Giulio Codogni

Fano Varieties in Mori Fibre Spaces

We show that being a general fibre of a Mori fibre space (MFS) is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities to be realised as a fibre of a Mori fibre space, which turn into a characterisation in the rigid case. We apply our criteria to figure out this property up to dimension 3 and on rational homogeneous spaces. The smooth toric case is studied and an interesting connection with K-semistability is also investigated.

The non-existence of stable Schottky forms

We show that there is no stable Siegel modular form that vanishes on every moduli space of curves.

Pluri-Canonical Models of Supersymmetric Curves

This paper is about pluri-canonical models of supersymmetric (susy) curves. Susy curves are generalisations of Riemann surfaces in the realm of super geometry. Their moduli space is a key object in supersymmetric string theory. We study the pluri-canonical models of a susy curve, and we make some considerations about Hilbert schemes and moduli spaces of susy curves.

A note on the fibres of Mori fibre spaces

We consider the problem of determining which Fano manifolds can be realised as fibres of a Mori fibre space. In particular, we study the case of toric varieties, Fano manifolds with high index and some Fano manifolds with high Picard rank.

The degree of the Gauss map of the theta divisor

We study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. We use this to obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus, and apply this bound in examples, and to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppavs by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension.