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


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

  1. Roughly speaking the family of “biliaison classes” of smooth subvarieties of lying on a hypersurface of degree s is limited.
  2. The family of smooth codimension 2 subvarieties of lying on a hypersurface of degree 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 , (the degree , the integer such that , the least degree, , of a hypersurface containing  ). In particular we prove: if 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