Projective Varieties with Cones as Tangential Sections

New Zealand Journal of Mathematics

Vol. 38, (2008), Pages 93-97

E. Ballico

Dept. of Mathematics

University of Trento

38050 Povo (TN)

ITALY

Abstract Let $X \subset P$n be an integral non-degenerate m-dimensional variety defined over an algebraically closed field $\mathbb K$. Assume the existence of a non-empty open subset U of Xreg such that TP $X \bigcap X$ is an (m − 1)-dimensional cone with vertex containing P. Here we prove that either X is a quadric hypersurface or char($\mathbb K$) = p > 0, n = m + 1, deg(X) = pe for some $e \ge 1$ and there is a codimension two linear subspace $W \subset P$n such that $W \subset T_{P} X$ for every $P \in X_{reg}$. We also give an "explicit" description (in terms of polynomial equations) of all examples arising in the latter case; dim(Sing(X)) = (m − 1) for every such X.

Keywords Tangent space, cone, quadric hypersurface, strange curve, strange variety

Classification (MSC2000) 14N05