Smooth divisors of projective hypersurfaces

Abstract

Let $X\subset P^n$ be a smooth codimension 2 subvariety. We first prove a “positivity lemma” (Lemma 1.1) which is a direct application of the positivity of $N_X(-1)$. Then we first derive two consequences:

1. Roughly speaking the family of “biliaison classes” of smooth subvarieties of $P^5$ lying on a hypersurface of degree s is limited.
2. The family of smooth codimension 2 subvarieties of $P^6$ 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 $X\subset P^n$, $n\ge 5$ (the degree $d$, the integer $e$ such that ${\omega }_X\backsimeq {\mathcal{O}}_X(e)$, the least degree, $s$, of a hypersurface containing $X$ ). In particular we prove: $s\ge n+1$ if $X$ is not a complete intersection.