Counting curves on P^r

Abstract: We will explain a complete solution to the following problem. If $(C,p_1,\ldots,p_n)$ is a general curve of genus $g$ and $x_1,\ldots,x_n$ are general points on $\mathbb{P}^r$, then how many degree $d$ maps $f:C\to\mathbb{P}^r$ are there with $f(p_i)=x_i$? These are the "Tevelev degrees" of projective space, which were previously known only when $r=1$, when $d$ is large compared to $g$, or virtually in Gromov-Witten theory. Time-permitting, we will also discuss some partial results when the conditions $f(p_i)=x_i$ are replaced by conditions $f(p_i) \in X_i$, where the $X_i$ are linear spaces of any dimension.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html