Matthew Satriano
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Research Interests
Matthew Satriano has a total of 40 co-authors. Here are the top co-authors by number of shared publications:
- Jason P. Bell
University of Waterloo
8 shared publications - David McKinnon
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
7 shared publications - Dragos Ghioca
University of British Columbia
5 shared publications - Oliver Pechenik
University of Waterloo
5 shared publications - Andrew P. Staal
Department of Pure Mathematics, University of Waterloo, Waterloo, Canada
3 shared publications - Colin Ingalls
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada
3 shared publications
Latest Publications
- Open Access
A cohomological interpretation for stringy Hodge numbers
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 $\mathcal{X}$.Open AccessA cohomological interpretation for stringy Hodge numbers
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 $\mathcal{X}$. - Open Access
$K$-Equivalence and Integral Cohomology
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.
Open Access$K$-Equivalence and Integral Cohomology
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.
- Open Access
Root stack valuative criterion for good moduli spaces
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.
Open AccessRoot stack valuative criterion for good moduli spaces
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.
- Open AccessAn arithmetic analog of Klein's classification of finite subgroups of $\mathrm{SL}_2(\mathbb{C})$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.Open AccessAn arithmetic analog of Klein's classification of finite subgroups of $\mathrm{SL}_2(\mathbb{C})$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.
- 4 April 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 378 (7) , 4509-4544
On stacky surfaces and noncommutative surfaces
Letbe an algebraically closed field of characteristic or zero. Let be a tame order of global dimension over a normal surface over such that is locally a direct summand of . We prove that there is a -gerbe over a smooth tame algebraic stack whose generic stabilizer is trivial, with coarse space such that the category of 1-twisted coherent sheaves on is equivalent to the category of coherent sheaves of modules on . Moreover, the stack is constructed explicitly through a sequence of root stacks, canonical stacks, and gerbes. This extends results of Reiten and Van den Bergh to finite characteristic and the global situation. As applications, in characteristic we prove that such orders are geometric noncommutative schemes in the sense of Orlov, and we study relations with Hochschild cohomology and Connes’ convolution algebra. - 4 April 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 378 (7) , 4509-4544
On stacky surfaces and noncommutative surfaces
Letbe an algebraically closed field of characteristic or zero. Let be a tame order of global dimension over a normal surface over such that is locally a direct summand of . We prove that there is a -gerbe over a smooth tame algebraic stack whose generic stabilizer is trivial, with coarse space such that the category of 1-twisted coherent sheaves on is equivalent to the category of coherent sheaves of modules on . Moreover, the stack is constructed explicitly through a sequence of root stacks, canonical stacks, and gerbes. This extends results of Reiten and Van den Bergh to finite characteristic and the global situation. As applications, in characteristic we prove that such orders are geometric noncommutative schemes in the sense of Orlov, and we study relations with Hochschild cohomology and Connes’ convolution algebra. - Open Access
On rational double points over nonclosed fields
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 $\textrm{SL}_{2,k}$.Open AccessOn rational double points over nonclosed fields
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 $\textrm{SL}_{2,k}$. - 10 March 2025
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 153 (05) , 1903-1915
Approximating rational points on surfaces
Let X X be a smooth projective algebraic variety over a number field k k and P ∈ X ( k ) P\in X(k) . McKinnon [J. Algebraic Geom. 16 (2007), pp. 257–303] conjectured that, in a precise sense, if rational points on X X are dense enough, then the best rational approximations to P P must lie on a curve. We present a strategy for deducing a slightly weaker conjecture from Vojta’s conjecture, and execute the strategy for the full conjecture for split surfaces.
- 10 March 2025
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 153 (05) , 1903-1915
Approximating rational points on surfaces
Let X X be a smooth projective algebraic variety over a number field k k and P ∈ X ( k ) P\in X(k) . McKinnon [J. Algebraic Geom. 16 (2007), pp. 257–303] conjectured that, in a precise sense, if rational points on X X are dense enough, then the best rational approximations to P P must lie on a curve. We present a strategy for deducing a slightly weaker conjecture from Vojta’s conjecture, and execute the strategy for the full conjecture for split surfaces.
- 22 January 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 378 (3) , 2247-2248
Corrigendum to “Approximating rational points on toric varieties”
This paper corrects an error in the statement and proof of Proposition 1.3 of [Trans. Amer. Math. Soc. 374 (2021), pp. 3557–3577].
- 22 January 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 378 (3) , 2247-2248
Corrigendum to “Approximating rational points on toric varieties”
This paper corrects an error in the statement and proof of Proposition 1.3 of [Trans. Amer. Math. Soc. 374 (2021), pp. 3557–3577].
- Open Access
Stringy Hodge numbers via crepant resolutions by Artin stacks
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.Open AccessStringy Hodge numbers via crepant resolutions by Artin stacks
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. - Open Access
- 3 October 2024
- Published by The Electronic Journal of Combinatorics in The Electronic Journal of Combinatorics
- Vol. 31 (4) , P4.43
On the Algebra Generated by Three Commuting Matrices: Combinatorial Cases
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.- 3 October 2024
- Published by The Electronic Journal of Combinatorics in The Electronic Journal of Combinatorics
- Vol. 31 (4) , P4.43
Open AccessOn the Algebra Generated by Three Commuting Matrices: Combinatorial Cases
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.