Smooth Divisors of Projective Hypersurfaces

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.