Smooth divisors of projective hypersurfaces

Abstract

Let X n ⊂ ℙ be a smooth codimension 2 subvariety. We first prove a “positivity lemma” (Lemma 1.1) which is a direct application of the positivity of NX (−1). Then we first derive two consequences:

1. Roughly speaking the family of “biliaison classes” of smooth subvarieties of ℙ5 lying on a hypersurface of degree s is limited.
2. The family of smooth codimension 2 subvarieties of ℙ6 lying on a hypersurface of degree s is limited.

The result in 1) is not effective, but 2) is. Then we obtain precise inequalities connecting the usual numerical invariants of a smooth subcanonical subvariety X  ⊂ ℙn, n ≥ 5 (the degree d, the integer e such that ΩX ⋍ OX (e), the least degree, s, of a hypersurface containing X ). In particular we prove: s ≥ n + 1 if X is not a complete intersection.