Vorträge in der Woche 05.06.2023 bis 11.06.2023

Mittwoch, 07.06.2023: Counting curves with fixed complex structure in P^r

Carl Lian (HU Berlin)

This is a report on work in progress. We will discuss the following problem: if (C,p_1,…,p_n) is a general pointed curve of genus g and X_1, …, X_n are general linear subspaces of P^r, then how many non-degenerate maps f from C to P^r are there with f(p_i) contained in X_i? A virtual answer in Gromov-Witten theory is easy to obtain, but is often not enumerative; the geometric counts are considerably more subtle. Our method proceeds through a series of degenerations on the moduli space of complete collineations. The case in which the X_i are points is that of geometric Tevelev degrees of P^r, which were previously known only when r=1 or when d is large; here, we explain a connection to torus orbit closures of Grassmannians and obtain a conceptual new proof of a result of Berget-Fink. Our approach to the general problem involves Coskun’s geometric Littlewood-Richardson rule for partial flag varieties in an essential way.