arXiv subjects

G. Codogni

Publications and source records attributed to G. Codogni.

2 recordsLinked to original sources

arXiv2017

The blow-up of $\mathbb{P}^4$ at 8 points and its Fano model, via vector bundles on a del Pezzo surface

Building on the work of Mukai, we explore the birational geometry of the moduli spaces M_{S,L} of semistable rank two torsion-free sheaves, with c_1=-K_S and c_2=2, on a polarized degree one del Pezzo surface (S,L); this is related to the birational geometry of the blow-up X of P^4 in 8 points. Our analysis is explicit and is obtained by looking at the variation of stability conditions. Then we provide a careful investigation of the blow-up X and of the moduli space Y=M_{S,-K_S}, which is a remarkable family of smooth Fano 4-folds. In particular we describe the relevant cones of divisors of Y, the group of automorphisms, and the base loci of the anticanonical and bianticanonical linear systems.

math.AG↗

arXiv2011

The non-existence of stable Schottky forms

Let $A_g^S$ be the Satake compactification of the moduli space $A_g$ of principally polarized abelian $g$-folds and $M_g^S$ the closure of the image of the moduli space $M_g$ of genus $g$ curves in $A_g$ under the Jacobian morphism. Then $A_g^S$ lies in the boundary of $A_{g+m}^S$ for any $m$. We prove that $M_{g+m}^S$ and $A_g^S$ do not meet transversely in $A_{g+m}^S$, but rather that their intersection contains the $m$th order infinitesimal neighbourhood of $M_g^S$ in $A_g^S$. We deduce that there is no non-trivial stable Siegel modular form that vanishes on $M_g$ for every $g$. In particular, given two inequivalent positive even unimodular quadratic forms $P$ and $Q$, there is a curve whose period matrix distinguishes between the theta series of $P$ and $Q$.

math.AG↗