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

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

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

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 $e+1$-th blowup dominates the $e$-th, locally or globally. It is shown that the answer is affirmative (globally for any $e$) 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

Universal flattening of Frobenius

For a variety $X$ of positive characteristic and a non-negative integer $e$, we define its $e$-th F-blowup to be the universal flattening of the $e$-iterated Frobenius of $X$. Thus we have the sequence (a set labeled by non-negative integers) of blowups of $X$. 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 $G$-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

The pp-cyclic McKay correspondence via motivic integration

We study the McKay correspondence for representations of the cyclic group of order $p$ in characteristic $p$. 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 $p$ 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