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 Xn ⊂ 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 ΩXOX (e), the least degree, s, of a hypersurface containing X ). In particular we prove: sn + 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