%Massimiliano Mella, Universita' di Ferrara

%" Birationally rigid quartics"

\documentclass{amsart}
%\usepackage{amsmath,amsthm,amssymb,xypic}
%\includeonly{intro, maps, contractions, plan, excl, refs, centres1, centres2}

%theorems and the like

\theoremstyle{plain}

\newtheorem{thm}{Theorem}

%operators

\DeclareMathOperator{\Bl}{Bl}
\DeclareMathOperator{\Cl}{Cl}
\DeclareMathOperator{\NE}{NE}
\DeclareMathOperator{\NEbar}{\overline{NE}}
\DeclareMathOperator{\rk}{rk}
\DeclareMathOperator{\cl}{cl}
\DeclareMathOperator{\Bsl}{Bsl}
\DeclareMathOperator{\hsp}{hsp}
\DeclareMathOperator{\Hilb}{Hilb}
\DeclareMathOperator{\hcf}{hcf}
\DeclareMathOperator{\red}{red}
\DeclareMathOperator{\cod}{cod}
\DeclareMathOperator{\mult}{mult}
\DeclareMathOperator{\di}{dim}
\DeclareMathOperator{\re}{real}
%\DeclareMathOperator{\for}{for}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\En}{End}
\DeclareMathOperator{\Exc}{Exc}
\DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\Diff}{Diff}
\DeclareMathOperator{\Sing}{Sing}
\DeclareMathOperator{\NonSing}{NonSing}
\DeclareMathOperator{\WCl}{WCl}


%othercommands

\newcommand{\QED}{\ifhmode\unskip\nobreak\fi\quad {\rm Q.E.D.}} % QED
\newcommand\recip[1]{\frac{1}{#1}}
\newcommand\Ga{\Gamma}
\newcommand\map{\dasharrow}
\newcommand\iso{\cong}
\newcommand\la{\lambda}
\newcommand{\wave}{\widetilde}
\newcommand{\f}{\varphi}
\newcommand{\A}{\mathcal{A}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\De}{\mathcal{D}}
\newcommand{\E}{\mathcal{E}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\G}{\mathbb{G}}
\renewcommand{\H}{\mathcal{H}}
\newcommand{\Hm}{\underline{H}}
\newcommand{\I}{\mathcal{I}}
\newcommand{\K}{\mathcal{K}}
\renewcommand{\L}{\mathcal{L}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\N}{\mathcal{N}}
\renewcommand{\O}{\mathcal{O}}
\renewcommand{\P}{\mathbb{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\s}{\mathbb{S}}
\newcommand{\T}{\mathbb{T}}
\newcommand{\Tangent}{$\mathbb{T}$angent }
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Wc}{\mathcal{W}}


\newcommand{\ra}{\text{rat}}

\title{The importance of being $\Q$-factorial}

\author{
Massimiliano Mella}
\address{M. Mella\\
Dipartimento di Matematica\\
Universit\`a di Ferrara\\
Via Machiavelli 35\\
44100 Ferrara Italia}
\email{mll@unife.it}
\date{March 2002}
\begin{document}
\maketitle
\section*{Abstract for the AMS-UMI meeting Pisa June 2002}
Consider a quartic threefold $X_4\subset P^4$ defined by $\det M=0$, 
where $M$ is a $4\times 4$
matrix of linear forms. One can define a map $f:X\dasharrow \P^3$ by 
the assignment
$P\mapsto(x_0:x_1:x_2:x_3)$, where $M_P(x_0,x_1,x_2,x_3)=0$. That is 
$(x_0,x_1,x_2,x_3)$ is a solution
of the linear system obtained substituting the coordinates of $P$ in $M$.
For $M$ sufficiently general such a map is well defined, that is for 
a general point $P$ $\rk M_P=3$,
  and birational. So that
  $f$ gives a rational parametrisation of $X$. The singularities of 
$X$ correspond to points where
the rank drops.
It is not difficult to show that, for a general $M$, the 
corresponding quartic has only simple double points
corresponding to points of rank 2. So that a general determinantal quartic
has only simple double points and it is rational.

On the other hand the famous result of Iskovskikh and Manin, \cite{IM},
states that a smooth quartic threefold is not birational to any Mori Space.
That is $X_4\subset \P^4$ is birationally rigid when $X$ is smooth. 
In the notation introduced in
\cite{CM} ${\mathcal P}(X)=\{ X\}$.

 From the point of view of birational classification of uniruled 
threefolds the above statements are somewhat
discouraging. Minimal Model Theory imposes to look at terminal 3-folds and
simple double points are the simplest possible terminal 
singularities. But still it is enough
to impose a bunch of them to change a rigid structure to a rational 
variety. The point I want to
stress in this talk is that the rationality of a determinantal 
quartic is due to the lack
of $\Q$-factoriality not to the presence of singularities.

\begin{thm} Let $X_4\subset \P^4$ be a $\Q$-factorial
quartic 3-fold with only simple double points as singularities. Then 
${\mathcal P}(X)=\{ X\}$, that is $X$
is birationally rigid.
\end{thm}

\begin{thebibliography}{99}
\bibitem[CM]{CM}
A.Corti and M. Mella, \textsl{Birational geometry of terminal quartic 
3-folds I}, preprint (2001)
\bibitem[IM]{IM}
V. A. Iskovskikh and Yu. I. Manin, \textsl{Three-dimensional quartics and
counterexamples to the L\"uroth problem}, Math. USSR Sb. {\bf15} (1971),
141--166
\end{thebibliography}
\end{document}