SISSA Algebraic Geometry Seminar – Fall 2023

This is the webpage of the SISSA Algebraic Geometry Seminar. This fall, we run a reading seminar on Deformation Theory with a view towards moduli spaces of sheaves and Quot schemes. We meet every Monday at 16:00 in Dubrovin Lecture room 136. Talks are up to 90 minutes long. Here is an outline of the program.

Detailed schedule

- 30 October 2023,
__Mahmoud Abdelrazek__(SISSA): Deforming affine schemes I - 6 November 2023,
__Dmitrii Rachenkov__(SISSA): Deforming affine schemes II - 13 November 2023,
__Ian Selvaggi__(SISSA): Hilb, Quot and moduli spaces of sheaves - 27 November 2023,
__Nicolò Bignami__(SISSA): Deformation functors and tangent-obstruction theories I - 4 December 2023,
__Elisa Vitale__(SISSA): Deformation functors and tangent-obstruction theories II - 11 December 2023,
__Armando Capasso__(Roma Tre): Deformation theory of simple sheaves, Atiyah classes and trace maps - 18 December 2023,
__Barbara Fantechi__(SISSA): Tangent and obstruction spaces for morphisms - 8 January 2024,
__Barbara Fantechi__(SISSA): Obstruction spaces and virtual classes

- 15 January 2024, Alberto Cobos Rábano (Sheffield): Reduced Gromov-Witten invariants in higher genus

*Abstract*. The Gromov-Witten invariants of projective spaces are not enumerative in positive genus. The reason is geometric: the moduli space of genus-g stable maps has several irreducible components, which contribute in the form of lower-genus GW invariants. In genus one, Vakil and Zinger constructed a blow-up of the moduli space of stable maps and used it to define reduced Gromov-Witten invariants, which correspond to curve-counts in the main component. I will present a new definition of reduced Gromov-Witten invariants of complete intersections in all genus using desingularizations of sheaves. This is joint work with E. Mann, C. Manolache and R. Picciotto and can be found in arXiv:2310.06727. - 22 January 2024, Michele Graffeo (SISSA): Nested variants of the Hilbert scheme of points

*Abstract*. Hilbert schemes are classical geometrical objects first introduced by Grothendieck. They parametrise closed subschemes of a given projective variety with fixed Hilbert Polynomial. In the seminar I will focus on variants of Hilbert schemes that parametrise nestings of zerodimensional schemes. Precisely I will describe the interplay between the combinatorics of the nestings viewed as posets functions and the geometry of the associated Hilbert scheme. This is a joint project with P. Lella, S. Monavari, A. Ricolfi, A. Sammartano.