ICTP Algebraic Geometry Seminars - Wednesday, 19 October, at 16:00 (Lothar Göttsche)/hybrid seminar

[ICTP Math Section]

ICTP ALGEBRAIC GEOMETRY SEMINARS 2022


Wednesday, 19 October, at 16:00


_Hybrid seminar_

Venue: for*in-person* attendees (Leonardo da Vinci, Luigi Stasi seminar 
room).

For *virtual* attendees: 
https://zoom.us/meeting/register/tJUvdumrpjMqH9TgK4-irQa1IZcvBANg3EGF 
After registering, you will receive a confirmation email containing 
information about joining the meeting.

_
_/_Kindly note that this seminar is divided in two parts of roughly 45 
minutes each. The first part is for more general audience, while the 
second part is more specialized._ /


*Speaker: Lothar Goettsche (ICTP)*




*Part 1: The Hilbert scheme of points*

We introduce the Hilbert scheme of points on surface, and talk about its 
invariants, its relations to other moduli spaces of sheaves and 
applications to enumerative geometry. Finally we introduce some 
techniques for working with these Hilbert schemes



*Part 2: (Refined) Verlinde and Segre formulas for Hilbert schemes of 
points*

This is joint work with Anton Mellit.
Segre and Verlinde numbers of Hilbert schemes of points have been 
studied for a long time.
The Segre numbers are evaluations of top Chern and Segre classes of 
so-called tautological bundles on Hilbert schemes of points.
The Verlinde numbers are the holomorphic Euler characteristics of line 
bundles on these Hilbert schemes.
We give the generating functions for the Segre and Verlinde numbers of 
Hilbert schemes of points.
The formula is proven for surfaces with K_S^2=0, and conjectured in 
general. Without restriction on K_S^2 we prove the conjectured 
Verlinde-Segre correspondence relating Segre and Verlinde numbers of 
Hilbert schemes. Finally we find a generating function for finer 
invariants, which specialize to both the Segre and Verlinde numbers, 
giving some kind of explanation of the Verlinde-Segre correspondence.



https://indico.ictp.it/event/10092/