Abstract

In this talk, I will describe recent work by G. Hitching and myself on the geometry of linearly stable coherent systems. I will begin by recalling the definition of linear stability for linear series on curves and its natural generalization to higher-rank vector bundles. In particular, I will provide a characterization of linear stability for scrolls, adapting the notion of a linearly stable variety as introduced by Mumford. We then apply this characterization to demonstrate the equivalence between the linear stability of a coherent system $(E,V)$ and the linear stability of the image scroll of the natural morphism $P(E^*) \to P(V^*)$. If time permits, I will outline a proof of Butler’s conjecture for specific linearly stable coherent systems $(E,V)$ of type $(r, d, r+2)$ on a curve $C$, where $r$ is the rank of $E$, $d$ is the degree of $E$, and $r+2$ is the dimension of global sections of the subspace $V$ of $H^0(C,E)$.

Location

Il seminario si terrà in Aula Beltrami, presso il Dipartimento di Matematica.