Explicit formulas express the dimension and degree of the singular subscheme of hypersurfaces in P^n via the graded Betti numbers of the Jacobian algebra, producing new restrictions on those Betti numbers and a dimension result for homologically strictly plus-one generated hypersurfaces.
Graded Betti numbers of the Jacobian algebra of surfaces in $\mathbb P^3$
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abstract
We compute an explicit closed formula for the Hilbert polynomial of the Jacobian algebra $M(f)$ of a reduced surface $X:f=0$ in $\mathbb P^3$ in terms of the graded Betti numbers of the algebra $M(f)$. When $X$ has only isolated singularities, a result by A. du Plessis and C. T. C. Wall yields new necessary condition 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. In the final section we construct four natural Jacobian syzygies for surfaces $X$ coming from pencils of surfaces.
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UNVERDICTED 2representative citing papers
Ziegler pairs of plane arrangements in P^3 have isomorphic intersection lattices but different Betti numbers for Jacobian algebra resolutions and relate to cones over Ziegler pairs of line arrangements in P^2.
citing papers explorer
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On the degree of the singular subscheme of hypersurfaces in ${\mathbb P}^n$
Explicit formulas express the dimension and degree of the singular subscheme of hypersurfaces in P^n via the graded Betti numbers of the Jacobian algebra, producing new restrictions on those Betti numbers and a dimension result for homologically strictly plus-one generated hypersurfaces.
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On the Jacobian algebras of Ziegler pairs of plane arrangements
Ziegler pairs of plane arrangements in P^3 have isomorphic intersection lattices but different Betti numbers for Jacobian algebra resolutions and relate to cones over Ziegler pairs of line arrangements in P^2.