2026/02/10 by Alexandru Dimca, Gabriel Sticlaru · 1 citation
#math.AG #math.AC
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 P3 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.