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hal-00672133, version 1

## Smooth Divisors of Projective Hypersurfaces

Philippe Ellia 1, Davide Franco 2, Laurent Gruson 3

Commentarii Mathematici Helvetici 83, 2 (2008) 371-385

Abstract: We work over an algebraically closed field of arbitrary characteristic. Ellingsrud-Peskine proved that smooth surfaces in P^4 are subject to strong limitations. Their whole argument is derived from the fact that the sectional genus of surfaces of degree d lying on a hypersurface of degree s varies in an interval of length \frac{d(s-1)^2}{2s}. The aim of the present paper is to show that for smooth codimension two subvarieties of P^n, n\geq 5, one can get a similar result with an interval whose length depends only on s. The main point is Lemma 1.1 whose proof is a direct application of the positivity of N_X(-1) (where N_X is the normal bundle of X in P^n). We get a series of (n-3) inequalities; the first one of which being in the paper of Ellingsrud-Peskine, the second was obtained in a preliminary version (arXiv:math.AG/0406497) by an essentially equivalent but more geometric argument. Then we first derive two consequences: 1) roughly speaking, (Thm. 2.1, Remark 2.2) 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 two subvarieties of P^6 lying on a hypersurface of degree s is limited (Thm. 1.4). The result quoted in 1) is not effective, but 2) is. In the last section we try to obtain precise inequalities connecting the usual numerical invariants of a smooth subcanonical subvariety X of P^n, n\geq 5 (the degree d, the speciality index e, the least degree, s, of an hypersurface containing X). In particular we prove (Thm. 3.12): s\geq n+1.

• 1:  Dipartimento di Matematica
• Università di Ferrara
• 2:  Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”
• Università degli studi di Napoli Federico II
• 3:  Laboratoire de Mathématiques de Versailles (LM-Versailles)
• CNRS : UMR8100 – Université de Versailles Saint-Quentin-en-Yvelines
• Domain : Mathematics/Algebraic Geometry
• Comment : (Submitted on 20 Jul 2005 (v1) – last revised 27 Jul 2006 (this version – v4))

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• Submitted on: Monday, 20 February 2012 15:42:27
• Updated on: Tuesday, 13 March 2012 10:30:20