Hypersurfaces in \(\mathbb{P}^{n}\) with 1-parameter symmetry groups. II (Q2655180)

The authors study hypersurfaces of degree \(d\) in \({\mathbb{P}^n({\mathbb C})}\) with isolated singularities, admitting a group of linear symmetries. The paper is a continuation of [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 456, No. 2002, 2515--2541 (2000; Zbl 0980.32011)], where the case of semi-simple groups was studied. Here they analyse the unipotent case. The possible groups are determined. There are four cases. In each case they discuss the geometry of the action, reduce the hypersurface to a normal form, find the singular points, study their nature, and calculate the Milnor numbers. The total Tjurina number is bounded by \((d - 1)^{n-2}(d ^{2} - 3d + 3)\). The hypersurface is called oversymmetric if this value is attained. In many cases \(\tau \) is calculated. The oversymmetric situations are characterised.