Positivity of the Poincar\'e bundle on the moduli space of vector bundles and its applications

TL;DR

This paper proves the positivity of the Poincaré bundle on moduli spaces of vector bundles over curves and explores its implications for derived categories and ACM bundles, advancing understanding of their geometric and categorical properties.

Contribution

It establishes the nefness of the Poincaré bundle on moduli spaces and applies this to embed derived categories and construct ACM bundles, extending prior results.

Findings

The Poincaré bundle induces nef vector bundles on moduli spaces.

Derived categories of curves embed into those of moduli spaces for large genus.

Constructs families of ACM bundles on moduli spaces.

Abstract

We prove that the normalized Poincar\'e bundle on the moduli space of stable rank $r$ vector bundles with a fixed determinant on a smooth projective curve $X$ induces a family of nef vector bundles on the moduli space. Two applications follow. We show that when the genus of $X$ is large, the derived category of $X$ is embedded into the derived category of the moduli space for arbitrary rank and coprime degree, which extends the results of Narasimhan, Fonarev-Kuznetsov, and Belmans-Mukhopadhyay. As the second application, we construct a family of ACM bundles on the moduli space. A key ingredient of our proof is the investigation of birational geometry of the moduli spaces of parabolic bundles.