,a_r)}={\mathbb P}(\operatorname{H}^0(\O_{{\mathbb P}^r}(a_0))\oplus\ldots \operatorname{H}^0(\O_{{\mathbb P}^r}(a_r))),$$
we shall suppose $1\leq a_0\leq\ldots\leq a_r$ and consider
$$X(a_0,\ldots, a_r)={\mathbb P}(\O_{{\mathbb P}^r}(a_0)\oplus\ldots \oplus\O_{{\mathbb P}^r}(a_r))\subset{\mathbb P}^{N(a_0,\ldots,a_r)}$$
embedded by the tautological line bundle $\O(1)$. This smooth manifold
is a ${\mathbb P}^r$-bundle over
${\mathbb P}^r$, which is projectively generated by the $r+1$ varieties  $\nu_{a_i}({\mathbb P}^r)$ lying in disjoint linear subspaces of
${\mathbb P}^{N(a_0,\ldots,a_r)}.$

The same calculations used in the proof of Theorem \ref{dualSab} above
prove that
$$X(a_0,\ldots, a_r)^*=V(\operatorname{Res}(f_0,\ldots, f_r)),$$
where $f_i$ is a generic polynomial of degree $a_i$ for $i=0,\ldots, r$. Moreover,
$$\deg(X(a_0,\ldots, a_r)^*)=a_0\cdots a_r\sum_{i=0}^r\frac{1}{a_i}$$
 by \eqref{degRes}.

In particular, if $a_0=\ldots=a_{r-1}=1$ and if $a_r=a\geq 2$, then $$X(1,\ldots, 1, a)^*\subset({\mathbb P}^{N(1,\ldots,1,a)})^*$$ is a hypersurface
of degree $r\cdot a+1$ with vanishing hessian which is  not
a cone.
\medskip

\begin{rem}{\rm
These are the first instances  of a general method, which  has been dubbed
{\it the Cayley Trick for mixed resultants} in \cite[Ch. 3, sections 2, 3, 4]{GKZ}, for calculating the explicit equations of dual varieties  of ${\mathbb P}^r$-bundles of the form ${\mathbb P}(\mathcal E)$  embedded by
the tautological line bundle $\O(1)$ with  $\mathcal E$ a very ample rank $r+1$ locally free sheaf over an irreducible projective variety
$X$ of dimension $r$.}
\end{rem}

We now introduce  and apply the
classical {\it Cayley Trick} and its dual variant, the so called {\it Dual Cayley Trick} to calculate explicitly the equations of some dual varieties.

\subsection{Cayley Trick}\label{notCayley}  Let $X\subset{\mathbb P}^N={\mathbb P}(V)$ be an irreducible non-degenerate variety of dimension $n\geq 1$.
Let $\mathbb G(r,{\mathbb P}(V))$ denote the Grassmann variety of $r$-dimensional projective subspaces of ${\mathbb P}(V)$. If $L={\mathbb P}(U)\subset{\mathbb P}(V)$
has dimension $r\geq 0$, then $L^\perp={\mathbb P}(\operatorname{Ann}(U))\subset{\mathbb P}(V^*)$ has dimension $N-r-1$ and we have a natural isomorphism
$\mathbb G(r,{\mathbb P}(V))\simeq\mathbb G (N-r-1,{\mathbb P}(V^*)),$ defined by sending $[L]$ to $[L^\perp]$. We have two natural rational maps:
$$q:{\mathbb P}(\mathbb K^{r+1}\otimes V)\dasharrow \mathbb G(r,{\mathbb P}(V))$$
and
$$p:{\mathbb P}(\mathbb K^{N-r}\otimes V^*)\dasharrow \mathbb G(r,{\mathbb P}(V)),$$
corresponding, respectively,  to the parametric equations and  to the
cartesian equations of a subspace $L={\mathbb P}(U)\subset {\mathbb P}(V)$. The rational maps
are defined on the open sets of elements of maximal rank and on these
open sets they are the quotient maps of the natural action via left multiplication of the group
of invertible matrices.
\medskip

Let $X\subset{\mathbb P}^{N}={\mathbb P}(V)$ be as above and let $e=\deg(X)\geq 2$.
Following \cite{GKZ}, let
$$Z(X)=\{[L]\in\mathbb G(N-n-1,{\mathbb P}(V)):\; L\cap X\neq\emptyset\}\subset\mathbb G(N-n-1,{\mathbb P}(V))$$
be the {\it associated hypersurface of $X$}. Indeed,  $$\operatorname{codim}(Z(X), \mathbb G(N-n-1,{\mathbb P}(V))=1,$$
see \cite[Proposition 2.2, Chap. 3]{GKZ}, and $Z(X)$ is given by a homogeneous element of degree $e=\deg(X)$ in the homogeneous coordinate ring of
$$\mathbb G(N-n-1,{\mathbb P}(V))\subset
{\mathbb P}(\Lambda^{N-n}V),$$ defined modulo Pl\" ucker relations and dubbed the {\it Chow form of $X$}.

\begin{example}{\rm
Let $f_0,\ldots, f_N\in\mathbb K[x_0,\ldots, x_N]_d$ be homogeneous forms of degree $d\geq 1$.
Then $\operatorname{Res} (f_0,\ldots, f_N)$ is  a homogeneous polynomial of degree $(N+1)d^N$ in
the $(N+1)\times {{N+d}\choose {d}}$ variables, which are the coefficients of the universal $f_i$'s
or equivalently the homogeneous coordinates  on ${\mathbb P}(\mathbb K[x_0,\ldots, x_N]_d).$

Under this assumption, if $[a_{i,j}]\in GL_{N+1}(\mathbb K)$ and if $$h_i=\sum_{j=0}^N a_{i,j}g_j,$$ then one proves that
\begin{equation}\label{invRes}
\operatorname{Res}(h_0,\ldots,h_N)=\det([a_{i,j}])^{d^N}\operatorname{Res}(g_0,\ldots, g_N).
\end{equation}

In particular, $\operatorname{Res}(f_0,\ldots, f_N)$ is an invariant for  the action by left multiplication of $SL_{N+1}(\mathbb K)$ on the set
of $(N+1)\times {{N+d}\choose {d}}$ matrices with entries in the coefficients of $f_0,\ldots, f_N$.

By the  {\it First
Theorem of Invariant Theory}  the polynomial $\operatorname{Res}(f_0,\ldots,f_N)$ can be written as a polynomial
of degree $d^N$ in the $(N+1)\times (N+1)$ minors of the $(N+1)\times {{N+d} \choose {d}}$ matrix associated to $\{f_0,\ldots, f_N\}$,
that is in the Pl\" ucker coordinates of the matrix. This polynomial is the (dual) Chow form of $\nu_d({\mathbb P}(V))\subset{\mathbb P}(S^d(V))={\mathbb P}^{N(d)}$,
defined modulo Pl\" ucker coordinates, and its  restriction  to $\mathbb G(N(d)-N-1,{\mathbb P}(S^d(V)))$ defines
$Z(\nu_d({\mathbb P}(V)))$ by the geometrical interpretation of the resultant.

Letting $$p:{\mathbb P}(\mathbb K^{N+1}\otimes (S^dV)^*)\dasharrow \mathbb G(N(d)-N-1,{\mathbb P}(S^d(V)))$$ be the natural map
defined above, we deduce
$$V(\operatorname{Res} (f_0,\ldots, f_N))=\overline{p^{-1}(Z(\nu_d({\mathbb P}(V)))}.$$

Let $y_0,\ldots, y_N$ be other variables and let $$y_0f_0+\cdots+y_Nf_N\in\mathbb K[x_0,\ldots, x_N, y_0,\ldots, y_N]_{d+1},$$
which is also a bihomogeneous polynomial of bidegree $(d,1)$. Then the  classical {\it Cayley Trick} is the formula:
\begin{equation}\label{Cayleytrick}
\mathrm{Res}(f_0,\ldots, f_N)=\Delta(y_0f_0+\cdots+y_Nf_N),
\end{equation}
a useful  remark which dates back to Cayley.

The geometrical translation of the Cayley Trick is the following: if
$${\mathbb P}(\mathbb K^{N+1})\times \nu_d({\mathbb P}(V))\subset {\mathbb P}^{(N+1)(N(d)+1)-1}={\mathbb P}(\mathbb K^{N+1}\otimes S^d(V))$$
is the Segre embedding of ${\mathbb P}^N\times\nu_d({\mathbb P}^N)$,
then
$$({\mathbb P}^N\times \nu_d({\mathbb P}(V)))^*=\overline{p^{-1}(Z(\nu_d({\mathbb P}(V))))}.$$
Indeed, formula \eqref{Cayleytrick} says that if a hyperplane $H\subset{\mathbb P}(\mathbb K^{N+1}\otimes S^d(V))$ contains
${\mathbb P}^N\times p$, $p\in\nu_d({\mathbb P}^N)$ (the condition on the left), then there exists $q\in {\mathbb P}^N$ such that $q\times T_p\nu_d({\mathbb P}^N)\subset H$
so that $H$ is tangent to ${\mathbb P}^N\times \nu_d({\mathbb P}^N)$ at $(q,p)$ (the condition on the right), yielding  $[H]\in({\mathbb P}^n\times\nu_d({\mathbb P}^N))^*$.}
\end{example}
\medskip

Nowadays the geometrical version of the Cayley Trick has been generalized by Gelfand, Weyman and Zelevinsky  to arbitrary irreducible varieties.
\medskip

\begin{theorem}\label{Cayleythm} {\rm (Cayley Trick, \cite[Theorem 2.7, Chap. 3]{GKZ})} Let $X\subset{\mathbb P}^N={\mathbb P}(V)$ be an irreducible non-degenerate variety of dimension $n\geq 1$ and let
${\mathbb P}^n\times X\subset{\mathbb P}(\mathbb K^{n+1}\otimes V)$ be the Segre embedding. Then
\begin{equation}\label{CayleyX}
({\mathbb P}^n\times X)^*=\overline{p^{-1}(Z(X))},
\end{equation}
where $p:{\mathbb P}(\mathbb K^{n+1}\otimes V^*)\dasharrow \mathbb G(N-n-1,{\mathbb P}(V))$ is the quotient map corresponding to cartesian
equations of linear subspaces of dimension $N-n-1$ of ${\mathbb P}(V)$.
\end{theorem}
\medskip

\subsection{Dual Cayley Trick}
We now present the  so called {\it Dual Cayley trick},  introduced  by Weyman and Zelevinsky in \cite{WZ}, see also \cite{Kohn}.

Let $\widetilde Y\subset{\mathbb P}^N={\mathbb P}(V)$ be an irreducible variety of dimension $n\geq 1$ such that $\widetilde Y^*\subset{\mathbb P}(V^*)$ has dimension $N-1-r$. Let ${\mathbb P}^r={\mathbb P}(T)$
and let
$${\mathbb P}^r\times \widetilde Y\subset{\mathbb P}^{(r+1)(N+1)-1}={\mathbb P}(T\otimes V)$$
be the Segre embedding of ${\mathbb P}(T)\times \widetilde Y$. Then by \cite[Corollary 3.3]{WZ} the dual variety
$({\mathbb P}^r\times \widetilde Y)^*\subset({\mathbb P}^{(r+1)(N+1)-1})^*$ is a hypersurface
$$V(f)\subset ({\mathbb P}^{(r+1)(N+1)-1})^*={\mathbb P}((T\otimes V)^*)={\mathbb P}(T^*\otimes V^*),$$
which can be computed in this way.

Let $$q:({\mathbb P}^{(r+1)(N+1)-1})^*={\mathbb P}(T^*\otimes V^*)\dasharrow \mathbb G(r,{\mathbb P}(V^*))$$
be the natural rational map defined above and corresponding to parametric equations of linear subspaces of dimension $r$ of ${\mathbb P}(V^*)$. This map, in the natural coordinates, sends
 a rank $r+1$ matrix $\mathbf X\in\mathbb K^{r+1,N+1}$ to its Pl\" ucker coordinates.  It is thus given by forms of degree $r+1$ in the natural coordinates.

Let $$Z(\widetilde Y^*)\subset  \mathbb G(r,{\mathbb P}(V^*))$$
be the  associated hypersurface of $\widetilde Y^*$.
Then \cite[Proposition 4.2.b]{WZ} yields the following  formula:
\begin{equation}\label{WZformula}
({\mathbb P}(T)\times \widetilde Y)^*=\overline{q^{-1}(Z(\widetilde Y^*))}.
\end{equation}

\subsection{Hypersurfaces with vanishing hessian constructed from cones with vertex a ${\mathbb P}^{r-1}$}
We now apply the Dual Cayley Trick to generalize part (ii) of  Theorem \ref{codimension}, giving also a different and more theoretical
proof of this result.

Let $r\geq 1$ be an integer and let $\widetilde Y\subset{\mathbb P}^N={\mathbb P}(V)$ be a cone with vertex a ${\mathbb P}^{r-1}={\mathbb P}(U)$ over a non-degenerate
variety
$Y\subset{\mathbb P}^{N-r}={\mathbb P}(W)$ without dual defect, that is $Y^*=V(g)\subset({\mathbb P}^{N-r})^*={\mathbb P}(W^*)$ is a hypersurface
with $g=g(z_0,\ldots, z_{N-r})\in \mathbb K[z_0,\ldots, z_{N-r}]_d$ for some $d\geq 2$.
By definition $V=U\oplus W$.

Let ${\mathbb P}^r={\mathbb P}(T)$ and let
$${\mathbb P}(T)\times \widetilde Y\subset{\mathbb P}(T)\times{\mathbb P}(V)\subset{\mathbb P}(T\otimes V)={\mathbb P}^{(r+1)(N+1)-1},$$
be the Segre embedding of ${\mathbb P}^r\times \widetilde Y$.

We can identify points of $({\mathbb P}^{(r+1)(N+1)-1})^*={\mathbb P}((T\otimes V)^*)$ with matrices
\begin{equation}\label{minors}
\mathbf X=\left[
\begin{tabular}{c|c}
A'&B'
\end{tabular}
\right],
\end{equation}
where $A'\in\mathbb K^{r+1,N-r+1}$ and $B'\in \mathbb K^{r+1,r}$. This decomposition corresponds
to the natural decomposition
$$\operatorname{Hom}(T,V^*) \simeq \operatorname{Hom}(T,W^*)\oplus\operatorname{Hom}(T,U^*)\simeq (T^*\otimes W^*)\oplus (T^*\otimes U^*).$$

The linear system of equations $B'=0_{(r+1)\times r}$ defines
the linear span ${\mathbb P}(\operatorname{Hom}(T, W^*))={\mathbb P}(T^*\otimes W^*)$ of  the Segre variety $$R={\mathbb P}(T^*)\times{\mathbb P}(W^*)\subset({\mathbb P}^{(r+1)(N+1)-1})^*.$$

There are exactly $N-r+1$ minors $B'_j$ of order $r+1$, obtained from $\mathbf X$ as in \eqref{minors} by adding the $j^{\th}$
column of $A'$ to $B'$, $j=0,\ldots, N-r$. Define
$$\phi_1:({\mathbb P}^{(r+1)(N+1)-1})^*\dasharrow {\mathbb P}^{(r+1)r-1}={\mathbb P}(\operatorname{Hom}(T,U^*)),$$
by
$$\phi_1(\mathbf X)=B' ,$$
and
$$\phi_2:({\mathbb P}^{(r+1)(N+1)-1})^*\dasharrow ({\mathbb P}^{N-r})^*={\mathbb P}(W^*),$$
by
$$\phi_2(\mathbf X)=(B'_0:\cdots:B'_{N-r})\in ({\mathbb P}^{N-r})^*.$$

The point $\phi_2(\mathbf X)$ is the intersection of ${\mathbb P}(W^*)$ with the $r$-dimensional linear subspace of ${\mathbb P}(V^*)$
corresponding to $\mathbf X$.
\medskip

\begin{theorem}\label{codimensionk} Let the hypothesis and the notation be as above,  let $$Z_{Y^*}=\overline{\nabla_g({\mathbb P}(W^*))}\subseteq{\mathbb P}(W)={\mathbb P}^{N-r},$$  let
$${\mathbb P}^r\times Z_{Y^*} \subset{\mathbb P}(T\otimes W)$$ be the Segre embedding, let $X=V(g(B'_0,\ldots,B'_{N-r}))\subset{\mathbb P}(T^*\otimes V^*)$ and let
$f=g(B'_0,\ldots,B'_{N-r}).$ Then:
\medskip
\begin{enumerate}
\item[(i)] $({\mathbb P}^r\times\widetilde Y)^*=V(g(B'_0,\ldots,B'_{N-r}))\subset{\mathbb P}(T^*\otimes V^*)$;
\medskip
\item[(ii)] $Z_X=\overline{\nabla_f({\mathbb P}(T^*\otimes V^*))}\subseteq S({\mathbb P}(T\otimes U),{\mathbb P}(T)\times Z_{Y^*})\subset{\mathbb P}(T\otimes V)$ and
$X$ has vanishing hessian.
\medskip
\end{enumerate}
\end{theorem}
\begin{proof} One can argue as in the proof of Theorem \ref{codimension} but we prefer
to deduce this more general result from \eqref{WZformula},  specializing the Dual Cayley Trick to our situation.

Recall that in this case  $\widetilde Y^*=Y^*\subset ({\mathbb P}^{N-r})^*={\mathbb P}(W^*)\subset{\mathbb P}(V^*)$. Moreover,   we can define a rational map
$$\psi:\mathbb G(r,{\mathbb P}(V^*))\dasharrow({\mathbb P}^{N-r})^*={\mathbb P}(W^*),$$
by
$$\psi([L])=[L\cap {\mathbb P}(W^*)].$$
The map $\psi$ is not defined along $\mathbb G(r, {\mathbb P}(W^*))\subset\mathbb G(r,{\mathbb P}(V^*))$ and   along
the Schubert cycles given by the  $[L]$'s  such that $\dim(L\cap {\mathbb P}(W^*))>0$.

Since, by hypothesis, $\widetilde Y^*=Y^*=V(g)$ is a hypersurface in ${\mathbb P}(W^*)$, it follows that
$$\overline{\psi^{-1}(Y^*)}=Z(\widetilde Y^*).$$

Using the coordinates introduced above, we deduce that
\begin{equation}\label{key}
\phi_2=\psi\circ q,
\end{equation}
where $q:{\mathbb P}(T^*\otimes V^*)\dasharrow \mathbb G(r,{\mathbb P}(V^*))$ is the natural rational map considered above.

Then \eqref{WZformula} gives
$$({\mathbb P}(T)\times \widetilde Y)^*=\overline{q^{-1}(Z(\widetilde Y^*))}.$$
Using \eqref{key} we get

$$\overline{q^{-1}(Z(\widetilde Y^*))}=\overline{q^{-1}(\psi^{-1}(\widetilde Y^*))}=\overline{\phi_2^{-1}(Y^*)}=V(g(B'_0,\ldots,B'_{N-r}))=X.$$

Let notation be as in  \eqref{minors}, let $a'_{i,j}$, with $i=0,\ldots, r$ and with $j=0,\ldots, N-r$, be the homogeneous coordinates corresponding to $A'$
and let $b'_{i,k}$ with $k=1,\ldots, r$ be the coordinates corresponding to $B'$. By definition of $B'_j$, $j=0,\ldots, N-r$, we deduce from  Laplace Formula applied to the first column of $B'_j$:
$$B'_j=\sum_{i=0}^r (-1)^i a_{i,j}C_i$$
yielding
$$\frac{\partial B'_j}{\partial a_{i,j}}=(-1)^i C_i.$$
Moreover, for $m\neq j$, we have
$$\frac{\partial B'_j}{\partial a_{i,m}}=0.$$

From
\begin{equation}\label{dfij}
\frac{\partial f}{\partial a_{i,j}}=(-1)^iC_i\frac{\partial g}{\partial z_j}(B'_0,\ldots, B'_{N-r})
\end{equation}
we deduce that, for every $i\neq k$ and for every $l\neq m$,
\begin{equation}\label{ik-ki}
\frac{\partial f}{\partial a_{i,l}}\frac{\partial f}{\partial a_{k,m}}-\frac{\partial f}{\partial a_{i,m}}\frac{\partial f}{\partial a_{k,l}}=0.
\end{equation}

Thus $X\subset{\mathbb P}(T^*\otimes V^*)$ has vanishing hessian since the partial derivates of $f$ are algebraically dependent and more precisely
\begin{equation}\label{finclusion}
Z_X=\overline{\nabla_f({\mathbb P}(T^*\otimes V^*))}\subseteq S({\mathbb P}(T\otimes U),{\mathbb P}(T)\times Z_{Y^*})\subset{\mathbb P}(T\otimes V).
\end{equation}
\end{proof}

Let us remark that for $r=1$ the base locus of $\psi$ is exactly $\mathbb G(1,({\mathbb P}(W^*))$ since a line cuts ${\mathbb P}(W^*)$ in one point if and only
if it is not contained in ${\mathbb P}(W^*)$. Thus in this case the expression of $\phi_2$ is, modulo the obvious identifications, that given in \eqref{phi2}.
\medskip

\begin{rem}\label{rhohess}{\rm
One could ask if equality holds in \eqref{finclusion}. Letting  $\rho=\operatorname{rk}(\operatorname{H} (g))=\dim(Z_{Y^*})+1$, it would be sufficient (indeed equivalent) to prove
that $$\operatorname{rk}(\operatorname{H}(f))-1=\dim(Z_X)=\dim(S({\mathbb P}(T\otimes U),{\mathbb P}(T)\times Z_{Y^*}))=r(r+1)+r+\rho-1,$$
i.e. that $\operatorname{rk}(\operatorname{H}(f))=r(r+1)+r+\rho.$

Since this analysis is quite delicate (and also intricate) we preferred to skip the details and to concentrate  on  the interesting connections with the Dual Cayley Trick in order to produce the new  examples, which  generalize to arbitrary $r\geq 2$  the case $r=1$ considered in part (ii) of Theorem \ref{codimension}. Last but not least, we point out that  Theorem \ref{codimension} is sufficient to construct examples of hypersurfaces with vanishing hessian with $\operatorname{codim}(X^*,Z_X)$ arbitrary large. Assuming equality in \eqref{finclusion},  one would deduce $\operatorname{codim}(X^*,Z_X)=\operatorname{codim}(Y, Z_{Y^*})+r^2$ and there is no
advantage in solving the previous equation instead of the simpler $\operatorname{codim}(X^*, Z_X)=\operatorname{codim}(Y, Z_{Y^*})+1$. }
\end{rem}

\subsection{The dual variety of ${\mathbb P}^1\times Y\subset{\mathbb P}^{2n+3}$ with $Y=V(f)\subset{\mathbb P}^{n+1}$ an irreducible hypersurface}

Let $X\subset{\mathbb P}^{N}={\mathbb P}(V)$ be an irreducible projective variety of dimension $n=\dim(X)$ and degree  $e\geq 2$. Let ${\mathbb P}^{n-i}={\mathbb P}(T)$,
$i\in\{0,\ldots, n\}$
and let
$${\mathbb P}^{n-i}\times X\subset{\mathbb P}^{(n-i+1)(N+1)+1}={\mathbb P}(T\otimes V)$$
be the Segre embedding of ${\mathbb P}(T)\times X$.
Let $Z_i(X)\subset \mathbb G(N-n+i-1,{\mathbb P}(V))$
be the {\it $i^{\th}$ higher associated variety of $X$} in the sense of \cite{GKZ}, i.e. it is
 the closure of the set
$$\{[L]\in\mathbb G(N-n+i-1,{\mathbb P}(V)): \exists\; x\in X_{\operatorname{reg}}: x\in L, \dim(L\cap T_xX)\geq i\} .$$
Clearly $Z_0(X)=Z(X)$ is the Chow hypersurface of $X$
and $$Z_n(X)=X^*\subset {\mathbb P}(V^*)=\mathbb G(N-1,{\mathbb P}(V)).$$
Let us recall that $Z_i(X)$ is a hypersurface in $\mathbb G(N-n+i-1,{\mathbb P}(V))$ if and only if $i\leq n-\operatorname{codim}(X^*)+1$, see \cite{GKZ, WZ} and
\cite{Kohn}.
In particular $Z_{n-1}(X)$ is a hypersurface if and only if  $\operatorname{codim}(X^*)\in\{1,2\}$.

Let $$p:({\mathbb P}^{(n-i+1)(N+1)-1})^*={\mathbb P}(T^*\otimes V^*)\dasharrow \mathbb G(N-n+i-1,{\mathbb P}(V))$$
be the rational map defined in Subsection \ref{notCayley} by considering the cartesian
equations of a linear subspace of ${\mathbb P}(V)$.
We defined also the rational map
$$q:({\mathbb P}^{(n-i+1)(N+1)-1})^*={\mathbb P}(T^*\otimes V^*)\dasharrow \mathbb G(n-i,{\mathbb P}(V^*))$$
associated to the representation of a linear subspace of ${\mathbb P}(V)^*$ via parametric equations.
Then \cite[Proposition 4.2.a]{WZ}, see also \cite{Kohn}, yields the following  formulas:
\begin{equation}\label{formulaiX}
({\mathbb P}^{n-i}\times X)^*=\overline{p^{-1}(Z_i(X))},
\end{equation}

\begin{equation}\label{formulaiXdual}
({\mathbb P}^{n-i}\times X)^*=\overline{q^{-1}(Z_{n-\operatorname{codim}(X^*)-i+1}(X^*))}.
\end{equation}
Suppose now that $Y\subset{\mathbb P}^{n+1}={\mathbb P}(V)$ is a hypersurface such that $Y^*\subset{\mathbb P}(V^*)$
is a hypersurface. Then the previous formulas give
\begin{equation}\label{p1hyp}
({\mathbb P}^1\times Y)^*=\overline{p^{-1}(Z_{n-1}(Y))}=\overline{q^{-1}(Z_1(Y^*))}.
\end{equation}

Let $Y\subset{\mathbb P}(V)={\mathbb P}^{n+1}$ be an irreducible hypersurface of degree at least two, then $Z_1(Y)\subset \mathbb G(1,{\mathbb P}(V))$ is a hypersurface
parametrizing the tangent lines to $Y$. It is given by a polynomial
 in the Pl\" ucker coordinates of $\mathbb G(1,{\mathbb P}(V))$.
 %
 To determine the degree $e$ of this polynomial let us
 remark that a general line
$L\subset  \mathbb G(1,{\mathbb P}(V))$  consists of the lines $l\subset {\mathbb P}(V)$ passing through a general point
$p\in{\mathbb P}(V)$ and contained in a general plane $\Pi\subset{\mathbb P}(V)$ with $p\in\Pi$.
Then $e=\#(L\cap Z_1(Y))$ equals the number of tangent lines to $Y\cap \Pi$ passing through $p$,
that is
$e=\deg((Y\cap \Pi)^*).$
\medskip

\begin{example}\label{Z1Fermat}{\rm Let $Y=V(x_0^2+\cdots+x_{n+1}^2)\subset{\mathbb P}^{n+1}$ be the Fermat quadric hypersurface.
Then  $Z_1(Y)\subset\mathbb G(1,{\mathbb P}^{n+1})$ is a hypersurface of degree 2 whose equation is
quadratic and, modulo Pl\"ucker relations, is the quadratic Fermat in the Pl\"ucker coordinates, that is
$$Z_1(V(x_0^2+\cdots+x_{n+1}^2))=V(\sum_{0\leq i<j\leq n+1}p_{i,j}^2)\subset \mathbb G(1,{\mathbb P}^{n+1}).$$
Let $f(x_0,\dots,x_{n+1})=x_0^2+x_1^2+\cdots+x_{n+1}^2$, then
 $Y^*=V(f(y_0,\ldots,y_{n+1}))\subset({\mathbb P}^{n+1})^*$. Let  $\mathbf a=(a_0:\cdots :a_{n+1})$ and $\mathbf b=(b_0:\cdots:b_{n+1})$
and let $(\mathbf a:\mathbf b)$ the natural coordinates on ${\mathbb P}^{2n+3}$ in such a way that the Pl\" ucker
coordinates $q_{i,j}$ of a matrix whose first row is $\mathbf a$ and whose second row is $\mathbf b$ are
$q_{i,j}=a_ib_j-a_jb_i$ for $0\leq i<j\leq n+1$. Then
$$Z_1(V(y_0^2+\cdots+y_{n+1}^2))=V(\sum_{0\leq i<j\leq n+1}q_{i,j}^2)\subset \mathbb G(1,({\mathbb P}^{n+1})^*)$$
and
$$({\mathbb P}^1\times Y)^*=\overline{q^{-1}(Z_1(Y^*))}=V(\sum_{0\leq i<j\leq n+1}(a_ib_j-a_jb_i)^2) \subset({\mathbb P}^{2n+3})^*.$$
By Lagrange's Identity
$$\sum_{0\leq i<j\leq n+1}(a_ib_j-a_jb_i)^2=||\mathbf a ||^2\cdot ||\mathbf b ||^2-(\mathbf a\bullet \mathbf b)^2.$$
Thus  the dual of ${\mathbb P}^1\times V(x_0^2+\cdots+x_{n+1}^2)\subset {\mathbb P}^{2n+3}$ has a  {\it Cauchy-Schwartz} equation:
$$({\mathbb P}^1\times V(x_0^2+\cdots+x_{n+1}^2))^*=V(||\mathbf a ||^2\cdot ||\mathbf b ||^2-(\mathbf a\bullet \mathbf b)^2).$$
This is a {\it homaloidal polynomial}, that is the associated polar map is a Cremona transformation of ${\mathbb P}^{2n+3}$.
In particular, the hessian of this quartic polynomial is different from zero and one can verify that it has a unique irreducible factor equal
to the polynomial itself.}
\end{example}
\medskip

\section{Duals of internal projections of Scorza Varieties from a point have vanishing hessian}

The series of varieties $${\mathbb P}^n\times {\mathbb P}^n\subset {\mathbb P}^{n^2+2n} \text{ (Segre embedded, $n\geq 2$),}$$

$$\nu_2({\mathbb P}^n)\subset{\mathbb P}^{\frac{n^2+3n}{2}} \text{ (quadratic Veronese embedding, $n\geq 2$),}$$

$$\mathbb G(1,{\mathbb P}^{2m+1})\subset {\mathbb P}^{2m^2+3m}  \text{ (Pl\" ucker embedding, $m\geq 2$)}$$
together with the Severi variety $E^{16}\subset{\mathbb P}^{26}$ are the so called {\it Scorza varieties.}
These varieties and their duals have a uniform description via linear algebra and via the theory of determinantal varieties
we now briefly recall.

\subsection{Generic determinantal Scorza varieties}

 Let $${\mathbb P}^{n^2+2n}={\mathbb P}(M_{(n+1)\times (n+1)}(\mathbb K)).$$
We shall indicate the generic matrix in $M_{(n+1)\times (n+1)}(\mathbb K)$
by
$$X=[x_{i,j}],$$
$i,j=0,\ldots, n$ and, by abusing notation, we shall also consider $$(x_{0,0}:\cdots:x_{n,n})$$ as homogeneous
coordinates on ${\mathbb P}^{n^2+2n}$. Analogously,  we shall indicate by $Y=[y_{i,j}]$ the matrices in the dual
space ${\mathbb P}((M_{(n+1)\times (n+1)}(\mathbb K))^*)$ in such a way that $(y_{0,0}:\cdots: y_{n,n})$  are homogeneous
coordinates dual to the previous ones.

For every $r=1,\ldots, n+1$ we can define the variety
$$X_r=\{[X]\;:\;\operatorname{rk}(X)\leq r\}\subset {\mathbb P}(M_{(n+1)\times (n+1)}(\mathbb K));$$
the variety $Y_r\subset {\mathbb P}((M_{(n+1)\times (n+1)}(\mathbb K))^*)$ is
defined in the same way.
With this notation we have $$X_1={\mathbb P}^n\times{\mathbb P}^n\subset{\mathbb P}^{n^2+2n}$$
Segre embedded and  $$X_n=V(\det(X))\subset{\mathbb P}^{n^2+2n}$$ is a hypersurface of degree $n+1$. For simplicity,
let
$$f=\det(X)\in\mathbb K[x_{i,j}]_{n+1}$$
and consider
$$\nabla_f:{\mathbb P}(M_{(n+1)\times (n+1)}(\mathbb K))\dasharrow {\mathbb P}((M_{(n+1)\times (n+1)}(\mathbb K))^*).$$

Letting
$$X^\#\in M_{(n+1)\times (n+1)}(\mathbb K)$$ be the matrix
defined by the Laplace formula:
\begin{equation}\label{Lapform}
X\cdot X^\#=\det(X)\cdot I_{(n+1)\times (n+1)}=X^\#\cdot X,
\end{equation}
the identity
\medskip
\begin{equation}\label{adjform}
(X^\#)^\#=\det(X)^{n-1}\cdot X
\end{equation}
shows that   $\nabla_f([X])=[X^\#]^t$ is  birational outside $X_n$.
The map is not  defined along  $X_{n-1}=\operatorname{Sing}(X_n)$ and its  ramification divisor  is given by the determinant of the Jacobian matrix of $\nabla_f$, which is $\operatorname{hess}(f)$. Since $f=\det(X)$ is an irreducible polynomial,
we deduce from \eqref{adjform} that the ramification divisor of $\nabla_f$ is supported on $X_n$, yielding

$$\operatorname{hess}(f)=\alpha\cdot f^{(n+1)(n-1)}$$
with $\alpha\in\mathbb K^*$. This property was  proved in a similar (but not identical) way   by B. Segre in \cite[Teorema 1]{Segre1}.
By evaluating
the previous identity on particular matrices B. Segre also deduced $\alpha=(-1)^{\frac{n(n-1)}{2}}n$,
see \cite[Teorema 1]{Segre1}.
\medskip

If $[X]\not\in X_n,$ then $[X^\#]\not\in Y_n$ while for $[X]\in X_n\setminus X_{n-1}$, $[X^\#]\in Y_1$ from which it easily
follows that
$$X_n^*=Y_1={\mathbb P}^n\times{\mathbb P}^n\subset {\mathbb P}( (M_{(n+1)\times (n+1)}(\mathbb K))^*).$$

By the definition of the Gauss map, for every $[X]\in X_n\setminus X_{n-1}$, we have that $\nabla_f([X])=[T_{[X]}X_n]$ and,
recalling that the closure of the fibers of the Gauss map are linear spaces, that $\overline{\nabla_f^{-1}([T_{[X]}X_n])}$
is linear space of dimension $n^2-1$.

We are now ready
to prove the next result.

\begin{prop}\label{conoPnPn} Let notation be as above, let $n\geq 2$ and let $[X]\in X_n\setminus X_{n-1}$. Then
$$\overline{\nabla_f(T_{[X]}X_n)}$$ is a hypersurface of degree $n$ which is a cone with  vertex
$$\overline{\nabla_f^{-1}([T_{[X]}X_n])}^\perp={\mathbb P}^{2n}$$ over the dual of a Segre variety  ${\mathbb P}^{n-1}\times{\mathbb P}^{n-1}$.
\end{prop}
\begin{proof} Since $X_n\setminus X_{n-1}$ is  homogeneous, it is sufficient to verify the assertion
for $X$ with $x_{i,j}=\delta_{i,j}$ for $i,j=0,\ldots, n-1$ and $x_{n,j}=x_{i,n}=0$ for every $i,j$. Then
$$X^\#=(0:0:\cdots:0:1) ,$$ so $T_{[X]}X_n$ has equation $x_{n,n}=0$.
Letting $h=\det(Y)$, \eqref{adjform} implies  $\nabla_f^{-1}=\nabla_h$ as rational maps. Then
$$\overline{\nabla_f(V(x_{n,n}))}=V(\frac{\partial h}{\partial y_{n,n}})$$
is the determinant of the $n\times n$ matrix with entries $y_{i,j}$, $i,j=0,\ldots, n-1$ which does not
depend on the $2n+1$ variables $y_{n,i}$ and $y_{j,n}$. Hence it is a cone with vertex $\overline{\nabla_f^{-1}([T_{[X]}X_n])}^\perp$ over the dual of the
Segre variety ${\mathbb P}^{n-1}\times{\mathbb P}^{n-1}\subset{\mathbb P}^{n^2-1}$ corresponding to the $n^2$ variables $y_{i,j}$,
$i,j=0,\ldots, n-1$.
\end{proof}
\medskip

\begin{cor}\label{cor4.2} Let notation be as above, let $n\geq 2$, let $$p\in Y_1={\mathbb P}^n\times {\mathbb P}^n\subset{\mathbb P}^{n^2+2n}={\mathbb P}((M_{(n+1)\times (n+1)}(\mathbb K))^*)$$ and let $T_n\subset{\mathbb P}^{n^2+2n-1}$ be the projection
of $Y_1={\mathbb P}^n\times{\mathbb P}^n$ from $p$. Let $q\in X_n\setminus X_{n-1}$ be such that  $T_qX_n=p^\perp$ and let
$$R_n=X_n\cap p^\perp\subset p^\perp={\mathbb P}^{n^2+2n-1} .$$ Then:
\medskip
\begin{enumerate}
\item[(i)] $R_n\subset{\mathbb P}^{n^2+2n-1}$ is an irreducible  hypersurface of degree $n+1$ with vanishing hessian and such that $R_n^*=T_n$.
\medskip
\item[(ii)] $Z_{R_n}\subset{\mathbb P}^{n^2+2n-1}$ is a hypersurface of degree $n$ which is a cone with vertex a ${\mathbb P}^{2n-1}$ over  the dual of a Segre variety ${\mathbb P}^{n-1}\times{\mathbb P}^{n-1}$.
\medskip
\item[(iii)] $\operatorname{codim}(R_n^*, Z_{R_n})=n^2-2.$
\end{enumerate}
\end{cor}
\begin{proof} The hypersurface $R_n\subset{\mathbb P}^{n^2+2n-1}$ is connected,  it is also normal by Serre's Criterion being non-singular in codimension 1
($\operatorname{Sing}(R)$ is the closure of the  contact locus ${\mathbb P}^{n^2-1}_q$ of $T_q X_n$
 defined above) and hence it is irreducible. The variety $T_n$ has no dual defect so that $T_n^*$ is an irreducible
hypersurface contained in $R_n$ (see for example \cite[Exercise 1.5.22]{Russo}), yielding $R_n=T_n^*$.   Since $\overline{\nabla_f(T_qX_n)}$ is a cone
such that
$p\in \vert( \overline{\nabla_f(T_qX_n)})$, we deduce that $Z_{R_n}$ is the projection of $\overline{\nabla_f(T_qX_n)}$ from $p$. Thus (ii) follows
from Proposition \ref{conoPnPn} and Lemma \ref{projectionHess}.
\end{proof}
\medskip

\begin{rem}\label{CuRS}{\rm The previous result has been discovered  for $n=2$ in \cite{GoRu}, see also \cite[Example 7.6.11]{Russo}.
Part (ii) has been proved algebraically also in \cite[Proposition 4.9]{MoSi}. By passing to a suitable linear section of $X_n$ obtained
by putting some more variable equal to zero in the matrix $X$ such that $f=\det(X)$, Cunha, Ramos and Simis produced explicit irreducible polynomials with vanishing hessian
and such that  $\operatorname{codim}(X^*,Z_X)$ is a function of $n$. These examples can be also described geometrically as the duals  of some
explicit projections of $Y_1$.
Clearly the examples in \cite{CuRaSi} are  of Gordan-Noether-Perazzo-Permutti-CRS type since one can  {\it separate the variables}
via Laplace formula for the expansion of the determinant.}
\end{rem}

\subsection{Symmetric determinantal Scorza varieties}
Let
$$W_n=\{S\in M_{(n+1)\times (n+1)}(\mathbb K)\;:\; S=S^t\}\subset M_{(n+1)\times (n+1)}(\mathbb K).$$
 \medskip

Although for $n\geq 1$ the subspace $W_n$ is not a subalgebra of  $M_{(n+1)\times (n+1)}(\mathbb K)$,  we have $S^\#\in W_n$ for every $S\in W_n$.
Let $S=[s_{i,j}]$ be the generic matrix in $W_n$ and let $(s_{0,0}:\cdots:s_{n,n})$ be the corresponding homogeneous coordinates on $${\mathbb P}(W_n)=
{\mathbb P}^{\frac{n^2+3n}{2}}.$$
Let
$$g=\det(S)\in\mathbb K[s_{i,j}]_{n+1}.$$
The operation $\#$  on $M_{(n+1)\times (n+1)}(\mathbb K)$ induces by restriction to $W_n$ a birational involution
$$\nabla_g:{\mathbb P}(W_n)\dasharrow{\mathbb P}(W_n^*),$$
defined by $\nabla_g([S])=[S^\#]^t.$

For every $r=1,\ldots, n+1$ we can define the variety
$$S_r=\{[S]\in{\mathbb P}(W_n)\;:\;\operatorname{rk}(S)\leq r\}\subset {\mathbb P}(W_n);$$
the variety $U_r\subset {\mathbb P}(W_n^*)$ is
defined in the same way.
With this notation we have $$S_1=\nu_2({\mathbb P}^n)\subset{\mathbb P}^{\frac{n^2+3n}{2}}={\mathbb P}(W_n)$$
Veronese  embedded and  that $$S_n=V(\det(S))\subset{\mathbb P}^{\frac{n^2+3n}{2}}$$ is a hypersurface of degree $n+1$.

The rational map $\nabla_g$ is not  defined along  $S_{n-1}=\operatorname{Sing}(S_n)$ and its  ramification divisor  is given by the determinant of the Jacobian matrix of $\nabla_g$, which is $\operatorname{hess}(g)$.  Since $g=\det(S)$ is an irreducible polynomial,
we deduce from \eqref{adjform} that the ramification divisor of $\nabla_g$ is supported on $S_n$, yielding
$$\operatorname{hess}(g)=\beta\cdot g^{\frac{(n+2)(n-1)}{2}}$$
with $\beta\in\mathbb K^*$. This property was  proved in a similar (but not identical) way   by B. Segre in \cite[Theorem 2]{Segre1}.
By evaluating
the previous identity on particular matrices B. Segre  deduced $\beta=(-1)^{\frac{n(n-1)}{2}}\cdot 2^{\frac{(n+1)n}{2}}\cdot n$,
see \cite[Teorema 2]{Segre1}.
\medskip

If $[S]\not\in S_n,$ then $[S^\#]\not\in U_n$ while for $[S]\in S_n\setminus S_{n-1}$, $[S^\#]\in U_1$. From this it
follows that
$$S_n^*=U_1=\nu_2({\mathbb P}^n)\subset {\mathbb P}^{\frac{n^2+3n}{2}}={\mathbb P}(W_n^*).$$
By the definition of the Gauss map, for every $[S]\in S_n\setminus S_{n-1}$, we have $\nabla_g([S])=[T_{[S]}S_n]$ and
$\overline{\nabla_g^{-1}([T_{[S]}S_n])}$
is an $((n^2+n-2)/{2})$-dimensional projective space.

 We are now ready
to prove the next result and its Corollary, whose proofs will be omitted being analogous to those presented in  Proposition \ref{conoPnPn}
and in Corollary \ref{cor4.2}.

\begin{prop}\label{conov2Pn} Let notation be as above, let $n\geq 2$ and let $[S]\in S_n\setminus S_{n-1}$. Then
$$\overline{\nabla_g(T_{[S]}S_n)}$$ is a hypersuface of degree $n$ which is a cone with  vertex
$$\overline{\nabla_f^{-1}([T_{[X]}X_n])}^\perp={\mathbb P}^{n}$$ over the dual of a Veronese variety  $\nu_2({\mathbb P}^{n-1})$.
\end{prop}
\medskip

\begin{cor}\label{cor4.5} Let notation be as above, let $n\geq 2$, let $$p\in S_1=\nu_2({\mathbb P}^n)\subset{\mathbb P}^{\frac{n^2+3n}{2}}={\mathbb P}(W_n^*)$$ and let $V_n\subset{\mathbb P}^{\frac{n^2+3n-2}{2}}$ be the projection
of $S_1=\nu_2({\mathbb P}^n)$ from $p$. Let $q\in S_n\setminus S_{n-1}$ be such that  $T_qS_n=p^\perp$ and let
$$Q_n=S_n\cap p^\perp\subset p^\perp={\mathbb P}^{\frac{n^2+3n-2}{2}} .$$ Then:
\medskip
\begin{enumerate}
\item[(i)] $Q_n\subset{\mathbb P}^{\frac{n^2+3n-2}{2}}$ is an irreducible  hypersurface of degree $n+1$ with vanishing hessian and such that $Q_n^*=V_n$.
\medskip
\item[(ii)] $Z_{Q_n}\subset{\mathbb P}^{\frac{n^2+3n-2}{2}}$ is a hypersurface of degree $n$ which is a cone with vertex a ${\mathbb P}^{n-1}$ over  the dual of a Veronese variety $\nu_2({\mathbb P}^{n-1})$.
\medskip
\item[(iii)] $\operatorname{codim}(Q_n^*, Z_{Q_n})=\frac{n^2+n-4}{2}.$
\end{enumerate}
\end{cor}
\medskip

\begin{rem}{\rm The varieties $V_n\subset{\mathbb P}^{\frac{n^2+3n-2}{2}}$ are smooth being isomorphic to $\operatorname{Bl}_p{\mathbb P}^n$ and their duals are hypersurfaces
with vanishing hessian. The classification of hypersurfaces with vanishing hessian whose dual is smooth
seems to be an intriguing question, also due to the lack of known examples.

The first element of the series, $V_2\subset {\mathbb P}^4$ is nothing but $S(1,2)$. The cubic hypersurface $S(1,2)^*\subset{\mathbb P}^4$, whose equation we
computed explicitly in Theorem \ref{dualSab}, is surely the easiest and simplest counterexample to Hesse's Claim. As far as we know, the fact that this example was the first member of an infinite series of hypersurfaces with vanishing hessian has been noticed
by us for the first time several years ago. The very recent paper \cite{CuRaSi2} deals with similar phenomena treated from a purely algebraic point of view.}
\end{rem}

\subsection{Skew-symmetric determinantal Scorza varieties}

Let
$$M_n=\{A\in M_{(n+1)\times (n+1)}(\mathbb K)\;:\; A=-A^t\}\subset M_{(n+1)\times (n+1)}(\mathbb K).$$
\medskip

For $n\geq 1$ the subspace $M_n$ is not a subalgebra of  $M_{(n+1)\times (n+1)}(\mathbb K)$ but  $A^\#\in M$ for every $A\in M$.
Let $A=[a_{i,j}]$ be the generic matrix in $M_n$ and let $(a_{0,1}:\cdots:a_{n-1,n})$ be the corresponding homogeneous coordinates on $${\mathbb P}(M_n)=
{\mathbb P}^{\frac{n^2+n-2}{2}}.$$
From now on suppose that $n+1=2m+2$ with $m\geq 2$ so that $\frac{n^2+n-2}{2}=2m^2+3m$ and $n=2m+1$.
Then
$$\det(A)=\operatorname{Pf}^2\in\mathbb K[a_{i,j}]_{2m+2},$$ with $\operatorname{Pf}\in \mathbb K[s_{i,j}]_{m+1}$.
The  operation $\#$  on $M_{(n+1)\times (n+1)}(\mathbb K)$ induces by restriction to $M_{2m+1}$ a birational involution
$$\nabla_{\operatorname{Pf}}:{\mathbb P}(M_{2m+1})\dasharrow{\mathbb P}(M_{2m+1}^*),$$
defined by $\nabla_{\operatorname{Pf}}([A])=[A^\#]^t.$

For every $r=1,\ldots, m+1$ we can define the variety
$$A_{2r}=\{[A]\in{\mathbb P}(M_{2m+1})\;:\;\operatorname{rk}(A)\leq 2r\}\subset {\mathbb P}(M_{2m+1});$$
the variety $C_{2r}\subset {\mathbb P}(M_{2m+1}^*)$ is
defined in the same way.
With this notation we have $$A_2=\mathbb G(1, {\mathbb P}^{2m+1})\subset{\mathbb P}^{2m^2+3m}={\mathbb P}(M_{2m+1})$$
Pl\" ucker  embedded and  $$A_{2m}=V(\operatorname{Pf})\subset{\mathbb P}^{2m^2+3m}$$ is a hypersurface of degree $m+1$.

The rational map $\nabla_{\operatorname{Pf}}$ is not  defined along  $A_{2m-2}=\operatorname{Sing}(A_{2m})$ and its  ramification divisor  is given by the determinant of the Jacobian matrix of $\nabla_{\operatorname{Pf}}$, which is $\operatorname{hess}(\operatorname{Pf})$.  Since $\operatorname{Pf}$ is an irreducible polynomial,
we deduce from \eqref{adjform} that the ramification divisor of $\nabla_{\operatorname{Pf}}$ is supported on $A_{2m}$, yielding
$$\operatorname{hess}(\operatorname{Pf})=\gamma\cdot \operatorname{Pf}^{(2m+1)(m-1)}$$
with $\gamma\in\mathbb K^*$. This property was  proved in a similar (but not identical) way   by B. Segre in \cite[Theorem 3]{Segre1}.
 By evaluating
the previous identity on particular matrices B. Segre  deduced $\gamma=(-1)^{m}\cdot m$,
see \cite[Teorema 3]{Segre1}.
\medskip

If $[A]\not\in A_{2m},$ then $[A^\#]\not\in C_{2m}$ while for $[A]\in A_{2m}\setminus A_{2m-2}$, $[A^\#]\in C_2$ so that
$$A_{2m}^*=C_2=\mathbb G(1,{\mathbb P}^{2m+1})\subset {\mathbb P}^{2m^2+3m}={\mathbb P}(M_{2m+2}^*).$$

By the definition of the Gauss map, for every $[A]\in A_{2m}\setminus A_{2m-2}$, we have $\nabla_{\operatorname{Pf}}([A])=[T_{[A]}A_{2m}]$ and
$\overline{\nabla_{\operatorname{Pf}}^{-1}([T_{[A]}A_{2m}])}$
is a $(2m^2-m-1)$-dimensional projective space.

We are now ready
to prove the next result and its Corollary, whose proofs will be omitted being analogous to those presented above.

\begin{prop}\label{conoGn} Let notation be as above, let $m\geq 2$ and let $[A]\in A_{2m}\setminus A_{2m-2}$. Then
$$\overline{\nabla_{\operatorname{Pf}}(T_{[A]}A_{2m})}$$ is a hypersuface of degree $m$ which is a cone with  vertex
$$\overline{\nabla_f^{-1}([T_{[A]}A_{2m}])}^\perp={\mathbb P}^{4m}$$ over the dual of a Grassmann variety   $\mathbb G(1,{\mathbb P}^{2m-1})$.
\end{prop}
\medskip

\begin{cor}\label{cor4.8} Let notation be as above, let $m\geq 2$, let $$p\in C_2=\mathbb G(1, {\mathbb P}^{2m+1})\subset{\mathbb P}^{2m^2+3m}={\mathbb P}(M_{2m+2}^*)$$ and let $G_m\subset{\mathbb P}^{2m^2+3m-1}$ be the projection
of $C_2$ from $p$. Let $q\in A_{2m}\setminus A_{2m-2}$ be such that  $T_qA_{2m}=p^\perp$ and let
$$F_m=A_{2m}\cap p^\perp\subset p^\perp={\mathbb P}^{2m^2+3m-1} .$$ Then:
\medskip
\begin{enumerate}
\item[(i)] $F_m\subset{\mathbb P}^{2m^2+3m-1}$ is an irreducible  hypersurface of degree $m+1$ with vanishing hessian and such that $F_m^*=G_m$.
\medskip
\item[(ii)] $Z_{F_m}\subset{\mathbb P}^{2m^2+3m-1}$ is a hypersurface of degree $m$ which is a cone with vertex a ${\mathbb P}^{4m-1}$ over the dual of  a Grassmann variety $\mathbb G(1,{\mathbb P}^{2m-1})$.
\medskip
\item[(iii)] $\operatorname{codim}(F_m^*, Z_{F_m})=2m^2-m-2.$
\end{enumerate}
\end{cor}
\medskip

\begin{rem}{\rm The three series of hypersurfaces $R_n$, $Q_n$ and $F_m$ are of Gordan-Noether-Perazzo-Permutti-CRS type and such that the duals of their polar images are of
the same type of their duals. Indeed, the first property easily  follows from Laplace formula and by the determinantal description of their equation in a suitable coordinate system
while $Z^*$ is of the same type by part (ii) of the previous Corollaries~\ref{cor4.2}, \ref{cor4.5}, and \ref{cor4.8}. }
\end{rem}

redpajama_set_name: RedPajamaArXiv


text:
House size is double that stipulated in 2010 court case: But Chester's engineering consultant says no housing restrictions apply
Part of the Greens' site plan
By Frances Ruth Harris
CHESTER — The Greens of Chester developer wants to build a sample house of 3,600 square feet, double the size of the 1,800-square-foot single-family house stipulated in a 2010 court case.
James Farr, the Town of Chester's building inspector, denied the developer's request for a permit to build the larger house on the construction site, where 455 residential units will be built in three phases.
In a letter to the town board about the permit denial, Farr referred to a passage in the State Environmental Quality Review that the units "will be a combination of one- and two-unit dwellings; 245 one-unit dwellings and 210 two-unit dwellings constructed in 105 structures. The typical lot size depicted for the one-unit dwellings is 3,600 square feet with a building floor area of 1,700 to 2,500 square feet in a two-story structure. The two-unit dwellings will each be located on lots of approximately 1,750 square feet and will have a typical building square footage of between 1,500 and 2.300 square feet in a two-story structure. There will be substantial acreage, over 80 acres, that will be held in common and be owned and maintained by a Homeowners' Association."
Town engineer denies restrictionsJehuda Landau, managing member of the Greens of Chester, wrote the letter to Farr requesting the permit. Landau referred The Chronicle to the New York City-based attorney Steven Barshov, who responded to Farr's denial.
Barshov shared an Oct. 31, 2018, letter from Philip Salerno, the town's engineering consultant, to Livy Schwartz, co-owner of Chester Developers Corp. Salerno in the letter says no housing size restrictions apply to the Greens property.
"The language in the project description was intended to provide a conceptual overview of the subdivision," Salerno states in his letter to Schwartz. "It was not intended, nor should it be interpreted, to establish development standards."
The development standards set down by the planning board "are limited to the establishment of minimum setback lines and minimum separations between buildings," Salerno wrote. He said the town's regulations on cluster developments allow the planning board to impose floor area ratios to prevent excessively large residences on smaller lots.
"Based upon my involvement in the review process and approval of this project...I can definitively state that no such restriction was enacted; nor was it intended," Salerno wrote.
Salerno said he reviewed engineering at the Greens and was "directly involved in the discussions with the planning board and the applicant regarding the project development."
Barshov told The Chronicle that a building's footprint "is determined by set back and bedroom size, not a hard square foot range. The floor area they put in is irrelevant because it doesn't apply to this project."
He referred to a statement made by land surveyor William Youngblood at the public hearing on the Greens.
"Probably 1,700 to 2,500 square feet for the single family home on a lot 3,600 square feet as a minimum," Youngblood said at the hearing. "They go up larger than that. The builder where you have two families in one building, even though they are single family, those lots are a minimum of 750 square feet. The range up higher than that. Those are the minimum."
Town board member Cindy Smith praised Farr at a town meeting last month.
"The point being that the building inspector is seriously looking at all the permit requests that come in," she said.
Supervisor Bob Valentine said the matter "will be addressed through the attorneys."
"We'll go from there," Smith said.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Retrieve the LANGID for French (Canada) on Windows Mobile 6? I'm writing a script using SOTI Package Manager and I'm trying to find the particular key I can use to retrieve what the current Windows Mobile Regional Setting is for Language.
If it's French (Canada) I want to display post-install script messages in French, otherwise I want them in English.
I've got the ability to check a registry value while running the post-install script by using the following syntax (for example):
set myVar=REG://HKEY_LOCAL_MACHINE\Windows CE Tools?VN=Platform

I've looked at the MSDN articles on using SetLocaleInfo() and MAKELANGID. I've gotta be honest in that I'm not sure how I'd go about figuring out what the French (Canada) LANGID actually is.
Anyone off the top of their head be able to tell me where I need to be checking for it or better yet, how you figured that out?

A: You can check the following value: HKEY_LOCAL_MACHINE\nls\DefaultLCID
The DWORD value will be the LCID of the currently active regional settings.
So for French (Canada) it would be 0x0c0c, and for English (Canada) it is 0x1009.

redpajama_set_name: RedPajamaStackExchange


text:
Lidměřice (deutsch Lidmeritz) ist eine Ortschaft der Minderstadt Olbramovice in Tschechien. Sie liegt südöstlich von Olbramovice und gehört zum Okres Znojmo.

Geographie
Das Längsangerdorf Lidměřice erstreckt sich linksseitig des Olbramovický potok (Wolframitzer Bach) in der Thaya-Schwarza-Senke. Lidměřice befindet sich in einer artesischen Zone eines Niederungsmoors, das sich bis Suchohrdly u Miroslavi und Damnice erstreckt.

Nachbarorte sind Kubšice im Norden, Šumice im Nordosten, Pohořelice im Osten, Branišovice und Vinohrádky im Südosten, Babice im Süden, Želovice im Westen sowie Olbramovice im Nordwesten.

Geschichte
Die erste urkundliche Erwähnung von Lidměřice erfolgte 1358, als Adam von Lidmeric seiner Frau Einkünfte auf den Hof verschrieb. Drei Jahre später verglich er sich mit seinen Brüdern Bertold, Peter und Heinrich über deren Anteil von zwei Äckern, zwei Lahn, zwei Gehöften und einer einradigen Mühle. Im Jahre 1378 veräußerte Heinrich von Haugwitz einen Hof in Lidmeritz an Thoman von Wistuben. Nachdem Andreas von Bítov das Dorf einschließlich eines Hofes von Markgraf Jobst von Mähren als Geschenk erhalten hatte, verschrieb er 1406 seiner Frau Margarethe darauf einen jährlichen Zins. Johann Lechwicky von Zástřizl überließ 1493 neun Lahn in Lidmeritz dem Stift Selau; den übrigen Teil des Dorfes ließ er 1503 dem Jaroš von Želenek intabulieren. Letzter trat den Besitz umgehend an Sigmund von Daubrawic ab, der ihn 1512 an Znata von Drahanowic und Johann von Herultic überschrieb. Johann von Drahanowic vererbte seinen Besitz in Lidmeritz 1533 an Smil Osowsky von Daubrawic. Im Jahre 1537 veräußerte Johann von Herultic auf Budkowitz seinen Anteil dem Besitzer des Gutes Bochtitz, Johann Kusy von Mukoděl. Burian Osowsky von Daubrawic verkaufte 1550 seinen Teil des Dorfes zusammen mit Rokytná an Sigmund Valecký von Mírov. Im Jahre 1567 tauschte Georg Valecký von Mírov die Herrschaft Mislitz mit allem Zubehör, darunter auch Lidmeritz und dem Markt Wolframitz, bei Wenzel Hodický von Hoditz gegen das Gut Hostim ein. Das Stift Selau verkaufte 1594 seinen Lidmeritzer Anteil an die Abtei Strahov. Während des Dreißigjährigen Krieges wurde Lidmeritz im Jahre 1619 durch die Truppen des kaiserlichen Feldmarschalls Dampierre verwüstet. Nach der Schlacht am Weißen Berg wurde der Besitz des Aufständischen Zdenek Hoditzky von Hoditz konfisziert; am 16. Dezember 1634 verkaufte König Ferdinand II.  das eingezogene Gut Wolframitz mit dem Markt Wolframitz, den Dörfern Aschmeritz, Kupschitz, Schömitz und Lidmeritz sowie der Wüstung Palikowitz für 48.000 mährische Gulden an Gundaker Fürst von Liechtenstein, der es mit seiner Herrschaft Krummau vereinigte, die fast 300 Jahre im Besitz des Hauses Liechtenstein verblieb.

Am 15. September 1687 verkaufte die Abtei Strahov ihren Lidmeritzer Anteil zusammen mit dem Dorf Jesram für 13.000 Rheinischer Gulden an das Zisterzienserinnenstift Maria Saal, das den neuerworbenen Besitz seinem Gut Frainspitz zuschlug. Nach der Aufhebung des Königinklosters Altbrünn fiel das Gut Frainspitz 1782 dem Religionsfonds zu. Am 8. Oktober 1807 ersteigerte die Vormundschaft des minderjährigen Prinzen Karl von Liechtenstein das Gut Frainspitz für 240.000 Gulden und schloss es an das kleine Majorat des Hauses Liechtenstein (Herrschaft Krummau) an. Damit waren beide Anteile von Lidmeritz vereinigt. 1793 gab es in Lidmeritz 27 Häuser, in denen 179 Menschen lebten; zum Dorf gehörten 61 ha Weingärten.

Im Jahre 1834 bestand das im Znaimer Kreis in einer sehr fruchtbaren Ebene gelegene Dorf Lidmeritz bzw. Ljdměřice aus 28 Häusern mit 181 deutschsprachigen Einwohnern. Haupterwerbsquelle bildeten die Landwirtschaft sowie der Obst- und Weinbau. Pfarr- und Schulort war Wolframitz. Bis zur Mitte des 19. Jahrhunderts blieb Lidmeritz der Fideikommiss-Primogeniturherrschaft Mährisch-Krummau untertänig.

Nach der Aufhebung der Patrimonialherrschaften bildete Lidmeritz / Lidměřice ab 1848 eine Gemeinde im Gerichtsbezirk Kromau. 1848 erfolgte der Bau eines steinernen Glockenturmes auf dem Anger. Ab 1869 gehörte Lidmeritz zum Bezirk Kromau; zu dieser Zeit hatte das Dorf 142 Einwohner und bestand aus 28 Häusern. Der Weinbau verlor zum Ende des 19. Jahrhunderts immer mehr an Bedeutung; die Weingärten hatten 1900 nur noch eine Ausdehnung von acht Hektar. Im Jahre 1900 lebten in Lidmeritz 158 Personen; 1910 waren es 150. In der ersten Hälfte des 20. Jahrhunderts wurden artesische Brunnen gebohrt. Nach dem Ersten Weltkrieg zerfiel der Vielvölkerstaat Österreich-Ungarn, das Dorf wurde 1918 Teil der neu gebildeten Tschechoslowakischen Republik.

Beim Zensus von 1921 lebten in den 33 Häusern von Lidmeritz 172 Personen, darunter 154 Deutsche und 17 Tschechen.

Die Elektrifizierung des Dorfes erfolgte 1928. Im Jahre 1930 bestand Lidmeritz aus 39 Häusern und hatte 153 Einwohner, darunter 124 Deutsche und 28 Tschechen. Nach dem Münchner Abkommen wurde die Gemeinde 1938 dem Großdeutschen Reich zugeschlagen und gehörte bis 1945 zum Kreis Znaim. 1939 lebten 150 Personen in Lidmeritz.

Nach dem Ende des Zweiten Weltkrieges kam Lidměřice 1945 zur Tschechoslowakei zurück, es erfolgte die Wiederherstellung der alten Bezirksstrukturen. 1946 wurden alle deutschsprachigen Bewohner vertrieben. Im Jahre 1949 erfolgte der Zusammenschluss der Gemeinden Babice, Lidměřice, Olbramovice und Želovice zu einer Gemeinde Olbramovice. Im Jahre 1950 hatte Lidměřice nur noch 123 Einwohner. Bei der Gebietsreform von 1960 wurde Lidměřice im Zuge der Aufhebung des Okres Moravský Krumlov dem Okres Znojmo zugeordnet; zugleich verlor das Dorf den Status eines Ortsteils. Der Katastralbezirk Lidměřice wurde 1966 dem Katastralbezirk Olbramovice u Moravského Krumlova zugeschlagen. Im Laufe des 20. Jahrhunderts wuchs Lidměřice entlang der Ortsverbindungsstraße mit Olbramovice zu einer Einheit zusammen. Von Želovice und Babice ist das Dorf nur durch den Bachlauf des Olbramovický potok getrennt.

Sehenswürdigkeiten
 Gemauerter Glockenturm, errichtet 1848
 Artesischer Brunnen, angelegt 1928

Literatur
 Historický lexikon obcí České republiky 1869–2005, Teil 1, S. 646

Weblinks
 Geschichte von Olbramovice
 Lidmeritz auf www.europas-mitte.de
 Lidmeritz auf suedmaehren.eu
 Lidmeritz auf suedmaehren.at

Einzelnachweise

Olbramovice

redpajama_set_name: RedPajamaWikipedia


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As the deadly Carr Fire continues to blaze through Northern California, many are left wondering how they can help the residents of Shasta County.
As the deadly Carr Fire continues to blaze through Northern California, destroying hundreds of buildings and forcing tens of thousands of people from their homes, many are left wondering how they can help the devastated residents of Shasta County.
With shelters filling up and volunteers working around-the-clock, those wanting to help should keep a few things in mind.
Care packages full of teddy bears, canned food and blankets often first come to mind when it comes to disaster relief. But while any donations are helpful, money is often the best way to help fire victims, said county officials.
"Money donations are the easiest to manage and provide the means to purchase the exact items in need," Emergency Operations Center spokeswoman Amy Travis said.
Needs of the affected community are often hard to predict this early in the disaster. Travis said crews are still assessing the level of damage.
She encouraged people to only donate to reputable, vetted organizations like the American Red Cross and the Salvation Army.
A local bank has also stepped up to help those directly impacted by the fire. Tri Counties Bank is accepting donations through its Carr Fire Fund 2018. Donors can stop by local bank branches or give online at GoFundMe.
The bank, which started the online fundraiser with a $25,000 deposit, has already raised more than $50,000. The funds will be distributed through the United Way of Northern California and local nonprofits to help fire victims with immediate needs.
After the fire triggered evacuations in Redding and the surrounding communities, the American Red Cross opened five emergency shelters, Red Cross spokesman Stephen Walsh said. With about total 800 people, one shelter is at capacity and others are starting to fill up.
Currently, the Red Cross is only soliciting for monetary donations. Walsh said with money, the organization is able to tailor its services depending on the disaster and needs of the victims. Funds are typically used to buy food, blankets and toiletries.
MORE: No shortage of volunteers helping animals as Carr Fire pummels Shasta Co.
Those returning to their blackened, damaged homes are also given kits with shovels, tarps and gloves to help them in recovery efforts.
The Salvation Army is taking the lead on accepting physical donations for victims. Captain Tim Danielson said brand new clothing, like t-shirts, socks and underwear, are especially needed for evacuees. The nonprofit is also accepting new pillows and blankets, baby items, water, sports drinks and cleaning supplies like Clorox wipes.
What helps disaster efforts the most, Danielson said, is listening to what is actually needed.
"That helps big time," he said. "The Northern California and Southern Oregon communities have really stepped up."
Haven Humane Society employees, who evacuated and took in hundreds of animals, are also asking for donations of kitten formula and bottles, cleaning supplies, puppy food, poop bags and dog crates. A full list can be found at the organization's Facebook page.


In the epoch of Amir Temur, the serious attention was paid to the instrumental music along with the other arts. The special "nogorakhona" and observation platforms for spectators - "tarobkhona" were installed above the gate of the Sohibkiran's palace. The musical instruments used in the military campaigns of Sohibkiran, in particular, percussive and wind instruments had the developed dynamic properties.
These are different "nogoras", which were widely used in the period of the Sassanians, copper "koranai" (karnai), "naiy rumi", "bug", "nodir" as well as the musical instruments used by the Turkic peoples for rituals and ceremonies - "Chabchig" and "shon". N. Fermer informed that "chabchig" had the special status "in military campaigns" of Sohibkiran (it directed the movement and was an original herald - Author) (1.4; 142). The military musicians were specially trained. The system "teacher - pupil" was widely applied. This tradition was continued by the Amir Temur's successors in Samarkand - Shahrukh, Ulugbeg, Husein Baiqara, in the period of Babur - in Bukhara, Samarkand, Khiva, Herat, Balkh, Kabul and the other cities.
Sohibkiran caused to gather the eminent musicians and artists to Samarkand. In particular, famous musicologist Abudkadir Murony was brought from Baghdad. He was an expert in the theory of music and created over 200 musical works, many of which were unfortunately lost. The works "Zar-al-Fath", "Charhor-zarb", "Davri shohi" and styles of Tasnifi khodja Abdul Kodir, Amali-taron, Amali-buston Amali-guliston and Amali-horiston are from them (1.3).
The styles of "Zar-al-Fath" and "Charhor-zarb" are still popular in Samarkand and Bukhara. Sohibkiran paid the serious attention to the professional skills of his musicians. He personally appointed the chiefs of musical groups. The public holidays and military parades (bozor-ishob) in the month of ramazan were held at the Samarkand bazaars at the high level.
French historian Lusien Kere wrote: "Besides various holidays, Amir Temur arranged the celebrations, which inspired the warriors. These celebrations were held as the troops parades. The solemn sounds of "nogora" accompanied the parade, observed personally by Sohibkiran. The celebrations lasted more than two days (1.2; 65)". Sohibkiran gave the military status to percussions. He wrote about that in his manuscript "Amir Temur tuzuklari" in the section about "distribution of the standards and nogoras": "I order to hand one standard and "nogora" to each the 12th senior emir. Amur-ul-umaro should be given the standard, "nogora", "tug" and "chortug". Mingboshi should be given "tug" and "karnai". (1.1; 82).


text:
Pappas to Host "Pastries with Pappas" to Meet with Constituents about Issues that Matter Most to Them
Apr 5, 2019 | Pappas
BLOOMINGDALE, Ill. – State Rep. Diane Pappas, D-Itasca, is hosting a meeting open to all constituents of the 45th District Saturday, April 13 from 10 a.m. to noon. "Pastries with Pappas" will take place at the Roselle Park District, 10 North Building, located at 10 N Roselle Rd. in Roselle.
"I want to have more one-on-one discussions with folks in the district about their specific concerns," said Pappas. "My goal is to make our district the best it can be, and the first step is to have regular conversation with people throughout the community so that I can best represent
them in Springfield."
Pappas is serving her first term, and is eager to hear from all the residing members of the district. "Pastries with Pappas" is set to be a monthly event for constituents and their families to have the opportunity to engage in conversations regarding legislative concerns.
"People need to know that their elected officials are supporting an agenda that serves the community, which is why I am inviting my neighbors across the district to meet with me over a pastry," said Pappas. "If anyone has any questions, concerns or ideas for legislation, I encourage them to attend this event."
Rep Diane Pappas
Springfield Office:
240-W Stratton Office Building
One Tiffany Pointe
Suite G3
Illinois House Democratic Caucus | Copyright © 2021


McKenna is a senior at Mt. Spokane High School. She loves to sing and recently performed at a local Farmer's Market. She also sings in the choir at school. Her mom showed me a few videos of her performing during her session - AMAZING! McKenna is working as a nanny this summer and can't wait for her trip to California. She has plans to go to Universal Studios and Harry Potter World. She is thinking about going to WWU and hopes to become an elementary school teacher (like her mama!).


text:
Whiskey Storm (Whiskey Falls Book 3)
Mercy West is a legend in "The Underground" secret society who aids women and children in fleeing when their lives are in danger and they need new identities and to start over in new places to keep them alive. Now Mercy finds herself in the place of many she has helped over the past two decades.
Secret of the Sonnets
Miranda Hatch is no ordinary Regency heroine. She's in her mid-twenties and would likely be considered a spinster. However, marriage was never as important to her as her education. Fortunately, she has parents who encourage her pursuit of knowledge, rather than dissuade her from it.
Meriden Park
Mary Grace was rescued as a girl in India and was raised by the Wellings in Calcutta. This half Indian half English girl has been raised as an English lady but knows her place in society as a lesser. Not a servant but not a lady, Mary Grace is caught in between two worlds belonging to none.
Tricked in October (Pineridge Book 2)
Starla
DeKruyf
Kelsey O'Henry's annual Halloween costume party is just around the corner, but it's the first year she's hosting it since her husband's passing. Kelsey's already got enough on her plate on top of running the bar, including caring for three children, watching over her alcoholic mother, and making sure she keeps up with paying the bills.
Cause for Elimination
Marla A.
Emily Conners' whole world flips upside down when she finds her boss in a stall with her head smashed in. Emily had already been through enough after her life-altering horse- riding accident, and now she must deal with the trauma of this too.


text:
1000 Wege, ins Gras zu beißen ist eine Doku-Soap des US-amerikanischen Fernsehsenders Spike. Die Sendung wird in deutscher Sprache auf DMAX ausgestrahlt. Sie behandelt ungewöhnliche und skurrile Todesfälle sowie die Vorgänge, die zum jeweiligen Todesfall führten. Einige der dargestellten Geschichten wurden für den Darwin-Award nominiert.

Inhalt
Laut Eigenaussage behandelt 1000 Wege, ins Gras zu beißen reale Ereignisse. Manche Geschichten wurden allerdings dramaturgisch abgeändert, um den Unterhaltungswert der Sendung zu steigern. Pro Folge werden in der Regel bis zu sieben (in der zweiten Staffel bis zu acht) ungewöhnliche Todesfälle behandelt. Hierzu kommen neben nachgestellten Szenen und Computeranimationen auch Experten verschiedenster Fachrichtungen zu Wort, welche die Geschehnisse und deren tödliche Auswirkungen erläutern.

Ein häufig wiederkehrendes Motiv ist, dass das Opfer eine falsche oder meistens auch dumme Entscheidung trifft und diese dann zum "verdienten" Tod führt. Außerdem werden die Opfer meistens als böse, gierig oder dämlich dargestellt, damit der Zuschauer nicht allzu viel Mitleid empfindet (Umweltverschmutzer stirbt an eigenem Gift; fieser Chef gibt vor, im Rollstuhl zu sitzen, und fällt damit in Aufzugsschacht; unfreundliche Stewardess wird aus dem Flugzeug gesaugt). Begleitet wird die ganze Sendung von einem Kommentator, der mit schwarzem Humor nicht geizt und zuweilen sogar recht zynisch auftritt. Weil oft Gewalt, Blut, vulgäre Sprache, illegale Aktivitäten oder auch sexuelle Handlungen gezeigt werden (wenn auch auf humorvolle Weise), wurde die Serie in den USA für Zuschauer ab 14 Jahren (TV-14) und in Deutschland ab 16 Jahren (FSK 16) freigegeben.

Bis zum Ende der ersten Staffel zeigte die jeweils letzte Geschichte Situationen, die eigentlich zum Tod hätten führen müssen, aber dennoch überlebt wurden. In diesen Überlebensfällen wurden Interviews zusätzlich mit Zeugen oder auch den Betroffenen selbst geführt, ein paar dieser Überlebensfälle wurden auch in Zerstört in Sekunden (ebenfalls auf DMAX) gezeigt.

Haftungsausschluss
Mit den folgenden ersten Sätzen machen die Schöpfer der Serie auf ihren Haftungsausschluss aufmerksam:

Folgen 1 und 2
 Original:
 "The stories you are about to see are true and based on actual events."
 "Names have been changed to protect the identities of the deceased."
 "WARNING: The deaths portrayed in this show are real and extremely graphic."
 Deutsch:
 "Die hier gezeigten Todesfälle basieren auf Vorfällen, die sich tatsächlich zugetragen haben."
 "Alle Namen wurden verändert, um die Identitäten der Verstorbenen zu schützen."
 "WARNUNG: Die gezeigten Todesfälle in dieser Show sind wahr und extrem schockierend."

Folgen 3 bis 12
 Original:
 "WARNING: The deaths portrayed in this show are real and extremely graphic."
 "Names have been changed to protect the identities of the deceased."
 "Do not attempt to try ANY of the actions depicted."
 "YOU WILL DIE!"
 Deutsch:
 "WARNUNG: Die gezeigten Todesfälle in dieser Show sind wahr und extrem schockierend."
 "Die Namen wurden geändert, um die Identitäten der Verstorbenen zu schützen."
 "Versuchen Sie nicht, auch nur eine der gezeigten Darstellungen nachzuahmen."
 "ES WÄRE IHR SICHERER TOD"

Ab Folge 13
Original:
 "WARNING: The stories portrayed in this show are based on real deaths and are extremely graphic."
 "Names have been changed to protect the identities of the deceased."
 "Do not attempt to try ANY of the actions depicted."
 "YOU WILL DIE!"
Deutsch:
 "WARNUNG: Dieses Programm zeigt schockierende Bilder von Todesfällen,
 die nicht für alle Zuschauer geeignet sind."
 "Die Namen der Toten wurden geändert"
 "Spielen sie nichts aus diesem Programm nach,
 ES WÄRE IHR SICHERER TOD"

Episoden

Todesarten

Anmerkungen
 Jeder Todesfall erhält eine Nummer sowie einen eigenen Namen. Auffallend dabei ist, dass die Nummern der Todesfälle der deutschen Serie nicht mit den Nummern der Originalserie übereinstimmen.

 Todesfälle mit Kindern als Opfer kommen in der Serie nicht vor.
 Die Serie enthält einige Fehler, zum Beispiel ein Klapphandy im Jahre 1993 und Euromünzen im Jahre 1978.

Siehe auch
 Liste ungewöhnlicher Todesfälle

Weblinks

Einzelnachweise

Fernsehserie (Vereinigte Staaten)
Doku-Soap
Anthologie-Fernsehserie
Fernsehsendung (DMAX)
Fernsehserie der 2000er Jahre
Fernsehserie der 2010er Jahre