\magnification=\magstep1
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\def\cocoa
{\hbox{\rm C\kern-.13em o\kern-.07
em C\kern-.13em o\kern-.15em A}}
\noindent Abstract of the talk
\centerline{\bf Koszul algebras and Gr\"obner bases of quadrics}
\medskip
\centerline{Aldo Conca (Universit\`a di Genova) }
\medskip
This is a joint work with M.E. Rossi, N.V.Trung and G.Valla. \smallskip
Let $R=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$ and let
$S=R/I$ where $I$ is a homogeneous ideal. The algebra $S$ is said to
be {\sl Koszul} if $K$ has a linear free resolution as an $S$-module, that is,
all the matrices in the resolution have entries of degree one. Furthermore, $S$
is {\sl quadratic} if $I$ is generated by polynomials of degree $2$
and it is {\sl G-quadratic} if there exists a coordinate system and a term order
such that $I$ is generated by a Gr\"obner basis of polynomials of degree $2$.
It is known that:
$$
\hbox{ G-quadratic} \ \ \Rightarrow \hbox{ Koszul } \Rightarrow \hbox{
quadratic }
$$
None of the two implications can be reversed in general.
Given a space of quadrics $V$ it is therefore natural to ask whether there
exist a coordinate system and a term order such that the ideal $(V)$ has a
Gr\"obner basis of quadrics. In my talk will discuss this problem. I will
present a (coordinate free) sufficient criterion for an ideal to have a
Gr\"obner basis of quadrics and some applications to the
study of three classes of algebras:\smallskip
\noindent
i) Algebras defined by spaces of quadrics of small codimension, \smallskip
\noindent
ii) Coordinate rings of set of points in projective spaces,\smallskip
\noindent
iii) Artinian Gorestein algebras with socle in degree $3$.
\smallskip
I will also illustrate how we have used \cocoa\ to deal with these problems.
\end