137 research outputs found
Toward motivic integration over wild Deligne-Mumford stacks
We discuss how the motivic integration will be generalized to wild
Deligne-Mumford stacks, that is, stabilizers may have order divisible by the
characteristic of the base or residue field. We pose several conjectures on
this topic. We also present some possible applications concerning stringy
invariants, resolution of singularities, and weighted counts of extensions of
local fields.Comment: 24 pages; minor corrections, added footnotes to mention subsequent
developments, to appear in the proceedings of the conference "Higher
Dimensional Algebraic Geometry - in honour of Professor Yujiro Kawamata's
sixtieth birthday" (ASPM
Non-adic formal schemes
Our purpose is to make a contribution to the foundation of the theory of
formal scheme. We are interested particularly in non-Noetherian or non-adic
formal schemes, which have been little studied. We redefine the formal scheme
as a proringed space and study its basic properties. We also find several
examples of non-adic formal schemes.Comment: 47 pages, This is a totally revised version of math.AG/060254
On subschemes of formal schemes
We think about what the subscheme of the formal scheme is. Differently form
the ordinary scheme, the formal scheme has different notions of ``subscheme''.
We lay a foundation for these notions and compare them. We also relate them to
singularities of foliations.Comment: 24 page
Densities of rational points and number fields
We relate the problem of counting number fields, in particular, Malle's
conjecture with the problem of counting rational points on singular Fano
varieties, in particular, Batyrev and Tschinkel's generalization of Manin's
conjecture.Comment: 22 pages. Comments are welcom
Universal flattening of Frobenius
For a variety of positive characteristic and a non-negative integer ,
we define its -th F-blowup to be the universal flattening of the
-iterated Frobenius of . Thus we have the sequence (a set labeled by
non-negative integers) of blowups of . Under some condition, the sequence
stabilizes and leads to a nice (for instance, minimal or crepant) resolution.
For tame quotient singularities, the sequence leads to the -Hilbert scheme.Comment: 25 pages; v4. a full revision, notations changed, the isomorphism of
the F-blowup and the G-Hilbert scheme has been generalized to the non-abelian
case, errors corrected, the introduction shortened. v5. minor revision, to
appear in the American Journal of Mathematic
Dimensions of jet schemes of log singularities
The aim of the paper is to characterize Kawamata log terminal singularities
and log canonical singularities by dimensions of jet schemes. It is a
generalization of Mustata's result.Comment: 7 page
Wilder McKay correspondences
A conjectural generalization of the McKay correspondence in terms of stringy
invariants to arbitrary characteristic, including the wild case, was recently
formulated by the author in the case where the given finite group linearly acts
on an affine space. In cases of very special groups and representations, the
conjecture has been verified and related stringy invariants have been
explicitly computed. In this paper, we try to generalize the conjecture and
computations to more complicated situations such as non-linear actions on
possibly singular spaces and non-permutation representations of non-abelian
groups.Comment: 42 pages. The title has been changed from the previous "The motivic
McKay correspondence for non-linear actions on possibly singular spaces". A
new added subject is computations of weight functions and masses for some
non-permutation representations. Comments are welcom
The -cyclic McKay correspondence via motivic integration
We study the McKay correspondence for representations of the cyclic group of
order in characteristic . The main tool is the motivic integration
generalized to quotient stacks associated to representations. Our version of
the change of variables formula leads to an explicit computation of the stringy
invariant of the quotient variety. A consequence is that a crepant resolution
of the quotient variety (if any) has topological Euler characteristic like
in the tame case. Also, we link a crepant resolution with a count of
Artin-Schreier extensions of the power series field with respect to weights
determined by ramification jumps and the representation.Comment: 44 pages, v3: The term "strongly Kawamata log terminal" has been
changed to "stringily Kawamata log terminal," as it is more consistent with
the definitio
Manin's conjecture vs. Malle's conjecture
By a heuristic argument, we relate two conjectures. One is a version of
Manin's conjecture about the distribution of rational points on a Fano variety.
We concern specific singular Fano varieties, namely quotients of projective
spaces by finite group actions, and their singularities play a key role. The
other conjecture is a generalization of Malle's conjecture about the
distribution of extensions of a number field. Main tools are several Dirichlet
series and previously obtained techniques, especially the untwisting, for the
counterpart over a local field.Comment: 28 pages. Any comments are welcome
On monotonicity of F-blowup sequences
For each variety in positive characteristic, there is a series of canonically
defined blowups, called F-blowups. We are interested in the question of whether
the -th blowup dominates the -th, locally or globally. It is shown that
the answer is affirmative (globally for any ) when the given variety is
F-pure. As a corollary, we obtain some result on the stability of the sequence
of F-blowups. We also give a sufficient condition for local domination.Comment: 10 pages, v.2: major revision. the title modified. the proof of the
main result simplified. a key argument in v.1 is now stated as Theorem 1.3.
the toric case is now explained with more details (Section 5), v.3: to appear
in Illinois J. Math., arguments in the toric case improved, It is proved that
the F-blowup does not preserve the F-purit
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