On the syzygies and Hodge theory of nodal hypersurfaces

Abstract

Abstract. We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and the author for the graded pieces with respect to the Hodge filtration of the top cohomology of the hypersurface complement in many new cases. A classical result by Severi on the position of the singularities of a nodal surface in P 3 is improved and applications to deformation theory of nodal surfaces are given.Dedicated to the memory of Alexandru Lascu, who was always searching for the Truth, and encouraging others to do the same.

Hodge Ideals

“…Note however that there exist situations where the bound in Corollary H can be improved: if D has only nodes, in [DS12, Corollary 2.2] the same bound ([ n 2 ] + 1)d − n − 1 is obtained when n is odd, but the better bound n 2 · d − n is shown to hold when n is even. See also [Dim13] for further interesting applications of such bounds.…”

Jacobian syzygies, stable reflexive sheaves, and Torelli properties for projective hypersurfaces with isolated singularities

“…It is interesting to note that even though the approaches in [9] and [19] are quite different, the condition that the singularities of V are weighted homogeneous plays a key role in both papers. While this inequality is the best possible in general, as one can see by considering hypersurfaces with a lot of singularities, see [10], [7], for situations when the hypersurface V has a small number of singularities this result is far from optimal. Our first result gives the following better bound in this case.…”

Nodal surfaces with obstructed deformations

“…We show that in the extreme case deg(f 3 ) = 4 these surfaces form the singular locus of the The author would like to thank Arnaud Beauville for pointing out the paper [3]. The author would thank the referee for many valuable suggestions to improve the exposition.…”

“…If d ≥ 3 then dim J d = 16 (see [3,Corollary 4.3] or [10, Proposition 3.3.8]). In this case one has a natural identification between J d /F and the tangent space to Aut(P 3 ).…”

“…The case d ≤ 7 was proven by Dimca [3] using different techniques. The above bound is sharp: Theorem 1.2.…”