arXiv subjects
Publications and source records attributed to Matthew Satriano.
Batyrev's non-negativity conjecture has been a major guiding conjecture in mirror symmetry, motivic integration, and the McKay correspondence. We show the conjecture is true in dimension at most 4 and give counter-examples in dimension at least 5.
We obtain a motivic and a cohomological McKay correspondence for finite linearly reductive group schemes in arbitrary characteristic. In particular, we prove that if $V$ is a finite dimensional vector space and $G$ is a finite linearly reductive subgroup scheme of $\mathrm{SL}(V)$, then the Euler number of any crepant resolution of $V/G$ is equal to the number of irreducible algebraic representations of $G$. We obtain these McKay correspondences as a consequence of a motivic change of variables formula applied to $[V/G] \to V/G$. If $G$ is non-reduced, as can happen in positive characteristic, the stack quotient $[V/G]$ is not Deligne-Mumford. Therefore in order to prove this change of variables formula and the resulting McKay correspondences, we generalize the authors' theory of motivic integration for Artin stacks to arbitrary characteristic, which may be of independent interest.
Batyrev's conjecture on the non-negativity of stringy Hodge numbers has been a fundamental open problem, guiding and motivating many beautiful mathematical results in motivic integration, mirror symmetry, and the McKay correspondence. Let $M_0$ be the coarse moduli space of rank 2 semistable bundles with trivial determinant over a fixed smooth projective genus 3 curve. Using a formula, obtained by Kiem and Kiem-Li, for the stringy $E$-function of $M_0$, we verify that $M_0 \times\mathbb{P}^1$ is a counter-example to Batyrev's conjecture.
We obtain a cohomological interpretation for Batyrev's stringy Hodge numbers in the full generality in which they are defined. In a previous paper, the second and third authors used motivic integration to define the stringy Hodge--Deligne invariant of a smooth Artin stack $\mathcal{X}$ and proved that when $\mathcal{X}$ is a crepant resolution of a variety $Y$ with log-terminal singularities, the generating function for the stringy Hodge numbers of $Y$ is equal to the stringy Hodge--Deligne invariant of $\mathcal{X}$. In this paper, we introduce a cohomology theory $H_{\mathrm{str}}^*(\mathcal{X})$ that computes the stringy Hodge--Deligne invariant of $\mathcal{X}$. Since, by previous work of the second and third authors, all varieties with log-terminal singularities admit a crepant resolution by an Artin stack, this gives a cohomological interpretation for stringy Hodge numbers of any variety with log-terminal singularities. We also show that in the special case where $\mathcal{X}$ is Deligne--Mumford, $H_{\mathrm{str}}^*(\mathcal{X})$ coincides with the orbifold cohomology of $\mathcal{X}$.
We introduce an integral version of the Hodge polynomial, which encodes the integral cohomology of smooth projective varieties. We prove it extends to a function which is well-defined on the Grothendieck ring of varieties and we obtain as a consequence that $K$-equivalent smooth projective varieties have isomorphic integral cohomology groups.
Motzkin and Taussky (and independently, Gerstenhaber) proved that the unital algebra generated by a pair of commuting $d\times d$ matrices over a field has dimension at most $d$. Since then, it has remained an open problem to determine whether the analogous statement is true for triples of matrices which pairwise commute. We answer this question for combinatorially-motivated classes of such triples.
We prove a root stack valuative criterion for good moduli space maps and for gerbes for reductive groups under some mild assumptions on the residue characteristic. We give several applications to parahoric extension for torsors, rational points on stacks, gerbes and homogeneous spaces, and the geometry of fibrations.
Let $K$ be a number field with ring of integers $\mathcal{O}_K$. We describe and classify finite, flat, and linearly reductive subgroup schemes of $\mathrm{SL}_2$ over $\mathrm{Spec}\:\mathcal{O}_K$. We also establish finiteness results for these group schemes, as well as density results for the associated quotient singularities.
We give an unconditional proof of the Coba conjecture for wonderful compactifications of adjoint type for semisimple Lie groups of type $A_n$. We also give a proof of a slightly weaker conjecture for wonderful compactifications of adjoint type for arbitrary Lie groups.
We compute the equations of all rational double point singularities and we determine their types over perfect ground fields $k$ that arise as quotient singularities by finite linearly reductive subgroup schemes of $\textrm{SL}_{2,k}$.
In a previous paper we showed that any variety with log-terminal singularities admits a crepant resolution by a smooth Artin stack. In this paper we prove the converse, thereby proving that a variety admits a crepant resolution by a smooth Artin stack if and only if it has log-terminal singularities. Furthermore if $\mathcal{X} \to Y$ is such a resolution, we obtain a formula for the stringy Hodge numbers of $Y$ in terms of (motivically) integrating an explicit weight function over twisted arcs of $\mathcal{X}$. That weight function takes only finitely many values, so we believe this result provides a plausible avenue for finding a long-sought cohomological interpretation for stringy Hodge numbers. Using that the resulting integral is defined intrinsically in terms of $\mathcal{X}$, we also obtain a notion of stringy Hodge numbers for smooth Artin stacks, that in particular, recovers Chen and Ruan's notion of orbifold Hodge numbers.
We show that any good moduli space $\pi : \mathcal{X} \to Y$ has a splitting after a proper, generically finite covering of $Y$. As an application we generalize Koll\'ar's ampleness lemma to give a criterion for projectivity of a good moduli space.
Soprunov and Soprunova introduced the notion of a good infinite family of toric codes. We prove that such good families do not exist by proving a more general Szemer\'edi-type result: for all $c\in(0,1]$ and all positive integers $N$, subsets of density at least $c$ in $\{0,1,\dots,N-1\}^n$ contain hypercubes of arbitrarily large dimension as $n$ grows.
Determining the limiting behaviour of the Jacobian as the underlying curve degenerates has been the subject of much interest. For nodal singularities, there are beautiful constructions of Caporaso as well as Pandharipande of compactified universal Jacobians over the moduli space of stable curves. Alexeev later obtained a canonical such compactification by extending the Torelli map out of the Deligne-Mumford compactification of $\mathcal{M}_{g,n}$. In contrast, Alexeev and Brunyate proved that the Torelli map does not extend over the cuspidal locus in Schubert's alternative compactification of pseudostable curves. In this paper, we consider curves with singularities that locally look like the axes in $m$-space, which we call axis-like singularities. We construct an alternative compactification of $\mathcal{M}_{g,n}$ consisting of curves with such singularities and prove that the Torelli map extends out of this compactification. Furthermore, for every alternative compactification in the sense of Smyth, we identify an axis-like locus over which the Torelli map extends.
Let $X$ be a smooth projective algebraic variety over a number field $k$ and $P$ in $X(k)$. In 2007, the second author conjectured that, in a precise sense, if rational points on $X$ are dense enough, then the best rational approximations to $P$ must lie on a curve. We present a strategy for deducing a slightly weaker conjecture from Vojta's conjecture, and execute this strategy for the full conjecture for split surfaces.
Gerstenhaber proved in 1961 that the unital algebra generated by a pair of commuting $d\times d$ matrices over a field has dimension at most $d$. It is an open problem whether the analogous statement is true for triples of matrices which pairwise commute. We answer this question for special classes of triples of matrices arising from combinatorial data.
We introduce a natural generalization of twisted maps, called \emph{warped maps}. While twisted maps play an important role in the study of Deligne--Mumford stacks, warped maps are better suited for studying Artin stacks. Heuristically, warped maps see the hidden proper-like behavior satisfied by good moduli space maps. Specifically, we show that every arc of a good moduli space admits a \emph{canonical} lift, in a warped sense, thereby proving a valuative criterion for good moduli spaces. Furthermore, we prove that warped maps to an Artin stack $\mathcal{X}$ are given by usual maps to an auxiliary Artin stack $\mathscr{W}(\mathcal{X})$, immediately obtaining a versatile framework for bootstrapping results about usual maps to the setting of warped maps. As an application we obtain a motivic change of variables formula which, given a stacky resolution of singularities $\mathcal{X} \to Y$, canonically expresses any given motivic integral over arcs of $Y$ as a certain motivic integral over warped arcs of $\mathcal{X}$. In particular, this yields a McKay correspondence for linearly reductive groups.
Let $\mathcal{X} \to Y$ be a birational modification of a variety by an Artin stack. In previous work, under the assumption that $\mathcal{X}$ is smooth, we proved a change of variables formula relating motivic integrals over arcs of $Y$ to motivic integrals over arcs of $\mathcal{X}$. In this paper, we extend that result to the case where $\mathcal{X}$ is singular. We may therefore apply this generalized formula to the so-called warping stack $\mathscr{W}(\mathcal{X})$ of $\mathcal{X}$, which may be singular even when $\mathcal{X}$ is smooth. We thus obtain a change of variables formula \emph{canonically} expressing any given motivic integral over arcs of $Y$ as a motivic integral over \emph{warped arcs} of $\mathcal{X}$.