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Hypersurfaces in Pn with 1-parameter symmetry groups II

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  • Department of Mathematical Sciences
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(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.
Original languageEnglish
JournalManuscripta Mathematica
Pages (from-to)111-143
Number of pages33
Publication statusPublished - 2010

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