Fibr\'es de Schwarzenberger et fibr\'es logarithmiques g\'en\'eralis\'es

TL;DR

This paper introduces a broad generalization of logarithmic and Schwarzenberger bundles over projective space, linking their isomorphism classes to geometric configurations like points and curves.

Contribution

It extends the theory of logarithmic and Schwarzenberger bundles to higher ranks and establishes criteria for their isomorphism based on geometric data.

Findings

Two logarithmic bundles are isomorphic iff associated to the same set of points or points on a curve of degree equal to the bundle's rank.

Generalization applies to bundles of rank greater than the dimension of the projective space.

Provides geometric criteria for bundle isomorphism in the case of the projective plane.

Abstract

We propose a generalization of logarithmic and Schwarzenberger bundles over $\P^n=\P^n(\C)$ when the rank is greater than $n$. The first ones are associated to finite sets of points on $\P^{n\vee}$ and the second ones to curves with degree greater than $n$ on $\P^{n\vee}$. On the projective plane we show that two logarithmic bundles are isomorphic if and only if they are associated to the same set of points or if the two sets of points belong to a curve of degree equal to the rank of the considered bundles.