Hypersurfaces in P n with 1-parameter symmetry groups II

Abstract

We assume V a hypersurface of degree d in \({P^n({\mathbb C})}\) with isolated singularities and not a cone, admitting a group G of linear symmetries. In earlier work we treated the case when G is semi-simple; here we analyse the unipotent case. Our first main result lists the possible groups G. In each case we discuss the geometry of the action, reduce V to a normal form, find the singular points, study their nature, and calculate the Milnor numbers. The Tjurina number τ(V) ≤ (d − 1)n–2(d2 − 3d + 3): we call V oversymmetric if this value is attained. We calculate τ in many cases, and characterise the oversymmetric situations. In particular, we list all the cases with dim(G) = 2 which are the oversymmetric cases with d = 3.