Bourbaki degree of pairs of projective surfaces

TL;DR

This paper introduces new invariants for pairs of projective surfaces, explores their properties using foliation theory, and provides classification results and counterexamples to existing conjectures.

Contribution

It defines the Bourbaki degree and invariant m for pairs of surfaces, connecting them with foliation theory and providing new classification insights.

Findings

Invariant m is bounded by a quadratic relation of degrees.

A nearly-free example induces an unstable tangent sheaf, countering a previous conjecture.

Results on minimal degree for syzygies and classification of specific surface pairs.

Abstract

The present work focuses on studying the logarithmic tangent sheaf associated with sequences of two homogeneous polynomials in four variables. We introduce two positive discrete invariants: the invariant m and the Bourbaki degree of a sequence, inspired by the framework of the Bourbaki degree recently developed for projective plane curves by Jardim-Nejad-Simis. The invariant m plays the role of the Tjurina number of plane projective curves and is bounded by a quadratic relation of the degrees. We establish results concerning the interplay of minimal degree for syzygies of the Jacobian matrix and the introduced discrete invariants. Our approach uses tools from foliation theory, taking advantage of the fact that the logarithmic sheaf is, up to a twist, the tangent sheaf of a codimension-one foliation in projective three-space. We provide examples and classification results for pencils of cubics and for pairs of a quadric and a cubic. In particular, one of the nearly-free examples induces an unstable, non-split tangent sheaf for a codimension-one foliation of degree 3, answering, in the negative, a conjecture of Calvo-Andrade, Correa and Jardim from 2018.