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Seminar by Ritwik Mukherjee (TATA Institute of Fundamental Research)

Title: Enumerative Geometry of rational cuspidal curves on del-Pezzo surfaces

2016.05.20 | Jane Jamshidi

Date Wed 25 May
Time 16:15 17:15
Location 1531-215 (Aud. D3)


Enumerative Geometry of rational curves is a classical question that is over a hundred years old. An extremely important question is "How many rational (genus 0) degree d curves are there in
CP^2 that pass through 3d-1 generic points?" Although this question was investigated in the nineteenth century, a complete solution to this problem was unknown until the early 90's, when Kontsevich-Manin
and Ruan-Tian announced a formula. In this talk we will discuss some natural generalizations of the above question; in particular we will be looking at rational curves on del-Pezzo surfaces that have a cuspidal singularity. We will describe a topological method to approach such questions. If time
permits, we will also explain the idea of how to enumerate genus one curves with a fixed complex structure by comparing it with the Symplectic Invariant of a manifold (which are essentially the number of curves that are solutions to the perturbed d-bar equation).