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.

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For line arrangements in P 2 with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal basis of the abelianization. We consider higher dimensional analogs of the above situation. For these analogs, we give purely combinatorial complete descriptions of the following topological invariants (over an arbitrary field): the twisted homology of the complement, with arbitrary rank one coefficients; the homology of the associated Milnor fiber and Alexander cover, including monodromy actions; the coinvariants of the first higher non-trivial homotopy group of the Alexander cover, with the induced monodromy action.

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We investigate the first-order Thom–Boardman singularity sets of the dual mapping for an arbitrary (and then for a generic) smooth hypersurface in the complex projective space ℙn. Our results focus on nonemptiness, connectedness, regular stratifications and numerical invariants for these sets.