PROJECTIVE VARIETIES WITH CONES AS TANGENTIAL SECTIONS

Abstract. Let X ⊂ Pn be an integral non-degenerate m-dimensional variety defined over an algebraically closed field K. Assume the existence of a non-empty open subset U of Xreg such that TPX ∩X is an (m − 1)-dimensional cone with vertex containing P. Here we prove that either X is a quadric hyper-surface or char(K) = p> 0, n = m+ 1, deg(X) = pe for some e ≥ 1 and there is a codimension two linear subspace W ⊂ Pn such that W ⊂ TPX for every P ∈ Xreg. 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. 1. Varieties With Cones As Tangential Sections Let X ⊂ Pn be an integral non-degenerate m-dimensional variety defined over an algebraically closed field K. For any P ∈ Xreg let TPX ⊂ Pn denote the Zariski tangent space of X at P. Here we prove the following result. Theorem 1.1. Let X ⊂ Pn, n ≥ 3, be an integral non-degenerate m-dimensional variety such that there is a non-empty open subset U of Xreg such that TPX ∩X i