arXiv subjects

Matthew Satriano

Publications and source records attributed to Matthew Satriano.

At least 55 records · Page 3Linked to original sources

arXiv2021

Stringy invariants and toric Artin stacks

We propose a conjectural framework for computing Gorenstein measures and stringy Hodge numbers in terms of motivic integration over arcs of smooth Artin stacks, and we verify this framework in the case of fantastacks, which are certain toric Artin stacks that provide (non-separated) resolutions of singularities for toric varieties. Specifically, let $\mathcal{X}$ be a smooth Artin stack admitting a good moduli space $π: \mathcal{X} \to X$, and assume that $X$ is a variety with log-terminal singularities, $π$ induces an isomorphism over a nonempty open subset of $X$, and the exceptional locus of $π$ has codimension at least 2. We conjecture a formula for the motivic measure for $\mathcal{X}$ in terms of the Gorenstein measure for $X$ and a function measuring the degree to which $π$ is non-separated. We also conjecture that if the stabilizers of $\mathcal{X}$ are special groups in the sense of Serre, then almost all arcs of $X$ lift to arcs of $\mathcal{X}$, and we explain how in this case, our conjectures imply a formula for the stringy Hodge numbers of $X$ in terms of a certain motivic integral over the arcs of $\mathcal{X}$. We prove these conjectures in the case where $\mathcal{X}$ is a fantastack.

math.AG↗

arXiv2021

On Dynamical Cancellation

Let $X$ be a projective variety and let $f$ be a dominant endomorphism of $X$, both of which are defined over a number field $K$. We consider a question of the second author, Meng, Shibata, and Zhang, which asks whether the tower of $K$-points $Y(K)\subseteq (f^{-1}(Y))(K)\subseteq (f^{-2}(Y))(K)\subseteq \cdots$ eventually stabilizes, where $Y\subset X$ is a subvariety invariant under $f$. We show this question has an affirmative answer when the map $f$ is étale. We also look at a related problem of showing that there is some integer $s_0$, depending only on $X$ and $K$, such that whenever $x, y \in X(K)$ have the property that $f^{s}(x) = f^{s}(y)$ for some $s \geq 0$, we necessarily have $f^{s_{0}}(x) = f^{s_{0}}(y)$. We prove this holds for étale morphisms of projective varieties, as well as self-morphisms of smooth projective curves. We also prove a more general cancellation theorem for polynomial maps on $\mathbb{P}^1$ where we allow for composition by multiple different maps $f_1,\dots,f_r$.

math.AG↗

arXiv2020

On curves with high multiplicity on $\mathbb{P}(a,b,c)$ for $\min(a,b,c)\leq4$

On a weighted projective surface $\mathbb{P}(a,b,c)$ with $\min(a,b,c)\leq 4$, we compute lower bounds for the {\em effective threshold} of an ample divisor, in other words, the highest multiplicity a section of the divisor can have at a specified point. We expect that these bounds are close to being sharp. This translates into finding divisor classes on the blowup of $\mathbb{P}(a,b,c)$ that generate a cone contained in, and probably close to, the effective cone.

math.AG↗

arXiv2020

Approximating rational points on toric varieties

Given a smooth projective variety $X$ over a number field $k$ and $P\in X(k)$, the first author conjectured that in a precise sense, any sequence that approximates $P$ sufficiently well must lie on a rational curve. We prove this conjecture for smooth split toric surfaces conditional on Vojta's conjecture. More generally, we show that if $X$ is a $\mathbb{Q}$-factorial terminal split toric variety of arbitrary dimension, then $P$ is better approximated by points on a rational curve than by any Zariski dense sequence.

math.AG↗

arXiv2020

Height Gap Conjectures, $D$-Finiteness, and Weak Dynamical Mordell-Lang

In previous work, the first author, Ghioca, and the third author introduced a broad dynamical framework giving rise to many classical sequences from number theory and algebraic combinatorics. Specifically, these are sequences of the form $f(Φ^n(x))$, where $Φ\colon X\to X$ and $f\colon X\to\mathbb{P}^1$ are rational maps defined over $\overline{\mathbb{Q}}$ and $x\in X(\overline{\mathbb{Q}})$ is a point whose forward orbit avoids the indeterminacy loci of $Φ$ and $f$. They conjectured that if the sequence is infinite, then $\limsup \frac{h(f(Φ^n(x)))}{\log n} > 0$. They also made a corresponding conjecture for $\liminf$ and showed that it implies the Dynamical Mordell-Lang Conjecture. In this paper, we prove the $\limsup$ conjecture as well as the $\liminf$ conjecture away from a set of density $0$. As applications, we prove results concerning the growth rate of coefficients of $D$-finite power series as well as the Dynamical Mordell-Lang Conjecture up to a set of density $0$.

math.NT↗

arXiv2020

What fraction of an $S_n$-orbit can lie on a hyperplane?

Consider the $S_n$-action on $\mathbb{R}^n$ given by permuting coordinates. This paper addresses the following problem: compute $\max_{v,H} |H\cap S_nv|$ as $H\subset\mathbb{R}^n$ ranges over all hyperplanes through the origin and $v\in\mathbb{R}^n$ ranges over all vectors with distinct coordinates that are not contained in the hyperplane $\sum x_i=0$. We conjecture that for $n\geq3$, the answer is $(n-1)!$ for odd $n$, and $n(n-2)!$ for even $n$. We prove that if $p$ is the largest prime with $p\leq n$, then $\max_{v,H} |H\cap S_nv|\leq \frac{n!}{p}$. In particular, this proves the conjecture when $n$ or $n-1$ is prime.

math.CO↗

arXiv2019

Higher arithmetic degrees of dominant rational self-maps

Suppose that $f \colon X \dashrightarrow X$ is a dominant rational self-map of a smooth projective variety defined over ${\overline{\mathbf Q}}$. Kawaguchi and Silverman conjectured that if $P \in X({\overline{\mathbf Q}})$ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $λ_1(f)$ of $f$ if the orbit of $P$ is Zariski dense in $X$. In this note, we extend the Kawaguchi-Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of $X$. We begin by defining a set of arithmetic degrees of $f$, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.

math.NT↗

arXiv2019

Dynamical Uniform Bounds for Fibers and a Gap Conjecture

We prove a uniform version of the Dynamical Mordell-Lang Conjecture for étale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined over a number field. More precisely, for our first result, we assume $X$ is a quasi-projective variety defined over a field $K$ of characteristic $0$, endowed with the action of an étale endomorphism $Φ$, and $f\colon X\to Y$ is a morphism with $Y$ a quasi-projective variety defined over $K$. Then for any $x\in X(K)$, if for each $y\in Y(K)$, the set $S_y:=\{n\in \mathbb{N}\colon f(Φ^n(x))=y\}$ is finite, then there exists a positive integer $N$ such that $\#S_y\le N$ for each $y\in Y(K)$. For our second result, we let $K$ be a number field, $f:X\dashrightarrow \mathbb{P}^1$ is a rational map, and $Φ$ is an arbitrary endomorphism of $X$. If $\mathcal{O}_Φ(x)$ denotes the forward orbit of $x$ under the action of $Φ$, then either $f(\mathcal{O}_Φ(x))$ is finite, or $\limsup_{n\to\infty} h(f(Φ^n(x)))/\log(n)>0$, where $h(\cdot)$ represents the usual logarithmic Weil height for algebraic points.

math.NT↗

arXiv2019

Towards an Intersection Chow Cohomology Theory for GIT Quotients

We study the Fulton-Macpherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are étale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore natural extensions of GIT quotients. Our main result is that, with rational coefficients, every operational class can be represented by a so-called topologically strong cycle on the corresponding stack. Moreover, this cycle is unique modulo rational equivalence on the stack. Using out methods, we prove that if $X$ is the good moduli space of a properly stable, smooth, Artin stack then the natural map from the Picard group of $X$ to the first operational Chow group of $X$ is surjective with rational coefficients.

math.AG↗

arXiv2019

Lifting tropical self intersections

We study the tropicalization of intersections of plane curves, under the assumption that they have the same tropicalization. We show that the set of tropical divisors that arise in this manner is a pure dimensional balanced polyhedral complex and compute its dimension. When the genus is at most 1, we show that all the tropical divisors that move in the expected dimension are realizable. As part of the proof, we introduce a combinatorial tool for explicitly constructing large families of realizable tropical divisors.

math.AG↗

arXiv2018

A rational map with infinitely many points of distinct arithmetic degrees

Let $f \colon X \dashrightarrow X$ be a dominant rational self-map of a smooth projective variety defined over $\overline{\mathbb Q}$. For each point $P\in X(\overline{\mathbb Q})$ whose forward $f$-orbit is well-defined, Silverman introduced the arithmetic degree $α_f(P)$, which measures the growth rate of the heights of the points $f^n(P)$. Kawaguchi and Silverman conjectured that $α_f(P)$ is well-defined and that, as $P$ varies, the set of values obtained by $α_f(P)$ is finite. Based on constructions of Bedford--Kim and McMullen, we give a counterexample to this conjecture when $X=\mathbb P^4$.

math.AG↗

arXiv2018

Galois closures of non-commutative rings and an application to Hermitian representations

Galois closures of commutative rank n ring extensions were introduced by Bhargava and the second author. In this paper, we generalize the construction to the case of non-commutative rings. We show that non-commutative Galois closures commute with base change and satisfy a product formula. As an application, we give a uniform construction of many of the representations arising in arithmetic invariant theory, including many Vinberg representations.

math.AG↗

arXiv2018

Canonical heights on hyper-Kähler varieties and the Kawaguchi-Silverman conjecture

The Kawaguchi--Silverman conjecture predicts that if $f\colon X \dashrightarrow X$ is a dominant rational-self map of a projective variety over $\overline{\mathbb{Q}}$, and $P$ is a $\overline{\mathbb{Q}}$-point of $X$ with Zariski-dense orbit, then the dynamical and arithmetic degrees of $f$ coincide: $λ_1(f) = α_f(P)$. We prove this conjecture in several higher-dimensional settings, including all endomorphisms of non-uniruled smooth projective threefolds with degree larger than $1$, and all endomorphisms of hyper-Kähler varieties in any dimension. In the latter case, we construct a canonical height function associated to any automorphism $f\colon X \to X$ of a hyper-Kähler variety defined over $\overline{\mathbb{Q}}$.

math.AG↗

arXiv2017

New classes of examples satisfying the three matrix analog of Gerstenhaber's theorem

In 1961, Gerstenhaber proved the following theorem: if k is a field and X and Y are commuting dxd matrices with entries in k, then the unital k-algebra generated by these matrices has dimension at most d. The analog of this statement for four or more commuting matrices is false. The three matrix version remains open. We use commutative-algebraic techniques to prove that the three matrix analog of Gerstenhaber's theorem is true for some new classes of examples. In particular, we translate this three commuting matrix statement into an equivalent statement about certain maps between modules, and prove that this commutative-algebraic reformulation is true in special cases. We end with ideas for an inductive approach intended to handle the three matrix analog of Gerstenhaber's theorem more generally.

math.AC↗

arXiv2017

Density of orbits of dominant regular self-maps of semiabelian varieties

We prove a conjecture of Medvedev and Scanlon in the case of regular morphisms of semiabelian varieties. That is, if $G$ is a semiabelian variety defined over an algebraically closed field $K$ of characteristic $0$, and $φ\colon G\to G$ is a dominant regular self-map of $G$ which is not necessarily a group homomorphism, we prove that one of the following holds: either there exists a non-constant rational fibration preserved by $φ$, or there exists a point $x\in G(K)$ whose $φ$-orbit is Zariski dense in $G$.

math.NT↗

arXiv2017

On a Dynamical Mordell-Lang Conjecture for Coherent Sheaves

We introduce a dynamical Mordell-Lang-type conjecture for coherent sheaves. When the sheaves are structure sheaves of closed subschemes, our conjecture becomes a statement about unlikely intersections. We prove an analogue of this conjecture for affinoid spaces, which we then use to prove our conjecture in the case of surfaces. These results rely on a module-theoretic variant of Strassman's theorem that we prove in the appendix.

math.AG↗

arXiv2017

Algebras, Synchronous Games and Chromatic Numbers of Graphs

We associate to each synchronous game an algebra whose representations determine if the game has a perfect deterministic strategy, perfect quantum strategy or one of several other perfect strategies. when applied to the graph coloring game, this leads to characterizations in terms of properties of an algebra of various quantum chromatic numbers that have been studied in the literature. This allows us to develop a correspondence between various chromatic numbers of a graph and ideals in this algebra which can then be approached via various Grobner basis methods.

math.OA↗

arXiv2016

On the Medvedev-Scanlon Conjecture for Minimal Threefolds of Non-Negative Kodaira Dimension

Motivated by work of Zhang from the early `90s, Medvedev and Scanlon formulated the following conjecture. Let $K$ be an algebraically closed field of characteristic $0$ and let $X$ be a quasiprojective variety defined over $K$ endowed with a dominant rational self-map $Φ$. Then there exists a point $α\in X(K)$ with Zariski dense orbit under $Φ$ if and only if $Φ$ preserves no nontrivial rational fibration, i.e., there exists no non-constant rational function $f\in K(X)$ such that $Φ^*(f)=f$. The Medvedev-Scanlon conjecture holds when $K$ is uncountable. The case where $K$ is countable (e.g., $K=\overline{\mathbb{Q}}$) is much more difficult; here the conjecture has only been proved in a small number of special cases. In this paper we show that the Medvedev-Scanlon conjecture holds for all varieties of positive Kodaira dimension, and explore the case of Kodaira dimension $0$. Our results are most complete in dimension $3$.

math.AG↗