EVENT DETAILS AND ABSTRACT

Title: Seshadri constants for vector bundles
Speaker: Mihai Fulger
Speaker Info: Universiy of Connecticut
Brief Description:
Special Note:
Abstract:

Denote by $X$ a projective variety, and by $x\in X$ a closed point. Let $L$ be a nef line bundle on $X$. The local positivity of $L$ at $x\in X$ is encoded by the Seshadri constant $\varepsilon(L;x)$. These useful invariants detect ampleness and the non-ample locus of nef divisors, and they measure asymptotic jet separation. The projective space $\mathbb P^n$ stands out as the only Fano $n$-fold $X$ with $\varepsilon(-K_X;x)\geq n+1$ for some $x\in X$. We investigate a more general positivity notion in the relative setting. For the bundle map $\mathbb P(V)\to X$, where $V$ is a vector bundle on $X$, we extract a Seshadri-type constant $\varepsilon(V;x)$. When $V$ is nef, it has been previously studied by Hacon. We observe many similarities to the case of line bundles. In particular we find an ampleness criterion, with the appropriate definition we can detect non-ample base loci, and interpret $\varepsilon(V;x)$ as measure of asymptotic jet separation. We conjecture that $\mathbb P^n$ is the only projective manifold such that $\varepsilon(TX;x)>0$ for some $x\in X$. We prove this for surfaces and for Fanos. This is all in joint work with Takumi Murayama.