Singularities with exact Poincaré complex but not quasihomogeneous. (English) Zbl 0708.14018

Pfister, Gerhard; Schönemann, Hans

Rev. Mat. Univ. Complutense Madr. 2, No. 2-3, 161-171 (1989).

For an isolated hypersurface singularity quasihomogeneity can be characterized by the exactness of the associated Poincaré complex as well as by saying that its Milnor number equals its Tjurina numer. This is due to K. Saito [Invent. Math. 14, 123-142 (1971; Zbl 0224.32011)]. For a complete intersection curve with isolated singularity this characterisation does not hold anymore: If such a singularity is quasihomogeneous, then its Milnor number equals its Tjurina number (which in fact is equivalent) and its Poincaré-complex is exact. On the other hand the authors show by computing Milnor and Tjurina numbers that the unimodal complete intersection singularities \(FT_{k,l}\) of Wall’s classification [C. T. C. Wall in Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 2, 625-640 (1983; Zbl 0519.58013)] are not quasihomogeneous but still have an exact Poincaré complex. To determine the numerical invariants they use Mora’s algorithm to compute a Gröbner base. The exactness of the complex in question is established by a result of Reiffen.

Reviewer: F.W.Knoeller

MSC:

14H20	Singularities of curves, local rings
14B05	Singularities in algebraic geometry
14M10	Complete intersections

Keywords:

quasihomogeneity of isolated singularity; complete intersection curve; Milnor number; Tjurina number; unimodal complete intersection singularities; Gröbner base