A splitting criterion for vector bundles on blowing ups of the plane

TL;DR

This paper provides a cohomological criterion to determine when a vector bundle on a blown-up projective plane is a pullback of a direct sum of line bundles, and characterizes certain rank 2 bundles on ${f P}^2$.

Contribution

It introduces a new cohomological criterion for identifying vector bundles that are pullbacks of line bundles on blow-ups of the plane, and characterizes specific rank 2 bundles.

Findings

Cohomological criterion for vector bundles on blow-ups

Characterizations of rank 2 bundles on ${f P}^2$

Conditions for bundles to be pullbacks of line bundles

Abstract

Let $f_s: X_s \to {\bf {P}}^2$ be the blowing-up of $s$ distinct points and $E$ a vector bundle on $X_s$. Here we give a cohomological criterio which is equivalent to $E \cong f_s^\ast (A)$ with $A$ a direct sum of line bundles. We also some cohomological characterizations of very particular rank 2 vector bundles on ${\bf {P}}^2$.