CMSA Algebraic Geometry in String Theory Seminar: Curve-counting with fixed domain (“Tevelev degrees”)

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February 1, 2022 9:30 am - 10:30 am

Carl Lian - Institut für Mathematik, Humboldt-Universität zu Berlin

We will consider the following problem: if (C,x_1,...,x_n) is a fixed general pointed curve, and X is a fixed target variety with general points y_1,...,y_n, then how many maps f:C -> X in a given homology class are there, such that f(x_i)=y_i? When considered virtually in Gromov-Witten theory, the answer may be expressed in terms of the quantum cohomology of X, leading to explicit formulas in some cases (Buch-Pandharipande). The geometric question is more subtle, though in the presence of sufficient positivity, it is expected that the virtual answers are enumerative. I will give an overview of recent progress on various aspects of this problem, including joint work with Farkas, Pandharipande, and Cela, as well as work of other authors.