PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 121, Number 4, August 1994 KOSZUL COMPLEXES AND HYPERSURFACE SINGULARITIES A. D. R. CHOUDARY AND A. DIMCA (Communicated by Louis J. Ratliff, Jr.) Abstract. The behavior of Poincaré series and Betti numbers of complex pro- jective hypersurfaces in terms of their singularities is compared. The PROBLEM Let f e£[xo, ... , x„] be a reduced homogeneous polynomial of degree d and fk=df/dxk, k = 0,l,...,n, its partial derivatives. Let K be the Koszul (homology) complex of the elements fo, ... , fn in the ring 5, where 5 = C[xo, ... , xn]. 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. Received by the editors December 16, 1991 and, in revised form, November 11, 1992. 1991MathematicsSubjectClassification. Primary 13D40, 14B05,14J70. © 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page 1009 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use