Theoretical analysis derives an explicit Hilbert polynomial formula for surface Jacobian algebras in projective three-space, indicating strict constraints on graded Betti numbers.
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 [Formula: see text] 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 [Formula: see text] is stated and support for it is provided. In the final section we construct four natural Jacobian syzygies for surfaces [Formula: see text] coming from pencils of surfaces.
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Dimca et al. (2026) studied this question.
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