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Enumerative Geometry of rational cuspidal curves on del-Pezzo surfaces

Ritwik Mukherjee (TATA Institute of Fundamental Research)
Onsdag, 25 maj, 2016, at 16:15, in Aud. D3 (1531-215)
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).
Organiseret af: QGM
Kontaktperson: Jørgen Ellegaard Andersen