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

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.