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

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 $n-2$ under some conditions.