# Projective Varieties with Cones as Tangential Sections

### From NZJM

**New Zealand Journal of Mathematics**

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

E. Ballico

Dept. of Mathematics

University of Trento

38050 Povo (TN)

ITALY

mailto:ballico@science.unitn.it

**Abstract** Let ^{n} be an integral non-degenerate
*m*-dimensional variety defined over an algebraically closed field
. Assume the existence of a non-empty open subset *U* of *X*_{reg} such that *T*_{P} is an (*m* − 1)-dimensional cone with vertex containing *P*. Here we prove that either *X* is a quadric hypersurface or char() = *p* > 0, *n* = *m* + 1, *d**e**g*(*X*) = *p*^{e} for some and there is a codimension two linear subspace ^{n} such that for every . We also give an "explicit" description (in terms of polynomial equations) of all examples arising
in the latter case; *d**i**m*(*S**i**n**g*(*X*)) = (*m* − 1) for every such *X*.

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

**Classification** (MSC2000) 14N05