Koszul complexes and hypersurface singularities

Abstract

The aim of this note is to consider relations between (i) the homology groups Hk(K) of the Koszul complex AT, and (ii) the singularities of the hypersurface V defined by the equation / = 0 in the complex projective space P". Note that Hq(K) is just the Milnor (or Jacobian) algebra of / given by M(f) = S/(fo,...,fn).Also note that all the homology groups Hk(K) are graded objects in a natural way (see §1 for details).For any graded object A we denote by Am its homogeneous component of degree m and by P(A) the corresponding Poincaré series, i.e., P(A)(t)=Y(àimAm)tm.To the best of our knowledge, the only general results relating (i) and (ii) are the following. Proposition 1. The following statements are equivalent:(i) Hk(K) = 0 for k > 0 and P(M(f))(t) = (l-td-[)"+x/(l-t)"+x.(ii) The hypersurface V is smooth.

On the syzygies and Alexander polynomials of nodal hypersurfaces

“…Let \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$S=\mathbb {C}[x_0,\ldots ,x_n]$\end{document} be the graded ring of polynomials in x 0 , …, x n with complex coefficients and denote by S r the vector space of homogeneous polynomials in S of degree r . For any polynomial f ∈ S r , we define the Jacobian ideal J f ⊂ S as the ideal spanned by the partial derivatives f 0 , …, f n of f with respect to x 0 , …, x n and the corresponding graded Milnor (or Jacobian ) algebra by The study of such Milnor algebras is related to the singularities of the corresponding projective hypersurface D : f = 0, see 3, as well as to the mixed Hodge theory of the hypersurface D and of its complement \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$U=\mathbb {P}^n \setminus D$\end{document}, see the foundational article by Griffiths 14 and also 6, 8, 9, 11. For mixed Hodge theory we refer to 19.…”

Free Divisors versus Stability and Coincidence Thresholds

“…The series coincides up to a point ct(D) and Choudary-Dimca Theorem [1] ensures that from an index st(D) we have stability.…”

On the Poincaré series of the Milnor algebra of arrangements

“…(2) This shows that in relation (1) only the term P ((z,) I'1Der(:, (2) This shows that in relation (1) only the term P ((z,) I'1Der(:,…”

“…Note also that from [6] it is possible to find that the leading coefficient of the Hilbert polynomial of the Milnor algebra of an arrangement depends only on the combinatorics. (For k = 3, this also follows from [2].) We shall try to find the Poincar~ series of the Milnor algebra of an arrangement by induction.…”