text:
\section*{Introduction}
The problem of rationality of Fano fourfolds,
with particular regard to the cases of
\emph{cubic fourfolds} and \emph{Gushel-Mukai fourfolds},
dates back to classical works; see \emph{e.g.}
\cite{Morin,Fano,Roth1949} (see also Table~\ref{tabella2} below).
Despite the great attention received and the numerous results obtained,
this is still an open problem for the general fourfold.
For instance, no examples of cubic fourfolds as well as of Gushel-Mukai fourfolds are known to be irrational,
and there are not so many constructions of rational examples.
The contribution of this paper is in the construction of
some new special examples of rational Gushel-Mukai fourfolds. This is
achieved by determining their equations through calculations
with \emph{Macaulay2} \cite{macaulay2}, using mainly the
packages \emph{SpecialFanoFourfolds} \cite{SpecialFanoFourfoldsSource} and \emph{Cremona} \cite{packageCremona}.
In particular, these packages provide the tools needed to verify the claims in the paper.
\medskip

Recall that by a result of Mukai \cite{mukai-biregularclassification} (extending
to all dimensions a result of Gushel proved in \cite{Gu}
only in dimension three),
a complex smooth prime Fano fourfold $X$ of degree $10$ and genus $6$, also known as \emph{Gushel-Mukai fourfold},
can be embedded in $\PP^8$
as a quadratic section of a $5$-dimensional linear section $Y\subset \PP^8$ of the
cone $C(\mathbb{G}(1,4))\subset\PP^{10}$ over the Grassmannian $\mathbb{G}(1,4)\subset\PP^9$
of lines in $\PP^4$.
There are two cases:
\begin{itemize}
 \item either $Y$ does not contain the vertex of the cone $C(\mathbb{G}(1,4))$,
 in which case $Y$ is isomorphic to a hyperplane section of $\mathbb{G}(1,4)$, and
 we have an embedding $\gamma_{X}:X\hookrightarrow \mathbb{G}(1,4)$;
 \item or otherwise $Y$ is isomorphic
          to a cone over a $4$-dimensional linear section  $Y_0$ of $\mathbb{G}(1,4)$,
          and we have a double cover $\gamma_{X}:X\to Y_0\subset\mathbb{G}(1,4)$.
\end{itemize}
The fourfolds in the second case are called \emph{Gushel fourfolds}, and they are specializations of fourfolds in the first case, called \emph{Mukai (or ordinary) fourfolds}.
In both cases, the map $\gamma_{X}$ from $X$ to $\mathbb{G}(1,4)$ is called the \emph{Gushel map}.
\medskip

By results proved in \cite{DIM} (see also \cite{DK1,DK3,DK2}),
Fano fourfolds as above
are parameterized (up to isomorphism) by the points of a coarse moduli space $\mathcal{M}_{4}$ of dimension $24$,
where the Gushel fourfolds correspond to the points of a closed irreducible subvariety
$\mathcal{M}_{4}^{G}\subset \mathcal{M}_{4}$ of codimension $2$.
A
fourfold $[X]\in \mathcal{M}_4$ is said to be \emph{special} (or \emph{Hodge-special}) if
it
contains a surface whose cohomology class does not lie in
 $\gamma_{X}^{\ast}(H^4(\mathbb{G}(1,4),\mathbb{Z}))$;
equivalently, $[X]$ is special if and only if $\mathrm{rk}(H^{2,2}(X)\cap H^4(X,\mathbb{Z}))\geq 3$.
A fourfold which corresponds to a very general point in $\mathcal M_4$ or in $\mathcal M_{4}^{G}$ is not special.
More precisely,
special fourfolds are parametrized by an infinite countable union
of hypersurfaces $\bigcup_d (\mathcal M_{4})_d\subset \mathcal M_4$,
labelled by the integers $d\geq 10$ with $d\equiv 0,2$, or $4$ (mod~$8)$.
If $d\equiv 0$ (mod~$4$) then the hypersurface $(\mathcal M_{4})_d$ is irreducible,
while if $d\equiv 2$ (mod~$8$) then it is the union of two irreducible
components $(\mathcal M_{4})_d'\cup (\mathcal M_{4})_d^{''}$.
When a fourfold
$[X]$ corresponds to a very general point of a component
of $(\mathcal M_{4})_d$, then the lattice $ H^{2,2}(X)\cap H^{4}(X,\mathbb Z)$
has rank $3$ and discriminant $d$. One says that $X$ has discriminant $d$ if $[X]\in (\mathcal M_{4})_d$.
\medskip

Suppose we have a fourfold $[X]\in \mathcal M_4$ containing an irreducible
surface $S$ of degree $\deg(S)$ and (sectional) genus $g(S)$, which has smooth normalization and  only a finite number $\delta$
of nodes as singularities. Let
 $a\sigma_{3,1} + b\sigma_{2,2}$ be the class of $\gamma_{X\ast}(S)$ in the Chow ring of $\mathbb{G}(1,4)$.
 The double point formula (see \cite[Theorem~9.3]{fulton-intersection} and also \cite[Section~7]{DIM}) gives
 the value of the self-intersection of $S$ in $X$:
\begin{equation}\label{selfIntersection}
(S)_X^2 = 3\,a+4\,b-2\,\deg(S)+4\,g(S)-12\,\chi(\mathcal O_S) + 2\,K_S^2 - 4 + 2\,\delta .
\end{equation}
Thus we have that $[X]\in (\mathcal M_4)_d$, where $d$ is
the discriminant of the lattice spanned by $(\gamma_X^\ast(\sigma_{1,1}), \gamma_X^{\ast}(\sigma_{2}), [S])$, that is
\begin{equation}\label{discriminant}
d = \mathrm{disc}\left(
     \begin{array}{c|c|c|c}
    & \gamma_X^\ast(\sigma_{1,1}) & \gamma_X^{\ast}(\sigma_{2}) & [S] \\ \hline
     \gamma_X^\ast(\sigma_{1,1}) & 2 & 2 & b \\ \hline
      \gamma_X^{\ast}(\sigma_{2}) & 2 & 4 & a \\ \hline
     \left[S\right] & b & a & (S)_X^2 \\
     \end{array}
     \right)= 4 (S)_X^2 -2 a^{2}+4 a b-4b^{2} .
\end{equation}
Moreover, when $d\equiv 2 \ (\mathrm{mod}\ 8)$, we have that
$[X]\in(\mathcal M_4)_d'$ if $a+b$ is even, and $[X]\in(\mathcal M_4)_d^{''}$
if $b$ is even.

\medskip

For some values of the discriminant $d$,
a fourfold $[X]\in (\mathcal M_4)_d$
has an \emph{associated} K3 surface of degree $d$;
and for others, it has an \emph{associated} cubic fourfold of discriminant $d$;
see \cite[Section~6]{DIM} for precise definitions.
The first values for which there is an associated K3 surface are:
\begin{equation*}
 { 10},\ {\color{Gray} 12,}\ {\color{Gray} 16,}\ {\color{Gray} 18,}\ { 20},\ {\color{Gray} 24,}\ { 26},\ {\color{Gray} 28,}\ {\color{Gray} 32,}\ { 34},\ {\color{Gray} 36,}\ {\color{Gray} 40,}\ {\color{Gray} 42,}\ {\color{Gray} 44,}\ {\color{Gray} 48,}\ { 50},\ { 52},\ {\color{Gray} 56,}\ { 58},\ {\color{Gray} 60,}\ldots
\end{equation*}
while, the first values for which there is an associated cubic fourfold are:
\begin{equation*}
 {\color{Gray} 10,}\ { 12},\ {\color{Gray} 16,}\ {\color{Gray} 18,}\ {\color{Gray} 20,}\ {\color{Gray} 24,}\ { 26},\ {\color{Gray} 28,}\ {\color{Gray} 32,}\ {\color{Gray} 34,}\ {\color{Gray} 36,}\ {\color{Gray} 40,}\ {\color{Gray} 42,}\ { 44},\ {\color{Gray} 48,}\ {\color{Gray} 50,}\ {\color{Gray} 52,}\ {\color{Gray} 56,}\ {\color{Gray} 58,}\ {\color{Gray} 60,}\ldots
\end{equation*}
The notion of associated K3 surface leads to the following conjecture,
which is analogous to the so-called
 \emph{Kuznetsov conjecture} for the rationality of cubic fourfolds (see \cite{kuz4fold,AT,kuz2,Levico,BRS,RS1,RS3}):
\begin{conjecture}
 A fourfold $[X]\in \mathcal M_4$ is rational if and only if
 it has an associated K3 surface, that is $[X]$ belongs to the infinite union:
 \begin{equation}\label{components}
  (\mathcal M_4)_{10}^{'} \cup (\mathcal M_4)_{10}^{''} \cup (\mathcal M_4)_{20} \cup (\mathcal M_4)_{26}^{'} \cup (\mathcal M_4)_{26}^{''} \cup (\mathcal M_4)_{34}^{'} \cup (\mathcal M_4)_{34}^{''} \cup \cdots
 \end{equation}
\end{conjecture}
The rationality for fourfolds in $(\mathcal{M}_4)_{10}^{'}$
is easy to show (see \cite[Proposition~7.3]{DIM}, \cite[Section~4.4]{ProcRS}, and also \cite[Example~1.1]{HoffSta}), and moreover
the associated K3 surface of degree $10$ appears naturally in the construction
of the birational map $\PP^4\dashrightarrow X$. The rationality
for the fourfolds in  $(\mathcal{M}_4)_{10}^{''}$ is classical:
it is achieved by Roth in \cite{Roth1949} as a consequence of a result of Enriques \cite{Enr,EinSh};
see also \cite[Proposition~7.5]{DIM}. In this last case, however, the role of the associated K3 surface
is not so clear.
      In the recent paper \cite{HoffSta}, it is showed that
a general fourfold in $(\mathcal{M}_4)_{20}$ (and hence \emph{every} by the main result in \cite{KontsevichTschinkelInventiones})
is rational. Again, the associated K3 surface of degree $20$
appears in the
explicit construction leading to rationality.
In conclusion, we have that every fourfold in the first three components of \eqref{components} is rational.

Restricting attention to the case of
 Gushel fourfolds, we point out that
using the same method presented in \cite{famGushelMukai},
just by replacing the role of the
smooth cubic scroll surface in \cite[Table~2]{famGushelMukai} with that of a cone over a twisted cubic curve,
one can find explicit  Gushel fourfolds in $(\mathcal{M}_4)_{10}^{'}$, $(\mathcal{M}_4)_{10}^{''}$,
and $(\mathcal{M}_4)_{20}$. In particular, the following three intersections
\begin{equation*}
  (\mathcal{M}_4)_{10}^{'}\cap \mathcal{M}_4^{G} ,\quad (\mathcal{M}_4)_{10}^{''}\cap \mathcal{M}_4^{G} ,\quad (\mathcal{M}_4)_{20}\cap \mathcal{M}_4^{G}
\end{equation*}
are not empty, and hence they parametrize  rational fourfolds.

As far as the author knows, no other fourfolds in $ \mathcal M_4$ as well as in $ \mathcal M_4^{G}$ are
known to be rational.
In the following of this paper, we explain how to find explicit equations and parameterizations
of rational fourfolds
in $(\mathcal{M}_4)_{26}''\cap \mathcal M_{4}^{G}$ and
$(\mathcal{M}_4)_{26}''\setminus \mathcal M_{4}^{G}$; see also Table~\ref{tabella1} for a summary.

\subsection*{Acknowledgements}
The author has benefited from discussions with Michele \-{Bolognesi}, Olivier Debarre, and Francesco Russo.

\section{Construction of new rational Gushel fourfolds}\label{sec Gushel}
In this section, we construct rational fourfolds
in $(\mathcal{M}_4)_{26}''\cap \mathcal M_{4}^{G}$. Here we briefly summarize the construction.
In Subsection~\ref{C26 rat},
starting with a general cubic fourfold $C$ in $\mathcal C_{26}$,
we explain how to determine the equations of a smooth surface
$S\subset \PP^8$ of degree $17$ and
sectional genus $11$, which is isomorphic
to a triple projection
of a minimal K3 surface of degree $26$ in $\PP^{14}$.
In Subsection~\ref{embedding in cone},
we embed the surface $S$ into a smooth quadratic section $X$
of a cone $Y$ over a smooth
$4$-dimensional linear section of $\mathbb{G}(1,4)$.
Then we deduce that $X$ is a fourfold in $(\mathcal{M}_4)_{26}''\cap \mathcal M_{4}^{G}$
since $\gamma_{X \ast}([S]) = 11 \sigma_{3,1} + 6 \sigma_{2,2}$
in the Chow ring of $\mathbb{G}(1,4)$.
In Subsection~\ref{subsec: count parameters 1},
we remark that the fourfold $X$
may not be a general
 point in $(\mathcal{M}_4)_{26}''\cap \mathcal M_{4}^{G}$.
In Subsection~\ref{congruence quintics},
we illustrate two different methods to deduce that the surface $S$ admits inside the cone $Y$
a \emph{congruence of $9$-secant rational normal quintic curves}: through
a general point of $Y$ there passes a unique rational normal quintic curve
which is $9$-secant to $S$ and contained in $Y$.
In Subsection~\ref{particular singular}, we show how from this congruence of quintic curves,
using in an essential way that $Y$ is not smooth,
we can deduce that $X$ is rational.
In Subsection~\ref{summary constr}, we
describe an explicit birational map between $X$ and the cubic fourfold $C$.

\subsection{Rationality of cubic fourfolds in \texorpdfstring{$\mathcal C_{26}$}{C26}}\label{C26 rat}
Here we recall some results from \cite{RS1,RS3} (see also \cite{ProcRS}) about the rationality of
special cubic fourfolds of discriminant $26$;
see also \cite{Hassett,Has00,Levico}  for general facts on cubic fourfolds.

Let $D\subset \PP^5$ be a septimic surface with one node, which is the projection
of a smooth del Pezzo surface of degree seven $D'\subset\PP^7$ from a general line
intersecting the secant variety of $D'$ at one point.
A cubic fourfold $C\subset\PP^5$ containing the surface $D$
has discriminant $26$, and more precisely
the locus $\mathcal C_{26}$ of cubic fourfolds of discriminant $26$
can be described as the closure inside the moduli space $\mathcal C$ of cubic fourfolds of the locus of fourfolds
containing such a surface.

The surface $D\subset\PP^5$ admits a \emph{congruence of $5$-secant conics}:
through a general point in $ \PP^5$ there passes a unique $5$-secant conic to $D$.
Moreover, the linear system $|H^0(\mathcal I^2_{D,\PP^5}(5))|$
of hypersurfaces of degree $5$ with points of multiplicity $2$ along $D$
gives a dominant map
\begin{equation}\label{KollarMap1}
  \PP^5\dashrightarrow Y_0\subset\PP^7,
\end{equation}
whose general fibers are the conic curves
of the congruence, and where $Y_0$ is a smooth $4$-dimensional linear section of $\mathbb{G}(1,4)\subset\PP^9$.
The restriction of the map \eqref{KollarMap1} to a general cubic fourfold $C$ through $D$ induces
a birational map $C\dashrightarrow Y_0$,
whose inverse
is defined by the linear system $|H^0(\mathcal I^2_{S_0,Y_0}(5))|$
of hypersurfaces in $Y_0$ of degree $5$ having points of
multiplicity $2$ along an irreducible surface $S_0\subset Y_0\subset\PP^7$
of degree $17$ and sectional genus $11$ cut out by the $5$ quadrics defining $Y_0$
and $13$ cubics.

It turns out that $S_0$ is the
projection of a surface $S\subset\PP^8$
from a special point $p$ on the secant variety of $S$,
where $S$ is a smooth surface of degree $17$,
sectional genus $11$, cut out by $12$ quadrics, and isomorphic
to a triple projection
of a minimal K3 surface of degree $26$ in $\PP^{14}$. The equations of $S$
can be determined from those of $S_0$
using
the
package \emph{IntegralClosure} \cite{IntegralClosureSource};
see also the function \texttt{associatedK3surface} from the package
\emph{SpecialFanoFourfolds} \cite{SpecialFanoFourfoldsSource}, which does most of this automatically.

\subsection{Gushel fourfolds in \texorpdfstring{$(\mathcal{M}_4)_{26}''$}{(M4)26''}: construction of a triple \texorpdfstring{$S\subset X \subset Y$}{S,X,Y}}\label{embedding in cone}

Continuing from the previous subsection,
let $\nu_p:\PP^8\dashrightarrow\PP^7$ denote the projection from the point $p$ such that $\nu_p(S) = S_0$, and let
$Y=\overline{\nu_p^{-1}(Y_0)}$ be the cone over $Y_0$ of vertex $p$.
Let $X\subset Y$ be a general quadratic section of $Y$ containing $S$.
Then $X$ is a smooth Gushel fourfold which belongs to $(\mathcal M_4)_{26}''$.

Indeed, the surface $S_0\subset Y_0\subset\mathbb{G}(1,4)$, which is equal to
the image of $S$ via the Gushel map of $X$, has class $11 \sigma_{3,1} + 6 \sigma_{2,2}$
in the Chow ring of $\mathbb{G}(1,4)$,
as one can  verify by simple calculations with Schubert cycles.
Then, from \eqref{selfIntersection} and \eqref{discriminant} it follows that
 $(S)_X^2 = 37$ and  $[X]\in  (\mathcal M_4)_{26}^{''}$,
 since one has $\deg(S)=17$, $g(S)=11$, $\chi(\mathcal O_S) = 2$, and $K_S^2 = -1$.
 This calculation can be performed automatically using the functions \texttt{discriminant}
 and \texttt{describe}
from the package \emph{SpecialFanoFourfolds}.

\subsection{Count of parameters from the triple \texorpdfstring{$S\subset X\subset Y$}{S,X,Y}}\label{subsec: count parameters 1}
Let $N_{S/X}$ and $N_{S/Y}$ denote, respectively,
the normal bundle of the surface $S$ in $X$, and of $S$ in $Y$.
A \emph{Macaulay2} calculation tells us that
$h^1(N_{S/Y})=0$, $h^0(N_{S/Y})=37$, and  $h^0(N_{S/X})=6$.
It follows that there exists a unique irreducible component $\mathcal S$
of the Hilbert scheme of $Y$ which contains $[S]$, and $\mathcal S$
is smooth at $[S]$ of dimension $37$.
Since we have $h^0(\mathcal I_{S,Y}(2))= 7$ (and this value is minimal on $\mathcal S$),
we deduce by the same
 semicontinuity argument
explained in \cite[Subsection~1.5]{famGushelMukai} that
inside the $39$-dimensional projective space $\mathbb{P}(H^0(\mathcal O_{Y}(2)))$
of quadratic sections of $Y$ the family of fourfolds containing a surface in $\mathcal S$
has codimension at most $39 - (37 + (7-1) - 6) = 2$.
This calculation can be performed automatically using the function \texttt{parameterCount}
from the package \emph{SpecialFanoFourfolds}.

\subsection{Congruence of \texorpdfstring{$9$}{9}-secant quintic curves to \texorpdfstring{$S\subset Y$}{S in Y}}\label{congruence quintics}
We claim that  the surface $S$ admits inside $Y$
a \emph{congruence of $9$-secant rational normal quintic curves}: through
a general point of $Y$ there passes a unique rational normal quintic curve
which is $9$-secant to $S$ and contained in $Y$.

This can be verified by considering the rational map
$\phi:Y\dashrightarrow \PP^6$ defined by the linear system $|H^0(\mathcal I_{S,Y}(2))|$
of quadratic sections through $S$, which is birational onto a non-normal sextic hypersurface $Z\subset \PP^6$.
If $p\in Y$ is a general point, then through the point $\phi(p)\in  Z$ there pass $72$ lines that are contained in $Z$.
These $72$ lines come from  $(2e-1)$-secant curves  to $S$ of degree $e\geq 1$ which pass through $p$ and are contained in $Y$.
Denoting by $n_e$ the number of such degree-$e$ curves, we have
$n_1=11$, $n_2 = 22$, $n_3 = 32$, $n_4 = 6$, $n_5 = 1$, and $n_e = 0 $ for $e\geq 6$.
This calculation can be performed automatically using the function \texttt{detectCongruence}
from the package \emph{SpecialFanoFourfolds}.

Alternately, using tools from the package \emph{Cremona}, one verifies that
 the linear system
$|H^0(\mathcal I^5_{S,Y}(9))|$ of hypersurfaces in $Y$
of degree $9$ with points of multiplicity $5$ along $S$
gives a  dominant rational map
\begin{equation}\label{kollarMap}
 Y\dashrightarrow X'\subset\PP^8,
\end{equation}
whose general fibers
are rational normal quintic curves, the curves of the congruence to $S$.
The image $X'$ is a Gushel fourfold, a smooth quadratic section of a
cone $Y'\subset\PP^8$ over a smooth $4$-dimensional linear section  of $\mathbb{G}(1,4)$.

The restriction of the map \eqref{kollarMap}  to $X$ induces a birational map
\begin{equation}\label{Map}
Y \supset X \stackrel{\simeq}{\dashrightarrow}  X' \subset Y' ,
\end{equation}
whose inverse is of the same type, \emph{i.e}, it is
the restriction to $X'$ of the
rational map $Y'\dashrightarrow X$ defined
by the linear system of
hypersurfaces in $Y'$
of degree $9$ with points of multiplicity $5$ along a smooth triple projection $S'\subset Y' $
of a minimal K3 surface of degree $26$.
In particular, the Gushel fourfold $X'$ also belongs to $(\mathcal M_4)_{26}^{''}$.

\subsection{Rationality of \texorpdfstring{$X$}{X} from the rationality of a particular rational singular fourfold \texorpdfstring{$\hat{X}$}{Xh} with \texorpdfstring{$S\subset \hat{X}\subset Y$}{S in X in Y}}\label{particular singular}
Let $\hat{X}\subset Y$ be a general quadratic section of $Y$
 containing the surface $S$ and the vertex $p$ of $Y$. Then $\hat{X}$ has $p$
as the only singularity and the restriction of the projection from $p$
induces a birational map $\hat{X}{\dashrightarrow} Y_0$, whose inverse is defined by the quadrics
through a minimal K3 surface $F$ of degree $10$.
Since $Y_0=\mathbb{G}(1,4)\cap \PP^7$ is rational (indeed, the projection from the unique $\sigma_{2,2}$-plane contained in it gives a birational map onto $\PP^4$), we have that  $\hat{X}$ is rational.
On the other hand, the restriction
of the map \eqref{kollarMap}  induces a birational map
\[
 Y \supset \hat{X} \stackrel{\simeq}{\dashrightarrow} X' \subset Y',
\]
due to the fact that $\hat{X}$ is \emph{transversal} to the congruence to $S$:
the quintic curve of the congruence passing through a general point of $\hat{X}$
is not contained in $\hat{X}$.
Therefore we deduce that also $X'$ and hence $X$ are rational.

\subsection{Summary construction}\label{summary constr}
Summing up we have the following diagram of birational maps, connecting explicitly
general cubic fourfolds in $\mathcal C_{26}$
to Gushel fourfolds in $(\mathcal M_{4})_{26}''$:
 \begin{equation}\label{mappe}
 \xymatrix{
 {\color{Gray} \PP^5} \ar@/^1.9pc/@{-->}@[Gray][rrd] & & & & {\color{Gray} Y}\ar@/^1.9pc/@{-->}@[Gray][rrd] & & {\color{Gray} Y'}\ar@/^1.9pc/@{-->}@[Gray][rrd] \\
  C\ar@{^{(}->}@[Gray][u]\ar@/^1.0pc/@{-->}[rr]^{|H^0(\mathcal{I}^{2}_{D}(5))|} && Y_0\ar@/^1.0pc/@{-->}[ll]^{|H^0(\mathcal{I}^{2}_{S_0}(5))|}
  \ar@/^1.0pc/@{-->}[rr]^{|H^0(\mathcal{I}_{F}(2))|} && \hat{X}\ar@{^{(}->}@[Gray][d]\ar@{^{(}->}@[Gray][u]\ar@/^1.0pc/@{-->}[ll]^{|H^0(\mathcal{I}_{p}(1))|}
  \ar@/^1.0pc/@{-->}[rr]^{|H^0(\mathcal{I}^{5}_{S}(9))|} && X'\ar@{^{(}->}@[Gray][u]\ar@{^{(}->}@[Gray][d]\ar@/^1.0pc/@{-->}[ll]^{|H^0(\mathcal{I}^{5}_{\hat{S}'}(9))|}
  \ar@/^1.0pc/@{-->}[rr]^{|H^0(\mathcal{I}^{5}_{S'}(9))|} && X\ar@{^{(}->}@[Gray][d]\ar@/^1.0pc/@{-->}[ll]^{|H^0(\mathcal{I}^{5}_{S}(9))|} \\
  & & & & { \color{Gray} Y } \ar@/^1.9pc/@{-->}@[Gray][llu] & & {\color{Gray} Y' } \ar@/^1.9pc/@{-->}@[Gray][llu] & & { \color{Gray} Y} \ar@/^1.9pc/@{-->}@[Gray][llu]
 }
\end{equation}
where $[C]\in \mathcal C_{26}$
is a general cubic fourfold of discriminant $26$ containing
a septimic one-nodal del Pezzo surface $D$;
$[X]$, $[X']\in (\mathcal M_{4})_{26}''\cap \mathcal M_{4}^{G}$ are (smooth) Gushel fourfolds
of discriminant $26$ contained, respectively, in cones $Y$ and $Y'$ over $Y_0=\mathbb{G}(1,4)\cap \PP^7$;
$\hat{X}\subset Y$ is a Gushel fourfold singular at $p$, where
$p$ is the vertex of $Y$;
$S\subset X\cap \hat{X}$ and $S',\hat{S}'\subset X'$
are smooth surfaces, isomorphic to triple projections of minimal K3 surfaces of degree $26$ (the intersection $S'\cap \hat{S}'$
consists of a twisted cubic curve and $31$ points);
$S_0\subset Y_0$ is the projection of $S$ from $p$;
and $F\subset Y_0$ is a minimal K3 surface
of degree $10$.

\subsection{Ancillary files}\label{computation 1}
We provide an ancillary file, named \texttt{gushel26.m2},
containing explicit equations for an example of map as \eqref{Map} over the finite field $\mathbb{F}_{10000019}$
(this is only necessary to reduce the size of the file).
After loading that file in \emph{Macaulay2},
some variables will be defined as follows:
\begin{description}
 \item[\texttt{X, X'}] two instances of the type \texttt{SpecialGushelMukaiFourfold}, respectively, the source and target of the map \eqref{Map};
 \item[\texttt{psi, psi'}] two instances of the type \texttt{RationalMap}, respectively,
 the birational map \eqref{Map} and its inverse (\texttt{psi'} is the same that \texttt{inverse psi});
 \item[\texttt{Psi, Psi'}] two instance of the type \texttt{RationalMap}, respectively,
 the dominant rational map \eqref{kollarMap} from $Y$ to $X'$ that extends \texttt{psi},
 and the analogous map from $Y'$ to $X$ that extends \texttt{psi'}.
\end{description}
For technical details about these types of data, we refer to the documentation of the packages \emph{Cremona} \cite{packageCremona} and
\emph{SpecialFanoFourfolds} \cite{SpecialFanoFourfoldsSource}.
We now show how to load the file
and extract some basic information from it (some output lines are omitted for brevity).
\begin{tcolorbox}[breakable=true,boxrule=0pt,opacityback=0.0,enhanced jigsaw]
{\footnotesize
\begin{Verbatim}[commandchars=&\[\]]
&colore[darkorange][$ M2 --no-preload]
&colore[output][Macaulay2, version 1.16]
&colore[darkorange][i1 :] &colore[airforceblue][needs] "&colore[bleudefrance][gushel26.m2]"
&colore[darkorange][i2 :] &colore[airforceblue][describe] X
&colore[circOut][o2 =] &colore[output][Special Gushel-Mukai fourfold of discriminant 26('')]
&colore[output][     containing a surface in PP^8 of degree 17 and sectional genus 11]
&colore[output][     cut out by 12 hypersurfaces of degree 2]
&colore[output][     and with class in G(1,4) given by 11*s_(3,1)+6*s_(2,2)]
&colore[output][     Type: Gushel (not ordinary)]
&colore[darkorange][i3 :] &colore[airforceblue][describe] Psi
&colore[circOut][o3 =] &colore[output][rational map defined by forms of degree 9]
&colore[output][     source variety: 5-dimensional variety of degree 5 in PP^8]
&colore[output][                     cut out by 5 hypersurfaces of degree 2]
&colore[output][     target variety: 4-dimensional variety of degree 10 in PP^8]
&colore[output][                     cut out by 6 hypersurfaces of degree 2]
&colore[output][     dominance: true]
&colore[output][     projective degrees: {5, 45, 211, 200, 50, 0}]
&colore[darkorange][i4 :] S = &colore[airforceblue][first ideals] X; &colore[commentoutput][-- the surface S]
&colore[darkorange][i5 :] h = &colore[airforceblue][detectCongruence](X,5); &colore[commentoutput][-- the congruence to S]
&colore[darkorange][i6 :] p = &colore[airforceblue][point] X; &colore[commentoutput][-- random point on X]
&colore[darkorange][i7 :] C = h p; &colore[commentoutput][-- 9-secant quintic curve to S passing through p]
\end{Verbatim}
}
\end{tcolorbox}

\section{Rationality
without passing through singular fourfolds}\label{particular smooth}
A count of parameters shows that
a general triple projection
 of a general minimal K3 surface of degree $26$
is contained in a one-dimensional family
of $5$-dimensional linear sections  of cones $C(\mathbb{G}(1,4))\subset\PP^{10}$.
So one expects that the Gushel fourfolds constructed in Section~\ref{sec Gushel}
can be deformed to ordinary fourfolds. Anyway, the simple argument
given in Subsection~\ref{particular singular} to deduce the rationality of the fourfold $X$
is not available in the ordinary case.
In this section, we remedy this with a construction that does not involve singular Gushel fourfolds.

Indeed, keeping the notation as in Section~\ref{sec Gushel}, here we construct
a special Gushel fourfold $\tilde{X}$ with $S\subset \tilde{X}\subset Y$,
uniquely determined by the embedding $S\subset Y$, which turns out to
be smooth and transversal to the congruence of $9$-secant quintic curves to $S$.
We then show that $\tilde{X}$ contains another ``simpler'' surface $T$ which admits
inside $Y$ a congruence of $5$-secant  cubic curves. So we deduce that $\tilde{X}$ is rational,
and hence that  $X$ is rational.

\subsection{Construction of \texorpdfstring{$\tilde{X}$}{X} and of a dominant map \texorpdfstring{$\tilde{X}\dashrightarrow\PP^2$}{X-->P2}}
Consider again the birational map introduced in Subsection~\ref{congruence quintics},
\[
\phi:Y\dashrightarrow Z\subset \PP^6 ,
\]
defined by the linear system $|H^0(\mathcal I_{S,Y}(2))|$,
where $Z=\overline{\phi(Y)}\subset\PP^6$ is a sextic hypersurface.
We take $\tilde{X}$ to be the top dimensional component of the (closure of the) exceptional locus of $\phi$, and the map $\tilde{X}\dashrightarrow\PP^2$ to be defined by the quadrics through the (closure of the) union of all $4$-secant conics to $S$ contained in $\tilde{X}$.
Let us provide some more detail.
\subsubsection{The fourfold $\tilde{X}$}
Using tools from the
package \emph{Cremona}, one verifies that the base locus scheme of the inverse map $\phi^{-1}:Z\dashrightarrow Y$
is the union of the following components:
\begin{itemize}
 \item a smooth cubic fourfold $\tilde{C}\subset\PP^5\subset\PP^6$ (with $[\tilde{C}]\in \mathcal{C}_{26}$, see Remark~\ref{remC26}); the fiber of $\phi$ at a general point of $\tilde{C}$
 consists of two points;
 \item a smooth cubic scroll surface $\Sigma_3\subset\PP^4\subset\PP^6$, which is double; the fiber of $\phi$ at a general point
 of $\Sigma_3$ is an irreducible conic curve which is $4$-secant to $S$;
 \item a  surface of degree $46$ cut out in $\PP^6$ by $81$ quintic hypersurfaces;
 the fiber of $\phi$ at a general point
 of this surface is a $2$-secant line to $S$.
\end{itemize}
Then, the  fourfold $\tilde{X}$ is taken to be $\overline{\phi^{-1}(\tilde{C})}$, so that
 the restriction of $\phi$
induces a generically finite map of degree $2$ from $\tilde{X}$ to $\tilde{C}$.
One
sees
that $\tilde{X}$ is a smooth Gushel fourfold which is transversal to the congruence to $S$
and in particular it is birational to $X'$ via the restriction of the map \eqref{kollarMap}.
Moreover, the inverse map $X'\dashrightarrow \tilde{X}$
is defined once again by the linear system $|H^0(\mathcal I^{5}_{\tilde{S}'}(9))|$,
where $\tilde{S}'\subset X'\subset Y'$
is a smooth triple projection of a minimal K3 surface of degree $26$. We stress that
this surface $\tilde{S}'\subset Y'$ and the fourfold $\tilde{X}$
  are uniquely determined by the embedding $S\subset Y$.

\subsubsection{The map $\tilde{X}\dashrightarrow\PP^2$}
The intersection $\tilde{C}\cap\Sigma_3$ is a twisted cubic curve, and the fiber of $\phi$
at a general point of this curve is a $4$-secant conic curve to $S$ contained in $\tilde{X}$.
The inverse image $R=\overline{\phi^{-1}(\tilde{C}\cap\Sigma_3)}$ is an irreducible surface contained in $\tilde{X}$
of degree $17$ and sectional genus $6$
cut out in $\PP^8$
by $9$ quadrics and $7$ cubics.
Let
\begin{equation}\label{MapEta}
     \eta:Y\dashrightarrow \PP^3
\end{equation}
be the rational map
defined by the linear system $|H^0(\mathcal I_{R,Y}(2))|$ of quadratic sections of $Y$ through $R$.
Since  $R\subset\tilde{X}$, the restriction of $\eta$ induces
another map
\begin{equation}\label{MapEta2}
     \eta|_{\tilde{X}}:\tilde{X}\dashrightarrow \PP^2 \subset\PP^3 .
\end{equation}
One sees that $\eta$ (resp., $\eta|_{\tilde{X}}$) is a dominant
map,
whose general fibers are surfaces like $S$, that is,
smooth triple projections
of minimal K3 surfaces of degree $26$.
Moreover the surface $S$
is recovered as a special fiber, and
all these fibers share the same twisted cubic curve contained in $S$.

\begin{remark}
 The projection of the surface $S$ from the plane spanned by a general $4$-secant
 conic to $S$ contained in $Y$ (computable as the the
 fiber of $\phi$ at a general point on $\Sigma_3$) is a smooth surface in $\PP^5$
 of degree $13$ and sectional genus $11$ cut out by $6$ cubics.
 This surface in $\PP^5$
 admits a congruence of $14$-secant rational normal quintic curves
 from which one can deduce the rationality
 for cubic fourfolds of discriminant $26$.
 (See also the example in \cite[Table~1, row~15]{RS3} of a nodal surface in $\PP^5$.)
\end{remark}

\subsection{Special fibers of the map \texorpdfstring{$\tilde{X}\dashrightarrow \PP^2$}{X-->P2}: construction of a surface \texorpdfstring{$T\subset\tilde{X}\subset Y$}{T in X in Y}}\label{Gushel fourfold T,Xtilde}
The $3$-dimensional projective space, image of the map $\eta$, contains a special plane $\Pi$
which intersects $\overline{\eta(\tilde{X})}\simeq\PP^2$ along a line $L$.
The fiber  of $\eta$ at a general point of $L$ (as well as of $\Pi$)
is an irreducible rational surface $T\subset Y\subset \PP^8$ of degree $11$, sectional genus $3$,
cut out in $\PP^8$ by $16$ quadrics, having smooth normalization and
 a node as the only singularity.
 In the Chow ring of $\mathbb{G}(1,4)$ we have
 $[\gamma_{\tilde{X}}(T)] = 7 \sigma_{3,1} + 4 \sigma_{2,2}$, so
 from  \eqref{selfIntersection} it follows that $(T)_{\tilde{X}}^2 = 19$ (this is also confirmed from the fact that two
 fibers of $\eta$ corresponding to two general points of $L$ intersects at $19$ points),
 and \eqref{discriminant} tells us that
 any smooth quadratic section of $Y$ containing $T$ is a Gushel fourfold
 of discriminant $26$, hence corresponding to a point of~$(\mathcal M_4)_{26}^{''}$.

 From another point of view, using the map $\eta$,
 we are able to obtain a degeneration of the surface $S$ as the union of the surface
 $T$ and a smooth surface $Q$ of degree $6$ and sectional genus $2$
 with $[\gamma_{\tilde{X} \ast}(Q)] = 2(2\sigma_{3,1} + \sigma_{2,2}) = 2 \sigma_2 \sigma_1^2$ and such that
 the intersection $T\cap Q$ is an irreducible curve of degree $7$ with $p_a=2$ and $p_g=1$.
 This surface $Q$ is contained in the base locus of $\eta$. More precisely,
 the support of the base locus of $\eta$ is the union of $Q$ with the surface $R$.

\begin{remark}\label{remC26}
The plane $\Pi$
 can be calculated as the image
 via $\eta$
of the $3$-dimensional linear space $\overline{\nu_p^{-1}(P)}$, where $\nu_p:Y\dashrightarrow Y_0$
is the projection from the vertex of $Y$, and $P$ is the unique $\sigma_{2,2}$ plane contained in $Y_0$.
Moreover,
the intersection $\overline{\nu_p^{-1}(P)}\cap \tilde{X}$
 is a quadric surface which is sent birationally by
 $\phi$ to a one-nodal septimic surface $\tilde{D}\subset\tilde{C}$
 as the surface $D\subset C$ considered in Subsection~\ref{C26 rat}. In particular, we also deduce that $\tilde{C}$
 is a cubic fourfold of discriminant $26$.
\end{remark}

\subsection{Count of parameters from the triple \texorpdfstring{$T\subset \tilde{X}\subset Y$}{T,X,Y}}
As in Subsection~\ref{subsec: count parameters 1}, we compute with \emph{Macaulay2} that
$h^1(N_{T/Y})=0$, $h^0(N_{T/Y})=29$, and  $h^0(N_{T/\tilde{X}})=2$.
Assuming that the Hilbert scheme $\mathrm{Hilb}_Y$ of $Y$ is smooth at $[T]$
(which is reasonable, but not guaranteed since $T$ is not a local complete intersection), we have that
$\mathrm{Hilb}_Y$ contains a unique irreducible component $\mathcal T$
 which contains $[T]$, and the dimension of $\mathcal T$
is $29$.
Therefore, since we have $h^0(\mathcal I_{T,Y}(2))= 11$ and this value is minimal,
we deduce that
inside the projective space $\mathbb{P}(H^0(\mathcal O_{Y}(2)))$
of quadratic sections of $Y$ the family of fourfolds containing a surface in $\mathcal T$
has codimension at most $39 - (29 + (11-1) - 2) = 2$.

 \subsection{Congruence of \texorpdfstring{$5$}{5}-secant cubic curves to \texorpdfstring{$T\subset Y$}{T in Y}  and rationality of \texorpdfstring{$\tilde{X}$}{X}}

 The surface $T\subset Y$  admits inside $Y$
 a congruence of $5$-secant twisted cubic curves,
 and $\tilde{X}$ is transversal to this congruence.

 Indeed one verifies that the linear system $|H^0(\mathcal I_{T,Y}^3(5))|$
 of
hypersurfaces in $Y$
of degree $5$ with points of multiplicity $3$ along
   $T$ gives a dominant rational map
 \begin{equation}\label{MapFromYtoW}
  Y\dashrightarrow W\subset\PP^{10}
 \end{equation}
 onto a smooth $4$-dimensional linear section $W$ of $\mathbb{G}(1,5)\subset\PP^{14}$,
 and whose general fibers are twisted cubic curves. The restriction
 of \eqref{MapFromYtoW} to $\tilde{X}$ induces a birational map
 \begin{equation}\label{MapFromXtildetoW}
  Y\supset \tilde{X}\stackrel{\simeq}{\dashrightarrow} W\subset\PP^{10} ,
 \end{equation}
 whose inverse is defined by the linear system of
 hypersurfaces of degree $5$ with points
 of multiplicity $3$ along a smooth surface $U\subset W$ of degree $21$ and sectional genus $13$, isomorphic
 to a double projection of a simple projection of a minimal K3 surface of degree $26$.

 We deduce the rationality of $\tilde{X}$ from that of $W$. Indeed,
 $W$ must contain a quintic del Pezzo surface, and
 it is classically known that
the linear system of hyperplanes through this surface gives a birational map $W\dashrightarrow\PP^4$.

 The congruence to $T\subset Y$ can be also verified by considering the
 map
 $
  Y\dashrightarrow \PP^{10},
 $
 defined by the linear system $|H^0(\mathcal I_{T,Y}(2))|$ of quadratic sections of $Y$ through $T$,
 which turns out to be  birational onto a fivefold of degree $20$ cut out by $7$ quadrics.
 Through the general point of this fivefold there pass $12$ lines, which come
 from seven $1$-secant lines to $T$, four $3$-secant conics to $T$,
 and one single $5$-secant twisted cubic to $T$.

 There is a further way to find the congruence to $T\subset Y$. Indeed,
 one has that the reducible surface $T\cup Q$
 considered in Subsection~\ref{Gushel fourfold T,Xtilde}, which is a degeneration of the surface $S$,
 admits  a congruence of $9$-secant quintic curves, exactly as $S$ does.
 In this degenerate case, the curves of the congruence
  split into $4$-secant conics to $T\cup Q$ and $5$-secant twisted cubics to $T$.

 \subsection{Summary construction} With the notation above introduced, we have the
 following diagram involving
 cubic fourfolds in $\mathcal{C}_{26}$ and Gushel fourfolds in $(\mathcal M_4)_{26}^{''}$, and where
 all the fourfolds are smooth.
  \begin{equation}\label{mappe2}
 \xymatrix{
  \tilde{C} & & {\color{Gray} Y}\ar@/^1.9pc/@{-->}@[Gray][rrd] & & {\color{Gray} Y'}\ar@/^1.9pc/@{-->}@[Gray][rrd] \\
   W
  \ar@/^1.0pc/@{-->}[rr]^{|H^0(\mathcal{I}^{3}_{U}(5))|} && \tilde{X}\ar@/_2.0pc/@{-->}[llu]_{2:1}^{H^0(\mathcal{I}_{S}(2))}\ar@{^{(}->}@[Gray][d]\ar@{^{(}->}@[Gray][u]\ar@/^1.0pc/@{-->}[ll]^{|H^0(\mathcal{I}^{3}_{T}(5))|}
  \ar@/^1.0pc/@{-->}[rr]^{|H^0(\mathcal{I}^{5}_{S}(9))|} && X'\ar@{^{(}->}@[Gray][u]\ar@{^{(}->}@[Gray][d]\ar@/^1.0pc/@{-->}[ll]^{|H^0(\mathcal{I}^{5}_{\tilde{S}'}(9))|}
  \ar@/^1.0pc/@{-->}[rr]^{|H^0(\mathcal{I}^{5}_{S'}(9))|} && X\ar@{^{(}->}@[Gray][d]\ar@/^1.0pc/@{-->}[ll]^{|H^0(\mathcal{I}^{5}_{S}(9))|} \\
   & & { \color{Gray} Y } \ar@/^1.9pc/@{-->}@[Gray][llu] & & {\color{Gray} Y' } \ar@/^1.9pc/@{-->}@[Gray][llu] & & { \color{Gray} Y} \ar@/^1.9pc/@{-->}@[Gray][llu]
 }
\end{equation}

 \subsection{Ancillary files}\label{computation 2}
 Continuing from Subsection~\ref{computation 1},
 after the loading of the ancillary file in \emph{Macaulay2},
 some other variables are defined as following:
 \begin{description}
  \item[\texttt{Xtilde}] an instance of the type \texttt{SpecialGushelMukaiFourfold},
  the Gushel fourfold corresponding to the pair $(T,\tilde{X})$
  constructed above;
  \item[\texttt{Eta, eta}] two instances of the type \texttt{RationalMap}, respectively,
  the map $Y\dashrightarrow\PP^3$ in \eqref{MapEta} and its restriction
  $\tilde{X}\dashrightarrow\PP^2\subset\PP^3$ in \eqref{MapEta2}.
 \item[\texttt{Psi2, psi2}] two instances of the type \texttt{RationalMap}, respectively,
 the map $Y\dashrightarrow W$ in \eqref{MapFromYtoW} and its restriction $\tilde{X}\dashrightarrow W$
 in \eqref{MapFromXtildetoW}.
 \end{description}
By way of example, we now compute a general and a special fiber of
the map \eqref{MapEta2}.
\begin{tcolorbox}[breakable=true,boxrule=0pt,opacityback=0.0,enhanced jigsaw]
{\footnotesize
\begin{Verbatim}[commandchars=&\[\]]
&colore[darkorange][i8 :] p = &colore[airforceblue][point image] eta; &colore[commentoutput][-- random point on the image of eta]
&colore[darkorange][i9 :] L = &colore[airforceblue][ideal] (&colore[airforceblue][gens target] eta)_{2,3}; &colore[commentoutput][-- the special line L in P^3 (Subsect. 2.2)]
&colore[darkorange][i10 :] q = &colore[airforceblue][point] L; &colore[commentoutput][-- random point on L]
&colore[darkorange][i11 :] &colore[airforceblue][describe](eta||p) &colore[commentoutput][-- fiber at p]
&colore[circOut][o11 =] &colore[output][source variety: surface of degree 17 and sectional genus 11 in PP^8]
&colore[output][                      cut out by 12 hypersurfaces of degree 2]
&colore[output][      target variety: one-point scheme in PP^3]
&colore[darkorange][i12 :] &colore[airforceblue][describe](eta||q) &colore[commentoutput][-- fiber at q]
&colore[circOut][o12 =] &colore[output][source variety: surface of degree 11 and sectional genus 3 in PP^8]
&colore[output][                      cut out by 16 hypersurfaces of degree 2]
&colore[output][      target variety: one-point scheme in PP^3]
\end{Verbatim}
}
\end{tcolorbox}

 \section{Construction of new rational Mukai fourfolds}\label{sec Mukai}
 In this section, we briefly explain how using
 the same method given in \cite{famGushelMukai}, one can
directly  construct a rational one-nodal surface  $T\subset Y$ of degree $11$ and
sectional genus $3$
as the one constructed in Section~\ref{particular smooth}. Actually, we do better than this:
We are able to construct a
rational one-nodal surface  $\breve{T}$ of degree $11$ and
sectional genus $3$
inside a smooth hyperplane section $\breve{Y}$ of $\mathbb{G}(1,4)$.
 This leads us to find out  rational ordinary fourfolds in $(\mathcal M_4)_{26}^{''}$.

\subsection{Construction of \texorpdfstring{$\breve{T}\subset\breve{Y}$}{T in Y}}
Let $E'\subset\PP^6$ be the image of the plane via the linear system
of quartic curves with one double point $q_0$ and $5$ simple base points $q_1,\ldots,q_5$.
Then $E'$
is a smooth surface of degree $7$ and sectional genus $2$ cut out by $8$ quadrics.\footnote{The
surface $E'\subset\PP^6$
is a hyperplane section of a so-called \emph{Edge variety} \cite{Edge}.
This surface also occurs in the classification of \emph{special birational transformations},
see \cite[Table~4, Case~VI]{Sta19}.}
Let $E\subset\PP^5$ be the projection of $E'$ from a general point
on the secant variety of $E'$.
Then $E$ is an irreducible surface of degree
$7$, sectional genus $2$, cut out by $2$ quadrics and $5$ cubics,
with a single node as the only singularity, and having normalization
isomorphic to $E'$.
Take $B\subset\PP^5$ to be
a general smooth cubic scroll surface which cuts $E$
along a quintic elliptic curve (such a curve is obtained as the image on $E'$ and hence on $E$ of a general plane cubic curve
passing through the six base points $q_0,\ldots,q_5$).

The linear system of quadrics through $B$ defines a birational map
\begin{equation}\label{SempleMap}
 \PP^5\dashrightarrow \breve{Y}\simeq \mathbb{G}(1,4)\cap\PP^8\subset\PP^8
\end{equation}
onto a smooth hyperplane section $\breve{Y}$ of $\mathbb{G}(1,4)\subset\PP^9$.
The restriction of this map to $E$ induces an isomorphism
between $E$ and a surface $\breve{T}\subset\breve{Y}$.

This surface $\breve{T}\subset\breve{Y}$  shares all the properties of the surface $T\subset Y$
constructed in Section~\ref{particular smooth}.
In particular, $\breve{T}\subset\breve{Y}$ is
a one-nodal irreducible surface
of degree $11$,
sectional genus $3$,
cut out in $\PP^8$ by $16$ quadrics,
and having class $7 \sigma_{3,1} + 4 \sigma_{2,2}$
 in the Chow ring of $\mathbb{G}(1,4)$.
 A general quadratic section of $\breve{Y}$ through $\breve{T}$
 gives an ordinary Gushel-Mukai fourfold in $(\mathcal M_4)_{26}^{''}$,
 and such a fourfold is birational to a smooth $4$-dimensional
 linear section of $\mathbb{G}(1,5)\subset\PP^{14}$
 via the linear system of quintic hypersurfaces
 with triple points along $\breve{T}$.

\begin{remark}
By counting parameters, one sees that the family of the
reducible surfaces $B\cup E\subset\PP^5$ has dimension $48$, and hence
 the family of the surfaces $\breve{T}\subset \breve{Y}$ obtained by this construction
has dimension $48 - 35 + 15 = 28 < 29 = h^0(N_{\breve{T}/\breve{Y}})$;
see also \cite[Remark~2.4]{famGushelMukai}.
\end{remark}

\begin{remark}
 Our ancillary file (see Subsections~\ref{computation 1} and \ref{computation 2})
 also provides an explicit example
 of ordinary Gushel-Mukai fourfold corresponding to the pair
  $(\breve{T},\breve{X})$, where $\breve{X}$ is a general quadratic section of $\breve{Y}$ through $\breve{T}$.
 The file includes
 the parameterization of $E'$, the nodal projection of $E'$ onto $E$, the map \eqref{SempleMap},
 the dominant map $\breve{Y}\dashrightarrow \mathbb{G}(1,5)\cap\PP^{10}$
 whose general fibers are $5$-secant twisted cubic curves to $\breve{T}$,
 and the restriction of this map
 to $\breve{X}$ together with its inverse.
\end{remark}

\section{Summary table of examples}
Table~\ref{tabella1} summarizes some information about the
   Gushel-Mukai fourfolds known to be rational.
 It includes all the examples we found in the literature,
 the example constructed in the present paper,
 and some few others of which we omit the details.
 In all cases, except the 4th, the rationality of the fourfold follows from
  a congruence  of $(2e-1)$-secant curves of degree $e\geq1$.
\begin{table}[htbp]
\renewcommand{\arraystretch}{1.0}
\centering
\tabcolsep=0.3pt
\begin{adjustbox}{width=\textwidth}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\rowcolor{gray!5.0}
Ref. & {\begin{tabular}{c} Surface $S\subset Y$  \end{tabular}} & $K_S^2$ & {\begin{tabular}{c} Class in \\ $\mathbb{G}(1,4)$\end{tabular}} & {\begin{tabular}{c} Locus in $\mathcal M_4$ \end{tabular}}  & $h^0(\mathcal I_{S/Y}(2))$ & $h^0(N_{S/Y})$ & $h^0(N_{S/X})$ &
  \scriptsize{\begin{tabular}{c} Curves of degree $e$ in $Y$\\ passing though a general \\ point of $Y$ and that  are \\ $(2e-1)$-secant to $S$ \\ for $e=1,2,3,4,5$  \end{tabular}} \\
\hline \hline
\cite{DIM} & {\begin{tabular}{c} Quadric surface   \end{tabular}} & $8$  & $\sigma_{3,1}+\sigma_{2,2}$ & $(\mathcal M_4)_{10}^{'}$ & $31$ & $8$ & $0$ & $1$, $0$, $0$, $0$, $0$  \\
\hline
\cite{ProcRS}  &  {\begin{tabular}{c} K3 surface of degree \\ $14$ and genus $8$  \end{tabular}}  & $0$  & $9 \sigma_{3,1}+ 5 \sigma_{2,2}$ & $(\mathcal M_4)_{10}^{'}$ & $10$ & $39$ & $10$ & $9$, $8$, $1$, $0$, $0$   \\
\hline
\cite{Roth1949} & {\begin{tabular}{c} Plane   \end{tabular}}  & $9$  & $\sigma_{3,1}$ & \begin{tabular}{c} codim $1$ in \\ $(\mathcal M_4)_{10}^{'}$ \end{tabular} & $34$ & $4$ & $0$ & $1$, $0$, $0$, $0$, $0$ \\
\hline
\cite{Roth1949} & {\begin{tabular}{c} Quintic del Pezzo \\ surface   \end{tabular}}  & $5$  & $3 \sigma_{3,1}+2 \sigma_{2,2}$ & $(\mathcal M_4)_{10}^{''}$ & $24$ & $18$ & $3$ & $3$, $0$, $0$, $0$, $0$  \\
\hline
\cite{HoffSta} & {\begin{tabular}{c} Rational surface of \\ degree $9$ and genus $2$  \end{tabular}}  & $5$ & $6 \sigma_{3,1}+3 \sigma_{2,2}$ & $(\mathcal M_4)_{20}$ & $14$ & $25$ & $0$ & $6$, $1$, $0$, $0$, $0$ \\
\hline
  & {\begin{tabular}{c} Rational surface of \\ degree $13$ and genus $6$  \end{tabular}}  & $-2$ & $8 \sigma_{3,1}+5 \sigma_{2,2}$ & \begin{tabular}{c} locus in \\ $(\mathcal M_4)_{20}$ \end{tabular} & $10$ & $33$ & $4$ & $8$, $7$, $1$, $0$, $0$ \\
\hline
\cite{famGushelMukai} & {\begin{tabular}{c} Septic scroll  \end{tabular}}  & $8$ & $4 \sigma_{3,1} + 3 \sigma_{2,2}$ & \begin{tabular}{c} codim $2$ in \\ $(\mathcal M_4)_{20}$ \end{tabular} & $16$ & $21$ & $0$ & $4$, $1$, $0$, $0$, $0$ \\
\hline
\S \ref{sec Gushel} & {\begin{tabular}{c} Triple projection of a \\ minimal K3 surface of \\ degree $26$ and genus $14$  \end{tabular}}  & $-1$ & $11 \sigma_{3,1} + 6 \sigma_{2,2}$ & \begin{tabular}{c} locus in \\ $(\mathcal M_4)_{26}^{''}\cap \mathcal M_4^{G}$ \end{tabular} & $7$ & $37$ & $6$ & $11$, $22$, $32$, $6$, $1$ \\
\hline
\S \ref{particular smooth},\ref{sec Mukai} & {\begin{tabular}{c} Rational $1$-nodal surface \\ of degree $11$ and genus $3$  \end{tabular}}  & $3$ & $7 \sigma_{3,1} + 4 \sigma_{2,2}$ & \begin{tabular}{c} locus in \\ $(\mathcal M_4)_{26}^{''}$ \end{tabular} & $11$ & $29$ & $2$ & $7$, $4$, $1$, $0$, $0$ \\
\hline
\end{tabular}
\end{adjustbox}
 \caption{Rational Gushel-Mukai fourfolds $X$
 obtained as quadratic sections of
  $Y\simeq C(\mathbb{G}(1,4))\cap\PP^8$ through  surfaces $S\subset Y$.}
 \label{tabella1}
\end{table}

\begin{table}[htbp]
\centering
\tabcolsep=1.2pt
\begin{adjustbox}{width=\textwidth}
\begin{tabular}{cccc}
\hline
\rowcolor{gray!5.0}
Fourfold & \begin{tabular}{c} Irrationality of \\ very general \end{tabular} & \begin{tabular}{c} Description of \\ the rational ones \end{tabular} & Birational map to $\PP^4$ \\
\hline
\hline
 Quadric hypersurface in $\PP^5$ & no & all & projection from a point on it \\
\hline
\rowcolor{yellow!25} Cubic hypersurface in $\PP^5$ & not known & \begin{tabular}{c} just many examples but \\ with a precise conjecture \end{tabular} & \begin{tabular}{c} some constructions in \\ \cite{Morin,Fano,Hassett,RS1,RS3} \end{tabular} \\
\hline
\rowcolor{yellow!8} Quartic hypersurface in $\PP^5$ & yes \cite{Totaro2015} & no known examples &  \\
\hline
 \begin{tabular}{c}Complete intersection of \\two  quadrics in $\PP^6$ \end{tabular} & no & all & projection from a line on it \\
\hline
\rowcolor{yellow!8} \begin{tabular}{c} Complete intersection of \\ a quadric and a cubic in $\PP^6$ \end{tabular} & yes \cite{NOtte} & no known examples & \\
\hline
 \rowcolor{yellow!8} \begin{tabular}{c} Complete intersection of \\ three quadrics in $\PP^7$ \end{tabular} & yes \cite{HPT2} & just some examples & \begin{tabular}{c} \emph{e.g.}, when the fourfold contains a plane \\ then one takes the projection from it  \end{tabular} \\
\hline
 Del Pezzo fourfold $\mathbb{G}(1,4)\cap\PP^7$ & no & all & projection from the unique $\sigma_{2,2}$-plane \\
\hline
\rowcolor{yellow!25} \begin{tabular}{c} Gushel-Mukai fourfold \end{tabular} & not known & \begin{tabular}{c} just the examples in \\ Table~\ref{tabella1}  \end{tabular} & \begin{tabular}{c} see Table~\ref{tabella1} \end{tabular} \\
\hline
 \begin{tabular}{c} Linear section in $\PP^9$ \\ of the spinorial  $\mathbb{S}^{10}\subset\PP^{15}$ \end{tabular}  & no & all \cite{Roth1949} & projection from a tangent space \\
\hline
 \begin{tabular}{c} Linear section in $\PP^{10}$ \\ of $\mathbb{G}(1,5)\subset\PP^{14}$ \end{tabular}  & no & all \cite{Roth1949} & \begin{tabular}{c} projection from the linear span of a \\ quintic del Pezzo surface contained in it  \end{tabular} \\
\hline
 \begin{tabular}{c} Linear section in $\PP^{11}$  of the \\ Lagrangian Grass.
$LG(3,6)\subset\PP^{13}$ \end{tabular} & no & all \cite{Roth1949} & \begin{tabular}{c} linear system of hyperplane sections \\ through a line and with one double point \end{tabular} \\
\hline
 \begin{tabular}{c} Fourfold of degree $18$, \\ genus $10$, and coindex $3$ in $\PP^{12}$ \end{tabular} & no & all \cite{Roth1949} & \begin{tabular}{c} linear system of hyperplane sections \\ through a conic and with one double point \end{tabular} \\
\hline
\end{tabular}
\end{adjustbox}
\caption{Rationality of smooth prime Fano fourfolds of coindex $\leq 3$, \cite{fujita-polarizedvarieties,mukai-biregularclassification}.}
\label{tabella2}
\end{table}

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redpajama_set_name: RedPajamaArXiv


Why would I care where Bucky is?
Troy is only making believe that he doesn't know.
Ramon had a ring on every finger of her left hand.
Raphael motioned for Rogue to answer.
Todd slipped some money into Pandora's hand.
Dory said Kirsten told him that.
Rebecca likes to sleep late.
Del had no idea how busy Presley was.
Carisa posted the picture on her wall.
Juliane may not want to do that, but he needs to.
Trey talked Gabriel into buying his old car.
Maria saw John and Roy holding hands.
Samir was hospitalized with symptoms of dehydration.
Sam and Jones like to gossip about celebrities.
Murph is very skilled in manual labor.
Jin pushed his plate away and said that he had eaten enough.
I didn't even hear Joon.
Sumitro doesn't have a lot of time.
Griff's mother died in 2013.
Archie wasn't wearing a seatbelt.
Julie won by three points.
Tad is neat, isn't he?
Sherri works as a waitress at a local cafe.
Brenda was elected class representative.
Skeeter said I was boring.
Why do you think Raja likes baseball so much?
Jason is an aspiring musician.
Part had lots of run-ins with the law when he was a teenager.


In today's technology-driven world, data center disruptions can be disastrous to a company's profits and reputation. So, when it comes to data centers, reliability and availability are the top priorities.
The only way to know if a data center truly meets the reliability goals required by your organization, is to properly execute a solid commission procedure. Parallel's definition of commissioning is not only component and system level testing, but starts at the beginning of the project with initial strategy and design.
We believe that data center design and ultimate construction should be based around the client's company's goals, objectives and reliability needs- not just for the present situation, but the future trajectory of the organization. Our rigorous process ensures the project requirements and objectives are reflected in the design, and we establish a process to make sure the design is built and functions as intended.
From system redundancies to site selection and internal and external building construction, Parallel Technologies identifies and evaluates risks and works with you to implement strategies that mitigate potential disruptions. Our data center design / build process means total accountability for your project's success.
We work with our clients to verify, validate and document that the facility and all of its systems (mechanical and electrical) are planned, designed, installed, tested, operated and maintained to meet the client's needs and desired Tier Level.


31st December 2016 - Minister Shane Ross supports Deputy Broughan's call for increase in points for mobile phone use. See Irish Times Report..
21st December 2016 - Clancy family tragedy leads to major change in learner driver law.
PARC Road Safety Group has organised a special ceremony to remember all road traffic victims in Loreto Sports Complex, Fermoy, Co Cork on Sunday 20 Nov. at 2pm. All are welcome.
17th November - Yet again Deputy Tommy Broughan reveals shocking facts on speeding and penalty points. Link to Shane Phelan Article, Irish Independent..
15th November 2016 - bereaved family thank PARC for producing Finding Your Way guide.
4 November 2016 - Deputy Broughan highlights scandalous Traffic Corps cutbacks. Irish Times article..
PARC helps man honour memory of his unborn grandchild..
Please Support 'European Day Without A Road Death' - Project EDWARD We in PARC Road Safety Group are proud to support "Project EDWARD" which aims to be a European Day Without A Road Death and takes place on Wednesday, the 21st September 2016.
"In whatever way we use the roads we can each play our part, a day at a time to ensure that nobody is killed or seriously injured. There is no acceptable number of deaths and injuries on our roads. We are all the loved ones of others."
I would like to put on public record the huge help PARC Road Safety Group has been to both the Enright and Walsh families over the last four years.
In 2012 both my daughter Mary and unborn granddaughter Mollie Enright were killed in an RTC. For the last four years I have campaigned to have my granddaughter Mollie recognised as a road traffic victim. THIS HAPPENED IN 2016 WITH THE ROAD STATS FOR 2012 BEING CHANGED FROM 161 TO 162. This is the first time in Ireland and Europe that an unborn child has been registered as a road traffic victim.
Without the huge assistance from PARC this might never have happened.
8 September 2016 - PARC very much welcomes the changes to the driver theory test from 10 October. We congratulate the RSA on this initiative. However we would also like to see questions on drug driving - perhaps these will follow legislation on drug driving presently before the Dail in the 2016 Road Traffic Bill. It is reasonable to state that these changes to the driver theory test will only be effective where we have much greater enforcement of drink and drug driving laws by An Garda Siochana.
30 August 2016 - After PARC's eight year long campaign An Garda Siochana finally waive the fees for copy witness statements provided to bereaved families of RTC victims. This will take effect from 1st January 2017.
PARC welcomes the news that the Garda Commissioner has announced her decision to waiver all fees for abstracts of Garda reports, copies of witness statements and sketches or maps of the scene in respect of a fatal collision for bereaved families. Up to now families had to pay €60 for the Garda Abstract report and €40 of each copy of a witness statement which in some instances could amount to thousands of euro. PARC campaigned since 2008 to have this unjust, insensitive and financial burden to be lifted from the shoulders of grieving families.
We are grateful to the Garda Commissioner for the opportunity to meet with her and her senior officials recently along with the bereaved to explain the pain and difficulties that families have to journey through. We are grateful for her time and empathy and her decision to remove these fees. PARC is happy that the Commissioner has opted for a compassionate approach to this issue.
We thank all those individuals who supported our campaign and stayed with us in our efforts through thick and thin over the past eight years.
www.parcroadsafety.ie PARC Road Safety Group is a national civil society organisation. It was founded by Susan Gray in 2006 after the death of her husband in a RTC on 27 December 2004 in Donegal. The group is made up of people who have been affected by road traffic collisions and those with an interest in road safety.
10 August 2016 - A tried and tested rehabilitation course is only effective when used in conjunction with the legal penalty related to the charge.
Link to Donegal Highland Radio news.
Link to Donegal Pro-Social Ireland Oireachtas Presentation ......Where are the research figures to support the claims?
3 August 2016 - Law to allow Garda cars to scan for uninsured drivers must be fast tracked.
Link to Donegal Highland Radio News today ..
24 July 2016 - Increase in drivers being disqualified but problems remain with incomplete recording of driver disqualifications. See report by Michael Brennan " Spike in driving bans under new speeding penalty points regime".
July 2016 - Garda Commissioner meets with PARC members of RTC victims.
Garda Commissioner meets with PARC members of RTC victims.
PARC Road Safety Group had a very satisfactory meeting with Garda Commissioner Noirin O' Sullivan, Deputy Commissioner John Twomey and their team at Garda Headquarters on Friday 15th July. The purpose of the meeting was to allow family members of road traffic collisions who were dissatisfied with their treatment by certain members of An Garda Síochána subsequent to their loved one's death to be heard. The Garda Commissioner listened very attentively to the families concerned and directed her team to assist them to ensure that they receive the excellent service from An Garda Síochána that she expects of them. The families were very pleased with the meeting and hope to hear from Garda HQ presently. We will continue to work closely with Garda Headquarters to resolve any outstanding issues including concerns around road safety.
30 June 2016 - PARC representatives attend The Policing Authority meeting with the Garda Commissioner today.
Statement following the Policing Authority Meeting with the Garda Commissioner below..
A recording of this meeting is available on Youtube click here.
15 June 2016 - The RSA is finalizing a proposal to the Department of Transport recommending that drivers disqualified for drink driving must resit and pass the theory and practical driving test before they are entitled to the return of their driving licence.
Link to Claire Byrne Live episode 19..
30 May 2016 - Representatives from PARC met with new Minister for Transport Shane Ross today to discuss road safety issues.
28 May 2016 - Drivers in Inishowen in Donegal tops the list in speeding offences as National Slow Down day concludes!
Link to Highland Radio news.. 27 May 2016 - Lives will be saved by Garda decision to increase GoSafe camera zones.
As from today there are now 1,031 sections of road identified as speed enforcement zones. The decision for additional camera zones was made following an extensive Garda analysis of road traffic collisions where speed was a contributory factor. The aim is to reduce the number of speed related collisions, save lives and reduce injuries on our roads. As of today's date, 13 more people have died on our roads compared to this date last year, with excessive and inappropriate speed being a major factor in many of these collisions.
Link to Irish Times article..
20 April 2016 - Minister Donohoe damns use of Court Poor Box.
16 April 2016 - PARC welcomes introduction of penalty points for worn tyres.
Link to Irish Independent report today.
23 March 2016 - Over half of learner drivers are not paying their Fixed Charge Notices while RSA figures show that unaccompanied learner drivers are involved in five times more fatal crashes than accompanied learners.
14 March 2016 - Bereaved families of RTC victims charged thousands for information relating to their loved one's death.
Link to report today by Fiona Gartland of the Irish Times.
22 February 2016 - The Social Democrats have fully endorsed and committed themselves to PARC road safety group's manifesto.
21 February 2016 - PARC still waiting for our Taoiseach to sign lifesaving manifesto!
Fianna Fail Spokesperson for Transport Timmy Dooley has signed PARC's road safety manifesto.
Sinn Fein Spokesperson for Finance Pearse Doherty signed our manifesto on his own behalf and on behalf of his party.
12 February 2016 - The following election candidates have been handed a copy of PARC's road safety manifesto today in Cork.
10 February - Micheál Martin is first Party Leader to sign lifesaving Road Safety Manifesto!
Micheál Martin has been the first party leader to sign PARC's manifesto for road safety which has been sent to all party leaders. We await the commitment of the other leaders for our 'Campaign Priorities in road Safety'.
4 February - RTE today show. Click here for link..
28 January - PARC member Ann Fogarty's interview with Ryan Tubridy today.
The Enright and Walsh families have spent the past 4 years looking for justice for their unborn baby Mollie Enright killed in a car crash in which her mother Mary Enright (nee Walsh) died on 26 March 2012. Until now the death of an unborn child killed in a road traffic collision was not recognised by the State.
Yesterday's ruling by Coroner Paul Morris changed all that. He will issue a death certificate for little Mollie.
This sends out a very important message to families of victims of Road Traffic Collisions - to have the courage and perseverance to seek justice in the name of their lost loved one. "Courage isn't having the strength to go on - it is going on when you don't have strength". Click here for link..
25 January - Summary of PARC's Road Safety Election Manifesto 2016.
PARC has drawn up a list of priorities for road safety that would substantially reduce the level of deaths and injuries on Irish roads. We must adopt a long-term vision for a transport system where no loss of life or injury is acceptable. Unless we have an improved system for apprehending and sanctioning law breakers we will never succeed in eliminating the main contributory factors to road deaths and injuries.
We will be asking all party leaders and candidates to sign a commitment to these priorities and will publish the names of those who support our campaign, on our website.
Equip all relevant state bodies with new technologies linked to a central database.
Provide effective recording of licences and precise reporting systems.
Eliminate barriers to successful prosecutions for drink driving.
Improve knowledge and understanding of road traffic collisions and casualties.
Enact legislation, inform Gardai and insist on correct procedures in court.
Consolidate the Road Traffic Acts.
Independent driving in training and on-road assessment on a pilot scheme basis.
Government publication of 'Finding Your Way'.
21 January - Fit for purpose integrated database/information sharing programme needed for road traffic offences. Link here..
20 January - Deputy Tommy Broughan already follows up our road safety manifesto query re serious injuries.
Parliamentary questions were submitted to the Minister for Health Mr Leo Varadkar by Deputy Broughan this week.
Deputy Tommy Broughan's Statement on PARC's Election Manifesto today. Link here..
19 January - Mark Hillard of the Irish Times reports on PARC's road safety election manifesto "Campaign Priorities for Road Safety". Link to report..
Congratulations to Mr Noel Brett former CEO of the Road Safety Authority on his appointment to the new Policing Authority of Ireland. We wish him well.


This week's featured invention patented by Pennsylvania inventors is within CPC Section D: Textiles & Paper.
Abstract: The invention relate to fibrous mats comprising chitosan nanofibers and, optionally, at least one filler material, at least one additive, or both. The invention also relates to methods of making same, and devices that include a fibrous mat comprising chitosan nanofibers.
This week's featured invention patented by a Pennsylvania organization is within CPC Section H: Electricity.
Abstract: A cable contact module of a cable connector assembly includes a dielectric frame, signal contacts, and a ground frame. The signal contacts, held by the dielectric frame, include contact beams that extend from a front of the dielectric frame and electrically connect to signal contact pads on a mating circuit card. The signal contacts are terminated to corresponding cables that extend from a rear of the dielectric frame. Each cable includes at least one center conductor housed within a cable shield. The ground frame includes a ground plate and integral ground beams. The ground plate is mounted to the dielectric frame and engages the cable shields of the cables. Each ground beam extends from a front edge of the ground plate into a void defined between adjacent pairs of the contact beams. The ground beams are configured to electrically connect to ground contact pads on the mating circuit card.


text:
Mayor Announces Laguna Honda Tops Nursing Home Ratings
Mayor's Press Office
MAYOR LEE ANNOUNCES LAGUNA HONDA TOPS
U.S. NURSING HOME RATINGS
City's Long Term Care & Community Rehabilitation Hospital
Reaches Top Rating for Quality
San Francisco, CA -- Mayor Edwin M. Lee today announced the U.S. Centers for Medicare and Medicaid Services raised the quality rating for San Francisco Department of Public Health's Laguna Honda Hospital and Rehabilitation Center to the highest rating of five stars. It is the third quality rating increase for Laguna Honda in two years.
"A five-star Laguna Honda confirms San Francisco's status as a health care leader," said Mayor Lee. "We have made real progress to meet the needs of seniors and residents with disabilities for long term care and community integration."
"It's a new era for Laguna Honda," said Laguna Honda Hospital and Rehabilitation Center Executive Administrator Mivic Hirose. "Laguna Honda is the most modern nursing facility in the country, and the federal recognition certifies that the quality of our care matches the quality of our built environment. San Franciscans can expect the best from us," she said.
"This is a testament to the public health vision of the City's voters," said DPH Director of Health Barbara Garcia. "When voters overwhelmingly agreed to rebuild Laguna Honda in 1999, they were making an investment in the future health of the city. The five-star rating is confirmation that the Department of Public Health has kept its promise to deliver a new center of excellence at Laguna Honda."
The Centers for Medicare and Medicaid Services Nursing Home Compare report rates facilities on three criteria: staffing, health inspections and quality measures. The hospital is rated at five stars on the staffing and quality measures categories. The health inspection rating is three stars. The hospital's goal is to attain five stars in all categories by 2014.
Laguna Honda is a branch of the San Francisco Department of Public Health, providing acute care, rehabilitation and skilled nursing services to a safety net population of 765 seniors and adults with disabilities.

redpajama_set_name: RedPajamaCommonCrawl


Creative Learning sits at the heart of Wells Maltings; we create hands-on learning opportunities with audiences, artists and academics, which provide schools and the wider community with new and exciting avenues of engagement with our programme of theatre, dance, music, film and visual arts.
Culture and art are integral parts of local development. From a social perspective, they are essential components in enhancing quality of life, as the more choices people have to participate in cultural and artistic events, the more they experience personal satisfaction. From an economic perspective, the development of the cultural sector in a rural area contributes to creating more jobs. It also has an impact on increasing revenues as a result of selling goods and services, or enhancing cultural tourism activities in rural areas.
From exciting, one-off taster events to our ambitious recent Driftwood programme with Wells Primary School and Alderman Peel High School, our work nurtures curiosity and enriches learning, which we believe is a lifelong entitlement for all.
Young People – by working with teachers to shape the learning offer and programme and through offering a combination of tailored, independent and outreach sessions, Wells Maltings is best placed to develop meaningful, sustainable and effective partnerships with key local learning audiences.
Families – plans are underway to work together with families to create workshops that parents and children can continue themselves at home to encourage curiosity and play. Also in the pipeline are grandparent and grandchildren workshops.
Our recent projects have included a sensitivity to the local landscape including the saltmarsh, beach and the surrounding countryside. We seek to include the interaction between cultural heritage, contemporary art and collective memory.
The Wells Maltings Learning Team seeks to preserve traditional or classical forms of arts and culture, exploring contemporary methods while incorporating individual unique expression.


text:
An Eccentric Novelist in the War – May '99 World War II Feature
Author Evelyn Waugh served honorably in the British Army as an SAS Commando.
By Paul S. Burdett, Jr.
Even at the still-vigorous age of 36, he could not be mistaken for a warrior. His physique already tended toward the pudgy. His ears were too large for the bubblish head that wore them. His bug eyes gave the impression of a man perpetually befuddled. His wispy moustache had no rakish appeal. His taste in suits was deplorable. Even worse, his parents had tagged him with the androgynous Christian name "Evelyn" (pronounced with a hard "E").
As acquaintance Mary Pakenham remarked, he "was one of the few people who was not made more distinguished-looking by wearing uniform." Yet, eccentric Evelyn Waugh would become an officer and play a part in some of the most hazardous, though little celebrated, British covert operations in the Mediterranean theater.
By the time Adolf Hitler ignited the European conflagration, Waugh had already built a considerable literary reputation on his deliciously dark-humored novels: Decline and Fall (1928), Vile Bodies (1930), Black Mischief (1932), A Handful of Dust (1934) and Scoop (1938). He had already fallen in love, married, divorced, converted to Catholicism, developed an affection for wine, remarried, become a father, and befriended some of the most "mucky" of the English high muck-a-mucks–including Randolph Churchill, the rambunctious son of the future prime minister.
His home and artistic life was nearly idyllic, yet he was not a happy man–bored with the superficial social whirl that he parodied with such sublime acidity in fiction, prone to bouts of depression due to the evil he clearly perceived across the English Channel, haunted by the spiritual impotence peculiar to approaching middle age. Waugh's ennui was later vented by his Brideshead Revisited (1944) protagonist, Charles Ryder: "Here at the age of thirty-nine I began to be old." Thus far he had been an observer of life. Now he wanted to do his bit.
Waugh pestered his well-heeled pals to wangle him a commission. "He had no use for cloud cuckoo war aims," his friend and biographer Christopher Sykes explained. "The only war aim was to win by killing great numbers of the enemy's population and preventing them killing great numbers of one's own; as for the utopians and air castles, they seemed to him not only silly but malign." Waugh was not bloodthirsty, he just had an accurate picture of victory's cost.
Alas, there were plenty of young chaps in '39 who would do very nicely. The longish-in-the tooth novelist finally received a post in the Royal Marines, thanks mostly to the pull of cigar-chomping First Sea Lord Winston Churchill, who appreciated Waugh's moxie. When Waugh appeared for his physical, the examining doctor ignored his unfavorable health report and brushed aside his poor eyesight, saying, "Anyway, most of your work will be in the dark."
Colonel Godfrey Lushington, Waugh's commander during training, rated him as having "any amount of moral courage"–which he had already displayed in his writing–"and [he] has self confidence when on subjects he knows. A little impatient, [but] with more military experience…he will make a first class Company Commander."
The colonel obviously did not spend much time with his pupils. Although Waugh was quickly promoted to captain, Christopher Sykes knew that Evelyn was "utterly unfit" to be a line officer. A jolly companion with his peers, Waugh was often curt and contemptuous with the other ranks. He had no idea how to lead men, and nothing in his training had prepared him for effective command. When he ranted at the men in language beyond their meager schooling, the lads usually responded, "What is Captain Wuff on about now?" The happy-go-lucky "writer chappie" was learning that regular duty was not his cup of tea, and his attitude began, ever so slightly, to sour. It was a subtle transformation that he later described with such pathos in the first volume of his war trilogy, Men At Arms.
Moreover, the action to which his battalion was assigned did not inspire dreams of glory. The men spent a week in South Wales cleaning up a deserted camp. Then, after much confusion, it was off to Cornwall. "Our task was to defend Liskeard," Waugh noted. "None of us can quite make out why anyone should want to attack it." Although he was finally in service, he began to despair of ever seeing real action.
Then, via Brendan Bracken, a longtime acquaintance of Waugh's who was now serving as Prime Minister Winston Churchill's parliamentary private secretary, came a ray of hope. Sir Roger Keyes, director of combined operations, was raising a volunteer Commando force under Lt. Col. Robert Laycock, another of Waugh's prewar London chums. "You ought to be with Bob Laycock's tough boys," Bracken advised. Waugh immediately agreed, since Laycock had a sterling reputation as a combat officer. The Marines had conducted an exercise with the Commandos at Liskeard, and Evelyn was smitten by the tough-looking Commando warriors.
Before Waugh's application could be approved, however, the Marines were hustled off to help Free French General Charles de Gaulle in Operation Menace, his expedition to seize Dakar, Senegal, from Vichy control. As Prime Minister Churchill later explained: "Our information was that a large portion of the French officers, officials, and traders in all these [African] territories had not despaired….To them General de Gaulle shone as a star in the pitch black night."
On September 14, 1940, Waugh's intelligence group landed at Freetown, Sierra Leone, 400 miles south of Dakar, and prepared for the main force's arrival. Waugh and his mates busied themselves depleting the local gin supply and surveying the girls on the beach. There was not much pertinent intelligence to gather.
When Menace kicked off on the 23rd, the Franco-British force was bedeviled by heavy fog and soon found out that the Vichy authorities had been strongly reinforced by very accurate shore batteries as well as naval and air assets. Two more attempts were made on the 24th and 25th, but, contrary to rosy expectations, the garrison put up a spirited defense, repelling de Gaulle's attempt to get ashore in strength.
The dejected invaders called it quits and retreated to Freetown to lick their wounds and drown their frustrations. As the prime minister later wrote, "To the world at large it seemed a glaring example of miscalculation, confusion, timidity, and muddle." To the lads in the saloons, it was a thorough "balls up," and Waugh griped to his wife Laura that "bloodshed had been avoided at the cost of honor."
Menace's failure was not the only thing eating at him. Waugh was considerably older than most of his comrades, and the Marines' youthful featherheadedness was growing tiresome. "I do wish I could meet an adult," he lamented to Laura. "They are all little boys…not one of them a mature man." This funk glared forth from the cynical Put Out More Flags, written during a dull stint as mail censor at Gibraltar.
In November, however, Waugh's spirits soared when the transfer to 8 Commando finally came through and he joined the Special Air Service (SAS) unit at Largs, Scotland, finding there a most congenial collection of "smart set" chaps. "All the officers have very long hair & lap dogs & cigars," he wrote to Laura. "I have done nothing so far except take a cuckoo clock to pieces & play a lot of ludo." The "tough boys" were not doing much that seemed so tough, still, "the whole thing was a delightful holiday from the Royal Marines."
His joy was tragically marred on the 30th, when his infant daughter Mary died shortly after birth. "Poor little girl," he grieved, the brevity of expression concealing the depth of his bereavement.
The war, however, did not allow much time for mourning and, after yet more mucking about, 8 Commando was finally shipped off to Egypt. Alas, no one at the Alexandria headquarters quite knew what to do with this new force, so Laycock's chaps were put on guard duty. But in April 1941, it looked like they were finally going to get "a biff" out in the desert. Headquarters could not figure out what Rommel was up to, so on April 19, 8 Commando unit was sent on a night raid against Bardia, behind the Afrika Korps lines in Libya. Waugh thought it might prove amusing.
As it turned out, the operation was little more than a Chinese fire drill. The first troops ashore accidentally killed their officer; another group wandered down the wrong wadi and were left behind and captured; one of the too-few boats was hopelessly grounded; and their only opposition was a pair of startled patrolling motorcyclists who skedaddled before anyone could draw a bead. The debacle, however, was not as demoralizing to Waugh as the ensuing cover-up. For his part, Waugh had to toe the headquarters line, but in his diary he vented considerable spleen about Colvin's incompetence, the juvenile conduct of the fiasco and his having to lie about it.
Waugh's disgust was leavened a bit in May 1941, when "Layforce," as Waugh called it, got another mission–this time to assist in the rescue of the Crete garrison. The British knew that German General Kurt Student's hardened paratroops of the 11th Air Corps were prepping to seize the strategic isle in a massive airborne assault (Operation Mercury). Back in Berlin, however General Franz Halder, chief of the German general staff, worried that "The delay for Barbarossa [the German invasion of the Soviet Union] resulting from this operation is very awkward."
As Waugh's unit steamed into Suda Bay aboard HMS Abdiel at midnight on May 26, "what they found…was nothing less than a vision of hell," wrote Waugh biographer Selina Hastings. "The harbour was clogged with the masts and funnels of sunken vessels, some of them still smoldering; the quay, heavily cratered by bomb attack, was littered with burned-out vehicles and abandoned stores, among which were slumped small groups of wounded men. While Laycock and his men were waiting to go ashore, a man wearing only shorts and a greatcoat burst into the captain's cabin, clearly half out of his mind with terror….Evelyn, openly contemptuous, observed with distaste, 'We took this to be an exceptionally cowardly fellow, but in a few hours realized that he was typical of British forces in the island.'"
Waugh displayed commendable fortitude and plain, old-fashioned guts in gathering useful intelligence, collecting groups of dazed Tommies and directing them safely to the embarkation area, and helping to organize and execute what Churchill called "a strong rear-guard action" that made a successful evacuation possible. Laycock was impressed by the "cool courage" Waugh displayed even during the nerve-shattering Junkers Ju-87 Stuka dive-bombing attacks, which Waugh merely thought monotonous and described as being "overdone, like everything German."
When Waugh returned to Alexandria, Chris Sykes noticed that he was very angry: "He said that he had never seen anything so degrading as the cowardice that infected the spirit of the army. He declared that Crete had been surrendered without need; that both the officers and men were hypnotized into defeatism by the continuous dive-bombing which with a little courage one could stand up to; that the fighting spirit of the British armed services was so meagre that we had not the slightest hope of defeating the Germans; that he had taken part in a military disgrace, a fact that he would remember with shame for the rest of his life."
Waugh had formerly viewed the war as a noble crusade against the evil represented by the swastika. Although he was far from being an expert intelligence analyst, he fully realized that there was no material reason for Crete's loss–it had been a failure of will, not resources.
For the next three years Waugh was placed on extended leave. The world is much richer for it because during that hiatus he penned his beloved masterpiece, Brideshead Revisited, in which he managed to come to grips with the erosion of sacred ideals and the cloying tawdriness of much that he had witnessed.
With practically all of Britain's resources tied down in Italy or committed to preparations for the cross-Channel invasion, Prime Minister Churchill still desired an active role in support of Marshal Tito's partisans in Yugoslavia and sent his own son, Randolph, to command the mission to Croatia. Randolph requested the service of his London drinking buddy, Evelyn Waugh.
Waugh's selection for this mission puzzled Frederick Lord Birkenhead at the time because he knew the novelist detested communism and Stalin in particular, "whom he believed…to be one of the arch-murderers of history." As Birkenhead put it, "To Evelyn, our alliance with Russia completely deprived the war of the crusading element with which he had first invested it, and our Partisan colleagues' derogatory remarks about the Germans…were to him merely a particularly nauseating example of Satan rebuking Sin. One could not fail to recognize that holding these views so strongly, he could be of little, if any use as a liaison officer with Communist allies."
Churchill thought it wise to help Tito, but that did not mean he trusted the Yugoslavian leader. He hoped that having Waugh along more as an observer than actual liaison officer could help balance reportage from the Croatian contingent–especially since the team's area expert, Stephen Clissold, was believed a bit too "pink" in his sympathies. Further, as Waugh noted in his diary, Randolph "asked me to go with him to Croatia in the belief that I should be able to heal the Great Schism between the Catholic and Orthodox churches."
Waugh was already recognized as one of the more prominent Roman Catholic laymen of his day. The religious schism in Croatia was indeed detrimental to partisan operations, since the feuding factions were battling each other almost as much as they were the Germans. Hence, if Waugh could bring his influence to bear to close this rift, he would surely help the war effort. Ironically, Waugh was available for this job because he had been a victim of the anti-Catholic purge of the Commandos by Lord "Shimi" Lovat, of later Pegasus Bridge fame on D-Day, and had been transferred to the Royal Horse Guards.
Waugh's part in this diplomatic effort is told in his fiction, via alter ego Crouchback, and the Yugoslavian episode is the focus of The End of the Battle. That operation would also be a sad, though not altogether unexpected, disappointment. For one thing, controlling Randolph Churchill was easier said than done. His father had had no luck on that score, and Waugh–who was much like Randolph in many respects–would be no more successful. Alas, what makes for rum-chumminess in the clubs of London does not necessarily auger as well in the close confines of wartime clandestinity.
While Waugh approached his duties seriously–even going entirely on the wagon during that time–Randolph Churchill partook, with gusto and great regularity, of the local beverages, rakija and grk (which could also be used to fuel a truck), and he succeeded in alienating most of the partisans he was supposed to assist. To keep himself in good spirits, Waugh fashioned a fantasy (one that later proved embarrassing to its author) that Tito was a woman. It was a most amusing story, but the Englishmen began to have "vague forebodings" about spreading the tale, especially after Tito himself got wind of the joke.
In any event, the unavoidable friction between Waugh and Churchill was not the main cause of the mission's failure. They tried to observe what they could. Their isolation in Topusco was not helpful, but the greatest obstacle was politics. If the prime minister did not trust Communists, they did not trust the British either. Hence, most of Waugh's and Randolph's efforts were wasted, except to gain a fairly accurate impression of the ruthless type of men who would take over Yugoslavia after the war.
For his part, Waugh was actually more successful than all the rest combined. At great risk to his own safety, he frequently evaded his Red watchdogs and contacted local Catholic clergymen, both Roman and Orthodox, to get a candid appreciation of Christian conditions in Croatia. His questions about Jews had long since already been answered by the Nazis.
He produced a very thorough and detailed report that–by March 1945–finally found its way to the Foreign Office. It was a remarkable and alarming document. Although he could not and certainly would not say that the churches had been treated well under the German occupation, they had at least been allowed to exist. Under Tito's Communists, however, they rapidly disappeared. As Communist power and territorial control increased, Christianity was eradicated–sometimes by the pen, most often by the sword. Moreover, the sectarian rift had been less religiously inspired than actively abetted by the Communists. It was clear that the persecution and murder of clergy, and the disappearances of lay faithful, were Tito's doing.
Unfortunately, Tito had by now become Britain's new friend, and no one wanted to hear Waugh's criticisms. At British Foreign Secretary Anthony Eden's direction, a campaign was mounted to discredit Waugh's report and its author, even going so far as to threaten Waugh with prosecution under the Official Secrets Act. But the truth has an inconvenient resistance to such baldfaced manipulation. The report was too detailed and consistent to be a fabrication, and whatever his other faults Waugh was widely known as a scrupulously honest man. Moreover, one can easily imagine the royal stink that his literary pals–most of whom, ironically, were Labor Party supporters–could have raised if such threats had been carried out.
In the end, as often happens in bureaucracies, the offensive document was buried in the Whitehall files, and thus ended the wartime escapades of "Captain Wuff." However, a reasonably fitting coda to this unsavory episode was provided by Sir Anthony Rhodes in The Vatican in the Age of the Dictators. After chastising the government for it shameful behavior in this affair and indicating the ethical muddles produced by the "enemy of my enemy is my friend" policy that produced the 1945 whitewash of Tito, Rhodes vindicated Waugh, saying that "events in the Communist 'satellite' countries since 1945, still not abated today–the arrests and persecution of priests–would seem to confirm that Captain Evelyn Waugh's diagnosis of Communist persecution of the Church…was broadly correct."
The story of Waugh's wartime service ends much as the novelist would conclude his story of Captain Crouchback in The End of the Battle. One could likewise say of Evelyn Waugh that things "turned out very conveniently." Back at home with Laura and their children, he wrote to his friend and new Labor parliament member, Tom Driberg, that he could–after all that had happened during the war–face the future "with fortitude" at least, if not joy. Evelyn Waugh died in 1966 at age 63. [ TOP ] [ Cover ]
HistoryNet Staff (1/30/2023) An Eccentric Novelist in the War – May '99 World War II Feature. HistoryNet Retrieved from https://www.historynet.com/an-eccentric-novelist-in-the-war-may-99-world-war-ii-feature/.
"An Eccentric Novelist in the War – May '99 World War II Feature."HistoryNet Staff - 1/30/2023, https://www.historynet.com/an-eccentric-novelist-in-the-war-may-99-world-war-ii-feature/
HistoryNet Staff 8/19/1999 An Eccentric Novelist in the War – May '99 World War II Feature., viewed 1/30/2023,<https://www.historynet.com/an-eccentric-novelist-in-the-war-may-99-world-war-ii-feature/>
HistoryNet Staff - An Eccentric Novelist in the War – May '99 World War II Feature. [Internet]. [Accessed 1/30/2023]. Available from: https://www.historynet.com/an-eccentric-novelist-in-the-war-may-99-world-war-ii-feature/
HistoryNet Staff. "An Eccentric Novelist in the War – May '99 World War II Feature." HistoryNet Staff - Accessed 1/30/2023. https://www.historynet.com/an-eccentric-novelist-in-the-war-may-99-world-war-ii-feature/
"An Eccentric Novelist in the War – May '99 World War II Feature." HistoryNet Staff [Online]. Available: https://www.historynet.com/an-eccentric-novelist-in-the-war-may-99-world-war-ii-feature/. [Accessed: 1/30/2023]


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FIS - Foundation for Infinite Survival, Inc. (Est. 1972)
Life-Extension & Control of Ageing Program
{ Berkeley, California }
www.fis.org
Register - mailing list
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Life-Extension Science & Technology
Among the many issues and conflicts which now confront us, the single most important challenge is the construction of a new philosophy - a philosophy which is based on fundamental principles and perennial and immutable truths, which is future oriented, captures the best potential of each individual, and organizes networks of association that will enable effective adaptation in an ever changing and seemingly chaotic world.
The Ideology of the Goal of Infinite Survival is presented as such a philosophy.
Based on the immutable and perennial truth of self-survival as the fundamental purpose of life, that idea is extended to its ultimate conception - immortality or infinite survival. Using that as the ultimate purpose or Goal, together with the Individual Self as the point of reference (or the "ontology") and Scientific Empiricism as the method for establishing knowledge (or the "epistemology"), within that ideological frame-work, a new conception of self and how to construct knowledge is created, which then generates an array of necessary and evolutionary objectives.
Our main enemy is our beliefs and the inherent defects of our nature.
"It is only through a change in human consciousness that the world will be transformed. The personal and the planetary are connected. As we expand our awareness of mind, body, psyche, and spirit, and bring that awareness actively into the world, so will the world be changed." Michael Toms - New Dimensions Radio.
History and the human condition are framed by advances in scientific knowledge; and history and the human condition are forced to change by the technologies which result from such advances in scientific knowledge. Nothing in the past will have been comparable to the force of Life-Extension Science and Technology.
We stand at a kairos in the evolution of human nature and individual potential, the key-stone of which is Life-Extension Science. The ultimate objective of this science is the invention of non-ageing humans through advances in regeneration biology.
Life-extension technology is emerging in Phases. Currently, practical applications in optimizing one's environment and personal behavior, together with some medical applications, should enable those people, who are so interested, to achieve a functional life-span of 80-90 years. In the course of that, some procedures may soon be available to slow biological ageing and thereby extend, further, the functional life-span.
Ultimately and much sooner than most people might imagine, developments in the field of molecular and cellular biology offer the prospect of restoring a person's body back to the condition which one had at the age of 20 ± 5 years of age. In such a state of biological vitality, performance would be optimal, disease would be minimal and, if it did occur, easier to treat effectively, and the potential life-expectancy would be greatly extended, rendering death to being mostly voluntary rather than imposed by one's own nature. The lead technology is stem cell biology and eumitogenic agents.
This is a bold pursuit which will appeal to only a small percentage of people at this time; but world-wide, their number is large and more than sufficient to bring this science into fruition, with their involvement.
The critical element is the plan and the people.
For further discussion the Philosophy, see this link. For further discussion of Life-Extension Science & Technology, see this link.


USPages - Retail Grocery - Safeway Inc.
USPages > Business Directory > Retail Grocery > Safeway Inc.
Safeway Inc. achieved revenue of $40,850.70 million in 2010 but posted a loss of $-1,097.50 million.
The company's ranking in 2010 was a decline from 2009 when it was ranked 50th on the Fortune 500. The Previous ranking was based on its 2009 revenue of $44,104 million (a change of -4 million compared to 2010) and profits of $965 million (a change of -966 million compared to 2010).
To many investors, profits are more important than revenue a company earns, If the Fortune 500 was ranked by profits instead of revenue then Safeway Inc. would be ranked 466th rather than 52nd due to the fact it made a loss of $-1,098 million in 2010.
Previous Page: the 51st ranked company MetLife, Inc.


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FGGA Research Seminar: When all else Defaults: Government as the Ultimate Debtor?
Lezing | Research Seminar
Waltraud Schelkle
No registration is needed for the seminar. After the seminar, refreshments will be served.
Turfmarkt 99
2511 DP Den Haag
In his monumental study of 'When all else fails: government as the ultimate risk manager', David Moss interpreted the history of public policy in the United States as the building of pervasive social safety nets for failing markets. Can we find a similar evolution in recent times of increasing household leveraging: do European welfare states come to act as ultimate financial risk managers of and for households? A recent example that the article documents were systemic bank bailouts that also protected savers. But in doing so, European welfare states have become the overstretched debtors of last resort. The massive deleveraging of private balance sheets in the wake of the Great Financial Crisis since 2008 has left a large legacy of public debt. Innovations in public debt management, such as perpetual and growth-indexed government bonds, can be understood as applying the entrenched wisdom of welfare state institutions to public finance.
Waltraud Schelkle is an Associate Professor in Political Economy at the European Institute of the London School of Economics and Political Science (LSE). Her research interests are the evolving economic governance of EMU and social policy reforms directed at financial markets. Her most recent book "The political economy of monetary solidarity: understanding the experiment of the euro" was published by Oxford University Press in 2017. She has published in journals such as the Journal of Common Market Studies, Review of International Political Economy and Journal of European Public Policy. Her current projects include work on the independence of central banks and on pension systems after the crisis.
political legitimacy
Waltraud Schelkle Associate Professor in Political Economy at the European Institute of the London School of Economics and Political Science (LSE)
Instituut Bestuurskunde
Take a look at the other Seminars in this series.


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This course is designed to provide the person who wishes to commence a career in providing support to people in Aged care / disability care health facilities and the community. Individual support workers will be given training in technical and procedural knowledge to provide person-centred support. They are taught factual, technical and procedural knowledge of the concepts and practices required.
Duration – 40 weeks (maximum 1 year) Support sessions by Health Careers Institute, Kochi, India.
Upon successful completion of the required units of competency, the students will be issued a testamur with record of results from IHNA.