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In this paper, we show that for every nonnilpotent hyperbolic map
on an infra-nilmanifold, the set
is cofinite in
. This is a generalization of a similar result for expanding maps in Lee and Zhao (J. Math. Soc. Japan 59(1) (2007), 179–184). Moreover, we prove that for every nilpotent map
on an infra-nilmanifold,
We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at nonpositive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the
-function defined by Arakawa and Kaneko. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.
be a Noetherian local ring of characteristic
. We introduce and study
-anti-nilpotent singularities, both are defined in terms of the Frobenius actions on the local cohomology modules of
supported at the maximal ideal. We prove that if
-anti-nilpotent for a nonzero divisor
, then so is
. We use these results to obtain new cases on the deformation of
be a field of characteristic
. We give a geometric proof that there are no smooth quartic surfaces
with more than 64 lines (predating work of Degtyarev which improves this bound to 60). We also exhibit a smooth quartic containing 60 lines which thus attains the record in characteristic
This paper concerns the classification of isogeny classes of
-divisible groups with saturated Newton polygons. Let
be a normal Noetherian scheme in positive characteristic
with a prime Weil divisor
-divisible group over
whose geometric fibers over
) have the same Newton polygon. Assume that the Newton polygon of
is saturated in that of
. Our main result (Corollary 1.1) says that
is isogenous to a
-divisible group over
whose geometric fibers are all minimal. As an application, we give a geometric proof of the unpolarized analogue of Oort’s conjecture (Oort, J. Amer. Math. Soc. 17(2) (2004), 267–296; 6.9).
Using the geometry of an almost del Pezzo threefold, we show that the moduli space
one-pointed ineffective spin hyperelliptic curves is rational for every
be a stable Chern character on
, and let
be the moduli space of Gieseker semistable sheaves on
with Chern character
. In this paper, we provide an approach to computing the effective cone of
. We find Brill–Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of
which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on