arXiv subjects
Publications and source records attributed to Takehiko Yasuda.
We construct an algorithm for the minimal model program in dimension three over a computable field of characteristic zero admitting a splitting algorithm, such as number fields and the field of algebraic numbers. As auxiliary results, we also construct algorithms for computing bigraded global Hom modules and for computing Stein factorization.
We study the McKay correspondence for the representations of certain wild-by-tame split metacyclic groups whose order is divisible by the characteristic of the base field. We calculate the stringy motive of the quotient variety and find a formula for its stringy Euler number. As a consequence, we prove that a crepant resolution of the quotient variety (provided one exists) does not in general have Euler characteristic equal to the number of conjugacy classes in $G$, in contrast to the classical case. In particular, we show it depends on the choice of representation as well as the group. As part of this, we compute the v-function associated to a $G$-representation, corresponding to a stacky local height function.
We study exceptional loci of F-blowups of normal toric varieties. In the $\Q$-factorial case, this study amounts to studying the exceptional loci of $G$-Hilbert schemes. We give a formula for the dimension of the center of a prime divisor on the F-blowup in terms of combinatorial data, together with an algorithm for computing it. Moreover, we study the relation between F-blowups and essential divisors for three-dimensional terminal singularities and canonical singularities. Finally, we give a simple condition ensuring that a prime divisor over the given toric variety has a positive-dimensional center on the F-blowup.
We study linear $α_p$-actions on affine spaces and the associated quotient singularities, using explicit stacky resolutions. We describe when the quotient singularities are log canonical, canonical or terminal, and we compute their stringy motivic invariants. The second author and Fabio Tonini conjectured that these invariants coincide with those of linear $\mathbb{Z}/p$-quotients: our approach reduces this conjecture to an equality of explicit multi-sets, which we check for a large number of primes using a computer software. A general proof of the equality of multi-sets is given in the appendix written by Linus Rösler.
The quotient variety associated to a permutation representation of a finite group has only canonical singularities in arbitrary characteristic. Moreover, the log pair associated to such a representation is Kawamata log terminal except in characteristic two, and log canonical in arbitrary characteristic.
In this paper, we explicitly describe the orbifold pseudo-effective cone of a split toric stack.
In this paper, we propose a new framework for studying the distribution of rational points on DM stacks of positive characteristic. Our primary focus is on wild stacks, which existing frameworks do not address. There was not even a satisfactory notion of heights for such stacks. First, we introduce a new kind of height function that extends the authors' idea from their preceding paper on characteristic-zero stacks. This new height function is more general and flexible than the previous one. Examples of the new height function include discriminants of torsors, minimal discriminants, and conductors of elliptic curves in characteristic three. Next, we formulate a generalization of the Batyrev-Manin conjecture for rational points of DM stacks in positive characteristic relative to this new type of height function. We provide several pieces of evidence for this generalization.
Let $F$ be a global field of characteristic $p > 0$ and $G$ a finite abelian $p$-group. In this paper we treat the question of counting $G$-torsors over $F$ for certain heights developed in [DY25].
We investigate the behavior of stringy motives under Galois quasi-étale covers. We prove that they descend under such covers in a sense defined via their Poincaré realizations. Further, we show that such descent is strict in the presence of ramification. As a corollary, we reduce the problem regarding the finiteness of the étale fundamental group of KLT singularities to a DCC property for their stringy motives. We verify such DCC property for surfaces in arbitrary characteristic. As an application, we give a characteristic-free proof for the finiteness of the étale fundamental group of log terminal surface singularities, which was unknown in equal characteristics 2 and 3 and in mixed characteristics.
We prove motivic versions of mass formulas by Krasner, Serre and Bhargava concerning (weighted) counts of extensions of local fields.
We investigate the relation between essential divisors and F-blowups, in particular, address the problem whether all essential divisors appear on the $e$-th F-blowup for large enough $e$. Focusing on the case of normal affine toric varieties, we establish a simple sufficient condition for a divisor over the given toric variety to appear on the normalized limit F-blowup as a prime divisor. As a corollary, we show that if a normal toric variety has a crepant resolution, then the above problem has a positive answer, provided that we use the notion of essential divisors in the sense of Bouvier and Gonzalez-Sprinberg. We also provide an example of toric threefold singularities for which a non-essential divisor appears on an F-blowup.
We define a new height function on rational points of a DM (Deligne-Mumford) stack over a number field. This generalizes a generalized discriminant of Ellenberg-Venkatesh, the height function recently introduced by Ellenberg-Satriano-Zureick-Brown (as far as DM stacks over number fields are concerned), and the quasi-toric height function on weighted projective stacks by Darda. Generalizing the Manin conjecture and the more general Batyrev-Manin conjecture, we formulate a few conjectures on the asymptotic behavior of the number of rational points of a DM stack with bounded height. To formulate the Batyrev-Manin conjecture for DM stacks, we introduce the orbifold versions of the so-called $a$- and $b$-invariants. When applied to the classifying stack of a finite group, these conjectures specialize to the Malle conjecture, except that we remove certain thin subsets from counting. More precisely, we remove breaking thin subsets, which have been studied in the case of varieties by people including Hassett, Tschinkel, Tanimoto, Lehmann and Sengupta, and can be generalized to DM stack thanks to our generalization of $a$- and $b$-invariants. The breaking thin subset enables us to reinterpret Klüners' counterexample to the Malle conjecture.
Split toric stacks over a number field $F$ are natural generalization of split toric varieties over $F$. Notable examples are weighted projective stacks. In our previous work, we defined heights on Deligne-Mumford stacks using so-called raised line bundles and made predictions on asymptotic formulas of the number of rational points of bounded height. In this paper, we prove that the number of rational points of any split toric stack of bounded height with respect to the anti-canonical raised line bundle satisfies one of our predictions, the Manin conjecture for Deligne-Mumford stacks.
We show basic properties of higher Jacobian matrices and higher Jacobian ideals for functions and apply it to obtain two results concerning singularities of functions. Firstly, we prove that a higher Nash blowup algebra is invariant under contact equivalences, which was recently conjectured by Hussain, Ma, Yau and Zuo. Secondly, we obtain an analogue of a result on motivic nearby cycles by Bussi, Joyce and Meinhardt.
We prove existence of non-commutative crepant resolutions (in the sense of van den Bergh) of quotient singularities by finite and linearly reductive group schemes in positive characteristic. In dimension two, we relate these to resolutions of singularities provided by G-Hilbert schemes and F-blowups. As an application, we establish and recover results concerning resolutions for toric singularities, as well as canonical, log terminal, and F-regular singularities in dimension 2.
We show that if a finitely generated prime algebra $Δ$ is a finitely generated maximal Cohen-Macaulay module over its centre $Z$, and has global dimension equal to $\dim Z$, then the pair given by its centre and ramification divisor is Kawamata log terminal.
A semicommutative finite group scheme is a finite group scheme which can be obtained from commutative finite group schemes by iterated performing semidirect products with commutative kernels and taking quotients by normal subgroups. In this article, for an étale tame semicommutative finite group scheme $G$, we give a lower bound on the number of connected $G$-torsors of bounded height (such as discriminant).
Let $F$ be a global field. Let $G$ be a non trivial finite étale tame $F$-group scheme. We define height functions on the set of $G$-torsors over $F,$ which generalize the usual heights such as discriminant. As an analogue of the Malle conjecture for group schemes, we formulate a conjecture on the asymptotic behavior of the number of $G$-torsors over $F$ of bounded height. This is a special case of our more general Stacky Batyrev-Manin conjecture from arXiv:2207.03645. The conjectured asymptotic is proven for the case $G$ is commutative. When $F$ is a number field, the leading constant is expressed as a product of certain arithmetic invariants of $G$ and a volume of a space attached to $G$. Moreover, an equidistribution property of $G$-torsors in the space is established.