On Rational Varieties Smooth Except at a Seminormal Singular Point

Abstract

We construct a class of projective rational varieties X of any dimension m ≥ 1, which are smooth except at a point O, with the projective space ℙ ^m as normalization, having smooth branches, and reduced projectivized tangent cone in O. The Hilbert function of X is considered and is explicitly computed when the point O is seminormal. Indeed, we study seminormality, obtaining necessary and sufficient conditions for O to be seminormal and show that in such case the tangent cone is reduced and seminormal.

Key Words:

• Branches
• Hilbert function
• Rational varieties
• Seminormality
• Tangent cone

2000 Mathematics Subject Classification:

• Primary: 14M20
• 14M05
• 14B05

ACKNOWLEDGMENT

The authors were supported in part by MURST and GNSAGA.

Notes

Communicated by C. Pedrini.