A closed formula expresses the Hilbert polynomial of the Jacobian algebra of a reduced surface in P^3 in terms of its graded Betti numbers, giving new necessary conditions when singularities are isolated.
Ellia, Quasi complete intersections and global Tjurina number of plane curves, J
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Graded Betti numbers of the Jacobian algebra of surfaces in $\mathbb P^3$
A closed formula expresses the Hilbert polynomial of the Jacobian algebra of a reduced surface in P^3 in terms of its graded Betti numbers, giving new necessary conditions when singularities are isolated.