Hypersurfaces in Pⁿ with 1-parameter symmetry groups II

TY - JOUR

T1 - Hypersurfaces in Pn with 1-parameter symmetry groups II

AU - Plessis, Andrew du

AU - Wall, C.T.C.

PY - 2010

Y1 - 2010

N2 - We assume V a hypersurface of degree d in 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(d 2 − 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.

AB - We assume V a hypersurface of degree d in 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(d 2 − 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.

U2 - 10.1007/s00229-009-0304-1

DO - 10.1007/s00229-009-0304-1

M3 - Journal article

SN - 0025-2611

VL - 131

SP - 111

EP - 143

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

IS - 1-2

ER -