The geometry of a desingularization Ym 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 pg(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 pg(Y) > 0 if d [ges ] 2n + 1 − m
and KY separates generic points if d [ges ] 2n + 2 − m
(see also [1, 4]). For Xn a generic complete intersection of type
(d1, …, dk) in any smooth polarized
(n + k)-fold M, Ein [6, 7] proved that
pg(Y) > 0 if
d1 + … + dk [ges ] 2n + k − m + 1
and Y is of general type if
d1 + … + dk [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 KY (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 KY 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 KY involving
KW (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 Yr 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.