Graded Betti numbers of the Jacobian algebra of surfaces in ℙ3

Source abstract

In this paper, we compute an explicit closed formula for the Hilbert polynomial of the Jacobian algebra [Formula: see text] of a reduced surface [Formula: see text] in [Formula: see text] in terms of the graded Betti numbers of the algebra [Formula: see text]. When X has only isolated singularities, two results by du Plessis and 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 four 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.