Abstract
For a reduced hypersurface V(f)⊆Pn of degree d, the Castelnuovo-Mumford regularity of the Milnor algebra M(f) is well understood when V(f) is smooth, as well as when V(f) has isolated singularities. We study the regularity of M(f) when V(f) has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by (d−2)(n+1), which is the degree of the Hessian polynomial of f. However, this is not always the case, and we prove that in Pn the regularity of the Milnor algebra can grow quadratically in d.
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