On a Special Class of Smooth Codimension Two Subvarieties
نویسنده
چکیده
We work on an algebraically closed field of characteristic zero. By Lefschetz’s theorem, a smooth codimension two subvariety X ⊂ P, n ≥ 4, which is not a complete intersection, lying on a hypersurface Σ, verifies dim(X ∩ Sing(Σ)) ≥ n− 4. In this paper we deal with a situation in which the singular locus of Σ is as large as can be, but, at the same time, the simplest possible: we assume Σ is an hypersurface of degree m with an (m-2)-uple linear subspace of codimension two. More generally, we are concerned with smooth codimension two subvarieties X ⊂ P , n ≥ 5. In the first part we consider smooth subcanonical threefolds X ⊂ P and we prove that if deg(X) ≤ 25, then X is a complete intersection (Prop. 2.2). In the second section we study a particular class of codimension two subvarieties and we prove the following result.
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