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

TL;DR

The paper derives an explicit closed formula for the Hilbert polynomial $P(M(f))$ of the Jacobian algebra $M(f)$ of a reduced surface $X=f=0$ in $\mathbb{P}^3$ in terms of its graded Betti numbers from the minimal resolution, and provides structural relations $p+r=q+3$ and $\sum d_i-\sum c_j+\sum b_k=d-1$. It shows that $P(M(f))$ is constant when $\dim\Sigma=0$ (isolated singularities) and linear when $\dim\Sigma=1$, with explicit coefficients; in the isolated case, it establishes new necessary bounds on the cubic sums of the Betti numbers via du Plessis–Wall and expresses $6\tau(X)$ in terms of Betti data. The work also discusses how these results compare to the plane-curve case, explains the subtleties of freeness for surfaces, and presents numerous concrete examples computed via computer algebra to illustrate the range of Betti-number patterns and Hilbert polynomials that can arise. Overall, the paper links algebraic invariants of the Jacobian algebra to the singularity geometry of surfaces in $\mathbb{P}^3$ and provides practical tools for verifying feasibility of Betti sequences.

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.

Theorems & Definitions (21)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• Lemma 2.2
• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Example 3.4
• Example 3.5
• Example 3.6