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

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, two results by A. du Plessis and C. T. C. Wall yield new necessary conditions 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. A natural conjecture on the smallest 4 exponents of $X$ is stated and support for it is provided. In the final section we construct four natural Jacobian syzygies for surfaces $X$ coming from pencils of surfaces.

v5. several new results and a Conjecture are added