Ample stable vector bundles on rational surfaces

Abstract: A theorem of Fulton says that ample vector bundles cannot be classified numerically. However, ampleness is open in families, and so producing a single ample bundle typically implies the existence of many more. If a bundle is both stable and ample, then it has stable and ample deformations. Le Potier suggests exploiting this fact and classifying those Chern characters for which the general stable bundle is ample (provided, say, the moduli space is irreducible). I will discuss recent progress on this problem on the minimal rational surfaces. I will give a complete classification of those Chern characters for which the general stable bundle is both ample and globally generated. I will also explain an "asymptotic" version of this result for bundles that aren't globally generated. This is joint work with Jack Huizenga.

algebraic geometry

Audience: researchers in the topic

Algebraic Geometry NorthEastern Series (AGNES)