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 d(s-1)^2/2s. The aim of the present paper is to show that for smooth codimension two subvarieties of P^n, n at least  5, one can get a similar result with an interval whose length depends only on s. The main point is a Lemma 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). As a consequence of our Lemma we get a series of (n-3) inequalities the first one of which being Lemme 1 of Ellingsrud-Peskine. The second Theorem was obtained in a preliminary version by an essentially equivalent but more geometric argument. Then we first derive two consequences: 1) roughly speaking, 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. 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 at least 5 (the degree d, the integer e such that the canonical sheaf of X is e times the hyperplane, the least degree, s, of an hypersurface containing X). In particular we prove thet s is less then or equal to n+1.

Smooth divisors of projective hypersurfaces.

FRANCO, DAVIDE;
2008

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 d(s-1)^2/2s. The aim of the present paper is to show that for smooth codimension two subvarieties of P^n, n at least  5, one can get a similar result with an interval whose length depends only on s. The main point is a Lemma 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). As a consequence of our Lemma we get a series of (n-3) inequalities the first one of which being Lemme 1 of Ellingsrud-Peskine. The second Theorem was obtained in a preliminary version by an essentially equivalent but more geometric argument. Then we first derive two consequences: 1) roughly speaking, 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. 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 at least 5 (the degree d, the integer e such that the canonical sheaf of X is e times the hyperplane, the least degree, s, of an hypersurface containing X). In particular we prove thet s is less then or equal to n+1.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11588/205214
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