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Enumerative Geometry of singular curves in a Linear System

Ritwik Mukherjee
(NISER, India)
Seminar
Wednesday, 3 July, 2019, at 14:15-15:15, in Aud. D3 (1531-215)
Abstract:
Enumerative geometry is a branch of mathematics that deals with the following question: "How many geometric objects are there that satisfy certain constraints". A well known class of enumerative question is to count curves in a linear system H^0(X,L) that have some prescribed singularities. In this talk we will describe a topological method to approach this problem. We will express the enumerative numbers as the Euler class of an appropriate bundle. We will then go on to explain how we compute the degenerate ontribution of the Euler class using a topological method. This method requires a detailed knowledge of what happens when different singularities collide and how the singular locus contributes to the Euler class. The method has been used to enumerate curves with delta nodes and one degenerate singularity, provided the total codiemnsion is at most eight (this is joint work with Somnath Basu). If time permits, we will discuss some questions we wish to investigate in the future, including whether this method applies to enumerating hypersurfaces in P^3 (and higher dimensional analogues).
Organised by: QGM
Contact person: Gergely Bérczi