# Smooth divisors of projective hypersurfaces

### Philippe Ellia

Università di Ferrara, Italy### Davide Franco

Università degli Studi di Napoli Federico II, Italy### Laurent Gruson

Université de Versailles-Saint Quentin en Yvelines, France

## Abstract

Let $X ⊂P_{n}$ 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:

- Roughly speaking the family of “biliaison classes” of smooth subvarieties of $P_{5}$ lying on a hypersurface of degree s is limited.
- 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 $X ⊂P_{n}$, $n≥5$ (the degree $d$, the integer $e$ such that $ω_{X}⋍O_{X} (e)$, the least degree, $s$, of a hypersurface containing $X$ ). In particular we prove: $s≥n +1$ if $X$ is not a complete intersection.

## Cite this article

Philippe Ellia, Davide Franco, Laurent Gruson, Smooth divisors of projective hypersurfaces. Comment. Math. Helv. 83 (2008), no. 2, pp. 371–385

DOI 10.4171/CMH/128