# Commentarii Mathematici Helvetici

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**Volume 83, Issue 2, 2008, pp. 371–385**

**DOI: 10.4171/CMH/128**

Published online: 2008-06-30

Smooth divisors of projective hypersurfaces

Philippe Ellia^{[1]}, Davide Franco

^{[2]}and Laurent Gruson

^{[3]}(1) Università di Ferrara, Italy

(2) Università degli Studi di Napoli Federico II, Italy

(3) Université de Versailles-Saint Quentin en Yvelines, France

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 `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.

`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.

*Keywords: *Smooth codimension two subvarieties, projective space,complete intersections, positivity

Ellia Philippe, Franco Davide, Gruson Laurent: Smooth divisors of projective hypersurfaces. *Comment. Math. Helv.* 83 (2008), 371-385. doi: 10.4171/CMH/128