Graded Betti numbers of the Jacobian algebra of surfaces in $\mathbb P^3$

TL;DR

This paper derives an explicit formula for the Hilbert polynomial of the Jacobian algebra of surfaces in projective 3-space, linking it to graded Betti numbers and exploring implications for surfaces with isolated singularities.

Contribution

It provides a new explicit formula for the Hilbert polynomial of the Jacobian algebra in terms of Betti numbers and introduces conditions for possible Betti number sets.

Findings

Explicit formula for Hilbert polynomial in terms of Betti numbers

Necessary conditions for Betti numbers of Jacobian algebras with isolated singularities

Construction of Jacobian syzygies from pencils of surfaces

Abstract

We compute an explicit closed formula for the Hilbert polynomial of the Jacobian algebra $M(f)$ of a reduced surface $X:f=0$ in $\mathbb P^3$ in terms of the graded Betti numbers of the algebra $M(f)$. When $X$ has only isolated singularities, a result by A. du Plessis and C. T. C. Wall yields new necessary condition for a set of positive integers to be the graded Betti numbers of the Jacobian algebra of such a surface. The comparison with the plane curve case is discussed in detail and additional information is given in the case of nodal surfaces. In the final section we construct four natural Jacobian syzygies for surfaces $X$ coming from pencils of surfaces.