A reminder: Algebraic Geometry seminar by C. Ciliberto

[Pietro De Poi]

Tomorrow,

*Wednesday 6th April, 2022 at 14:30*

venue: *sala riunioni* - Università degli Studi di Udine - Dipartimento 
di Scienze Matematiche, Informatiche e Fisiche

Prof. Ciro Ciliberto (Univ. Roma "Tor Vergata") will give a talk on

_Extensions of canonical curves and double covers __
___
ABSTRACT:  A variety of dimension $n$ is said to be extendable $r$ times 
if it is the space section of a variety of dimension $n+r$ which is not 
a cone. I will recall some general facts about extendability, with 
special regard for extensions of canonical curves to $K3$ surfaces and 
Fano 3-folds. Then I will focus on double covers and on their 
extendability properties.  In particular I will consider $K3$ surfaces 
of genus 2, that are double covers of the
plane branched over a general sextic. A first results is that the 
general curve in the linear system pull back of plane curves of degree 
$k\geq 7$ lies on a unique $K3$ surface, so it is only once extendable. 
A second result is that, by contrast, if $k\leq 6$ the general such 
curve is extendable to a higher dimensional variety. In fact in the 
cases $k=4,5,6$, this gives the existence of singular index $k$ Fano 
varieties of dimensions 8, 5, 3, genera 17, 26, 37,
and indices 6, 3, 1 respectively. For $k = 6$ one recovers the Fano 
variety $\P(3, 1, 1, 1)$, one of two Fano threefolds with canonical 
Gorenstein singularities with the maximal genus 37, found by Prokhorov.
A further result is that this latter variety is no further extendable.
For $k=4$ and $5$ these Fano varieties have been identified by Totaro.


//Within the project "PRIN 2017JTLHJR Geometric, algebraic and analytic 
methods in arithmetic" activities.



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