Subvarieties of generic hypersurfaces in any variety

Abstract

The geometry of a desingularization Y^m of an arbitrary subvariety of a generic hypersurface Xⁿ in an ambient variety W (e.g. W = ℙⁿ⁺¹) has received much attention over the past decade or so. Clemens [5] has proved that for m = 1, n = 2, W = ℙ³ and X of degree d, Y has genus g [ges ] 1 + d(d − 5)/2; Xu [13, 14] improved this to g [ges ] d(d −3)/2 − 2 for d [ges ] 5 and showed that if equality holds for d [ges ] 6 then Y is planar; he also gave some lower bounds on the geometric genus p_g(Y) in case m = n − 1. Voisin [10, 11] proved that, for X of degree d in ℙⁿ⁺¹, n [ges ] 3, m [les ] n − 2 then then p_g(Y) > 0 if d [ges ] 2n + 1 − m and K_Y separates generic points if d [ges ] 2n + 2 − m (see also [1, 4]). For Xⁿ a generic complete intersection of type (d₁, …, d_k) in any smooth polarized (n + k)-fold M, Ein [6, 7] proved that p_g(Y) > 0 if d₁ + … + d_k [ges ] 2n + k − m + 1 and Y is of general type if d₁ + … + d_k [ges ] 2n + k − m + 2.

In [2] the first two authors applied the classical method of focal loci of families as in Segre [8] and Ciliberto–Sernesi [3] (really normal bundle considerations) to give a new proof of one of the main results of Xu in [13, 14] and to extend the lower bounds for the genus in the cases of general surfaces in a component of the Noether–Lefschetz locus in ℙ³ and of general projectively Cohen–Macaulay surfaces in ℙ⁴.

In this paper we introduce some notions of ‘filling’ subvarieties and use them to prove two new results which inter alia give another perspective on the above results, especially the genus bounds. The general philosophy is as usual that as Y moves with X, sections of K_Y (or some twist) can be produced through differentiation; but here this is implemented by exploiting and extending an elementary but perhaps surprising technique in the spirit of classical projective geometry which goes back to [2] and which gives a useful lower bound, depending on the dimension of the projective span of Y, on the number of independent sections of K_Y produced by the differentiation process.

As to our results, in Theorem 1 below we extend and refine in the above sense most of the above-quoted results (not including Voisin's), by giving a lower bound on the number of sections of a certain twist of K_Y involving K_W (for a recent extension, including the proof of a conjecture of Clemens, see [15]). Our second result (Theorem 2), based on the notion of ‘r-filling’ subvariety, indicates an apparently new and unexpected direction as it deals with some higher-order tensors on Y, manifested in the form of an effective divisor on a Cartesian product Y^r having certain vanishing order on a diagonal locus as well as on a ‘double point’ locus associated to the map Y → X. As one application, we conclude a lower bound on the number of quadrics (and higher-degree hypersurfaces) containing certain ‘adjoint-type’ projective images of Y (and even on the dimension of the kernel of certain ‘symmetric Gaussian’ maps) (Corollary 2·1 below). Our feeling, however, is that the latter is only the tip of an iceberg and we hope to explore further in this direction in the future.