Fibr\'es vectoriels de rang deux sur $\P^2$ provenant d'un rev\^etement double

TL;DR

This paper studies rank two vector bundles on the projective plane arising from double covers, showing their jumping lines relate to the branch curve and providing explicit classifications for certain cases.

Contribution

It establishes a link between jumping lines of vector bundles and tangent lines to branch curves, especially characterizing bundles with bitangent lines for quartic branches.

Findings

Jumping lines are r-tangent to the branch curve for double covers.

For r=2, vector bundles have jumping lines exactly as the bitangent lines to the branch quartic.

Provides classification of vector bundles with specific tangent properties.

Abstract

Since Schwarzenberger and his celebrated paper called "Vector bundles on the projective plane" we know that any rank two vector bundle on $\P^2$ is a direct image of a line bundle on a double covering of the plane. This theorem suggests to study the rank two vector bundles according to the branch curve of the covering which they come from. Thus, in the first part we prove that, given a double covering ramified over an irreducible curve $C_{2r}$ with degree $2r$, the jumping lines of fixed order (order depending on $r$ and on the parity of the rank two vector bundle) of the direct images vector bundles are necessarely $r$-tangent to $C_{2r}$. In the second part we concentrate on the case $r=2$. Then we give a list of vector bundles for which the jumping lines are exactly the bitangent lines to the branch quartic.