Logarithmic sheaves of complete intersections

TL;DR

This paper introduces logarithmic tangent sheaves for complete intersections, analyzes their properties, and classifies free pencils of quadrics, providing new insights into the structure of these sheaves and related foliations.

Contribution

It defines and studies logarithmic sheaves for complete intersections, classifies free pencils of quadrics, and constructs examples of non free pencils of surfaces in P3.

Findings

Complete classification of free pencils of quadrics.

Construction of examples of non free pencils of surfaces in P3.

Analysis of stability and local freeness of logarithmic sheaves.

Abstract

We define logarithmic tangent sheaves associated with complete intersections in connection with Jacobian syzygies and distributions. We analyse the notions of local freeness, freeness and stability of these sheaves. We carry out a complete study of logarithmic sheaves associated with pencils of quadrics and compute their projective dimension from the classical invariants such as the Segre symbol and new invariants (splitting type and degree vector) designed for the classification of irregular pencils. This leads to a complete classification of free (equivalently, locally free) pencils of quadrics. Finally we produce examples of locally free, non free pencils of surfaces in P3 of any degree k at least 3, answering (in the negative) a question of Calvo-Andrade, Cerveau, Giraldo and Lins Neto about codimension foliations on P3 .