The subject of these lectures is enumerative geometry of curves on

algebraic surfaces and its relation to Hilbert schemes of points, real

algebraic geometry and tropical geometry.

The generating function of the Euler numbers of Hilbert schemes of

points on curves has been used by Pandharipande and Thomas to give a

mathematical definition of Gopakumar-Vafa invariants, and by

Kool-Shende-Thomas to give a proof of a conjecture of mine which

describes the generating function of the numbers of δ-nodal singular

curves in a linear system of dimension δ on any surface. Shende and me

gave a conjectural refinement of this conjecture. Here the number of

curves is replaced by a polynomial, whose meaning is still mysterious,

but it can be seen to be related to tropical geometry and real algebraic

geometry. In this series of lectures we want to explain this circle of

ideas, and also use the opportunity to introduce a number of important

concepts and techniques, generating functions, cobordism ring,

localization, Severi degrees, Welschinger invariants, tropical geometry,

Heisenberg algebra.

## Refined curve counting IV

Research Group:

Prof.

Lothar Goettsche

Institution:

ICTP

Schedule:

Friday, May 31, 2013 - 14:00 to 15:00

Location:

Luigi Stasi Seminar Room, ICTP

Abstract: