Corpus ID: 119178952

# Koszul complexes and spectra of projective hypersurfaces with isolated singularities

@article{Dimca2012KoszulCA,
title={Koszul complexes and spectra of projective hypersurfaces with isolated singularities},
author={Alexandru Dimca and Morihiko Saito},
journal={arXiv: Algebraic Geometry},
year={2012}
}
• Published 2012
• Mathematics
• arXiv: Algebraic Geometry
For a projective hypersurface with isolated singularities, we generalize some well-known results in the nonsingular case due to Griffiths, Scherk, Steenbrink, Varchenko, and others. They showed, for instance, a relation between the mixed Hodge structure on the vanishing cohomology and the Gauss-Manin system filtered by shifted Brieskorn lattices of a defining homogeneous polynomial by using the V-filtration of Kashiwara and Malgrange. Numerically this implied an identity between the Steenbrink… Expand
20 Citations
Generalization of theorems of Griffiths and Steenbrink to hypersurfaces with ordinary double points
• Mathematics
• 2014
Let Y be a hypersurface in projective space having only ordinary double points as singularities. We prove a variant of a conjecture of L. Wotzlaw on an algebraic description of the graded quotientsExpand
Computing Milnor fiber monodromy for some projective hypersurfaces
• Mathematics
• 2017
We describe an algorithm computing the monodromy and the pole order filtration on the top Milnor fiber cohomology of hypersurfaces in $\mathbb{P}^n$ whose pole order spectral sequence degenerates atExpand
Bernstein-Sato polynomials and graded Milnor algebras for projective hypersurfaces with weighted homogeneous isolated singularities
We present a quite efficient method to calculate the roots of the Bernstein-Sato polynomials of homogeneous polynomials if the associated projective hypersurfaces have only weighted homogeneousExpand
Bernstein-Sato polynomials for projective hypersurfaces with weighted homogeneous isolated singularities
We present a quite efficient method to compute the roots of Bernstein-Sato polynomial of a homogeneous polynomial if the associated projective hypersurface has only weighted homogeneous isolatedExpand
Syzygies of Jacobian ideals and weighted homogeneous singularities
• Computer Science, Mathematics
• J. Symb. Comput.
• 2016
It is shown that these singularities are weighted homogeneous if and only if the Koszul syzygies among the partial derivatives of an equation for V are exactly the syzyGies with a generic first component vanishing on the singular locus subscheme of V. Expand
On the syzygies and Hodge theory of nodal hypersurfaces
We give 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 ofExpand
Hodge ideals and spectrum of isolated hypersurface singularities
• Mathematics
• 2019
We introduce Hodge ideal spectrum for isolated hypersurface singularities to see the difference between the Hodge ideals and the microlocal $V$-filtration modulo the Jacobian ideal. We compare theExpand
Zeta functions of projective hypersurfaces with ordinary double points
• Mathematics
• 2021
We extend the approach Abbott, Kedlaya and Roe to computation of the zeta function of a projective hypersurface with τ isolated ordinary double points over a finite field Fq given by the reduction ofExpand
Computing the monodromy and pole order filtration on Milnor fiber cohomology of plane curves
• Mathematics, Computer Science
• J. Symb. Comput.
• 2019
An algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of any reduced projective plane curve C, which has some non weighted homogeneous singularities, has been described. Expand
Roots of Bernstein-Sato polynomials for projective hypersurfaces with general hyperplane sections having weighted homogeneous isolated singularities
For homogeneous polynomials of $n$ variables with singular locus dimension at most 2, we give a new method to compute the roots of Bernstein-Sato polynomials supported at the origin if certainExpand

#### References

SHOWING 1-10 OF 47 REFERENCES
Brieskorn modules and Gauss–Manin systems for non-isolated hypersurface singularities
• Mathematics
• 2004
AbstractWe study the Brieskorn modules associated to a germ of a holomorphic function with non-isolated singularities and show that the Brieskorn module has naturally the structure of a module overExpand
Multiplier ideals, b-function, and spectrum of a hypersurface singularity
We prove that certain roots of the Bernstein-Sato polynomial (i.e. b-function) are jumping coefficients up to a sign, showing a partial converse of a theorem of L. Ein, R. Lazarsfeld, K.E. Smith, andExpand
A generalization of Griffiths' theorem on rational integrals, II
• Mathematics
• 2007
We show that the Hodge and pole order filtrations are globally different for sufficiently general singular projective hypersurfaces in case the degree is 3 or 4 assuming the dimension of theExpand
On the mixed Hodge structure on the cohomology of the Milnor fibre
• Mathematics
• 1985
In [18], the second author introduced a mixed Hodge structure on the cohomology of the Milnor fibre of an isolated hypersurface singularity. (For the definition of a mixed Hodge structure, el. [17,Expand
Die monodromie der isolierten singularitäten von hyperflächen
J. Milnor recently introduced the local Picard-Lefschetz-monodromy of an isolated singularity of a hypersurface. This is an important tool in the investigation of the topology of singularities. TheExpand
On the Periods of Certain Rational Integrals: II
In this section we want to re-prove the results of ?? 4 and 8 using sheaf cohomology. One reason for doing this is to clarify the discussion in those paragraphs and, in particular, to show howExpand
Intersection form for quasi-homogeneous singularities
We consider quasi-homogeneous polynomials with an isolated singular point at the origin. We calculate the mixed Hodge structure of the cohomology of the Milnor fiber and give a proof for a conjectureExpand
Syzygies of Jacobian ideals and defects of linear systems
Our main result describes the relation between the syzygies involving the first order partial derivatives $f_0,...,f_n$ of a homogeneous polynomial $f\in \C[x_0,...x_n]$ and the defect of the linearExpand
Jumping coefficients and spectrum of a hyperplane arrangement
• Mathematics
• 2009
In an earlier version of this paper written by the second named author, we showed that the jumping coefficients of a hyperplane arrangement depend only on the combinatorial data of the arrangement asExpand
The local cohomology of the jacobian ring
We study the 0-th local cohomology module of the jacobian ring of a singular reduced complex projective hypersurface X, by relating it to the sheaf of logarithmic vector field along X. We investigateExpand