On the degree of the singular subscheme of hypersurfaces in ${\mathbb P}^n$

TL;DR

This paper provides explicit formulas for the dimension and degree of the singular subscheme of hypersurfaces in projective space, linking them to graded Betti numbers of the Jacobian algebra.

Contribution

It introduces formulas relating singular subscheme invariants to Betti numbers and defines homologically strictly plus-one generated hypersurfaces with specific singular locus properties.

Findings

Formulas for dimension and degree of singular subscheme in terms of Betti numbers

New restrictions on graded Betti numbers of Jacobian algebras

Characterization of hypersurfaces with singular locus of dimension n-2

Abstract

Explicit formulas determining the dimension and the degree of the singular subscheme of hypersurfaces in ${\mathbb P}^n$ are given in terms of the graded Betti numbers of the minimal free resolution of the corresponding Jacobian algebra. This gives in particular new restrictions which must be satisfied by such graded Betti numbers. We define a homologically strictly plus-one generated hypersurface, and show that such a hypersurface has a singular locus of dimension $n-2$ when its degree is not too low.