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Refined curve counting

Lothar Goettsche
Thursday, April 18, 2013 - 15:00 to 16:00
Luigi Stasi Seminar Room, ICTP

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.

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