Quasi-complete intersections in P2 and syzygies

TL;DR

This paper investigates the algebraic and geometric properties of singular curves in the projective plane, focusing on the relationships between their Jacobian schemes, Betti numbers, and quasi-complete intersection structures.

Contribution

It establishes new connections between the Betti numbers of certain modules and the invariants of singular curves, extending to quasi-complete intersections of specific types.

Findings

Relations between Betti numbers and curve invariants established

Results apply to quasi-complete intersections of type (s; s; s)

Insights into the structure of Jacobian schemes for singular curves

Abstract

Let C \in P2 be a reduced, singular curve of degree d and equation f = 0. Let \Sigma denote the jacobian subscheme of C. We have 0 -> E -> 3.O -> I_\Sigma(d-1) -> 0 (the surjection is given by the partials of f). We study the relationships between the Betti numbers of the module H^0_*(E) and the integers, d; \tau, where \tau = deg(\Sigma). We observe that our results apply to any quasi-complete intersection of type (s; s; s).