Bourbaki degree of pairs of projective surfaces
Felipe Monteiro
TL;DR
This work develops a Bourbaki-degree framework for pairs of projective surfaces in $\mathbb{P}^3$ by studying the kernel ${\mathcal T}_{\sigma}$ and cokernel ${\mathcal Q}_{\sigma}$ of the Jacobian map $\nabla\sigma$ for $\sigma=(f,g)$. It introduces the discrete invariants $m(\sigma)=\mathrm{ch}_2({\mathcal Q}_{\sigma})$ and ${\rm Bour}(\sigma)$, derives a key formula ${\rm Bour}(\sigma)=e(e-d)+m_0-m(\sigma)$ with $d=d_f+d_g$ and $m_0=d_f^2+d_g^2+d_f d_g$, and studies freeness, compressibility, and stability of ${\mathcal T}_{\sigma}$ through connections to codimension-one foliations. The paper provides sharp bounds and complete classifications for pencils of cubics and for mixed-degree pairs $(d_f,d_g)=(1,2)$, including several counterexamples that answer longstanding questions about foliation stability. By relating the Bourbaki degree to plane curves and Bourbaki schemes, it delivers a unified, scheme-theoretic view of singularity data, syzygies, and Chern classes in this higher-dimensional setting, with explicit constructions and Macaulay2-based computations supporting the classifications.
Abstract
The logarithmic tangent sheaf associated to an algebraically independent sequence of homogeneous polynomials - defined as the kernel of the associated Jacobian matrix - naturally generalizes the classical logarithmic tangent sheaf of a divisor in a projective space to the case of subvarieties defined by more than one equation. As is the case for divisors, one may investigate the freeness of such sequences, and other weaker notions. The present work focuses on 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. 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 polynomials, relating stability and Chern classes with the discrete invariants introduced, while classifying free and nearly-free cases. 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.