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