Logarithmic sheaves of complete intersections
Daniele Faenzi, Marcos Jardim, Jean Vallès, Alan Muniz
TL;DR
This work defines the logarithmic tangent sheaf ${\mathcal{T}}_\sigma$ for complete intersections and investigates local freeness, freeness, and stability, tying these properties to Jacobian syzygies and associated distributions. It develops a detailed theory for pencils of quadrics, introducing Segre symbols and degree vectors to classify regular and irregular pencils, and derives precise criteria for (semi)stability and the projective dimension of ${\mathcal{T}}_\sigma$. The authors provide a full classification of free pencils of quadrics, show that freeness and local freeness coincide in this setting, and construct locally free but non-free examples in higher degrees; they also connect to rational foliations, yielding both stability results and negative answers to conjectures about tangent sheaf splitting. The results illuminate how logarithmic tangent sheaves arise from geometric data such as Segre type and singular loci, with implications for foliations and vector bundle phenomena on projective spaces.
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 .