Projective Varieties with Cones as Tangential Sections


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New Zealand Journal of Mathematics

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

E. Ballico

Dept. of Mathematics

University of Trento

38050 Povo (TN)


Abstract Let X \subset Pn 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 Pn 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

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