Motivated by Kontsevich's graph complexes, this paper gives a systematic study of matroid complexes. We construct deletion and contraction bicomplexes on the vector space spanned by matroid classes equipped with ground-set orientations, organizing the several naturally arising variants into a single unified framework. We show that direct sum and restriction-contraction make this space into a connected graded Hopf algebra extending Schmitt's matroid Hopf algebra, and use the resulting dg-algebra structure to prove broad acyclicity results. We compute the total, simple, loopless, regular, binary, and ternary matroid complexes through ground-set size 9, and the connected quotient of the simple loopless regular complex through ground-set size 15. These computations detect nontrivial homology and lead to a conjectural description in terms of odd-wheel matroids.

We explore the relationship between multigraded Castelnuovo-Mumford regularity, truncations, Betti numbers, and virtual resolutions on a product of projective spaces X. After proving a uniqueness theorem for certain virtual resolutions, we show that the multigraded regularity region of a module M is determined by the minimal graded free resolutions of the truncations M >= d for d is an element of Pic X. Further, by relating the minimal graded free resolutions of M and M >= d we provide a new bound on multigraded regularity of Min terms of its Betti numbers. Using this characterization of regularity and this bound we also compute the multigraded Castelnuovo-Mumford regularity for a wide class of complete intersections in products of projective spaces. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Multigraded Castelnuovo--Mumford regularity of a module $M$ over the total coordinate ring $S$ of a smooth projective toric variety $X$ is a region $\operatorname{reg} M \subset \operatorname{Pic} X$ invariant under translation by the nef cone $\operatorname{Nef} X$. We prove that the multigraded regularity of a finitely generated faithful module is contained in a translate of $\operatorname{Nef} X$ determined by the degrees of the generators of $M$, and thus contains only finitely many minimal elements. We show that this condition can fail even for cyclic modules if $M$ has torsion and the rank of the Picard group is at least two. As an application, we exhibit asymptotic bounds for the multigraded regularity of powers of ideals. For $I$ an ideal in $S$, we bound $\operatorname{reg}(I^n)$ by proving that it contains a translate of $\operatorname{reg} S$ and is contained in a translate of $\operatorname{Nef} X$, where each bound translates by a fixed vector as $n$ increases.

We introduce the Seshadri region of a subvariety, a convex region packaging the classical Seshadri constants with respect to every line bundle simultaneously. We develop the theory of Seshadri regions as a measure of positivity along subvarieties and apply it to determine asymptotic Castelnuovo-Mumford regularity for ideal powers and symmetric powers on smooth projective toric varieties.

This paper provides a rigorous study of tropicalizations of locally symmetric varieties. We give applications beyond tropical geometry, to the cohomology of moduli spaces as well as to the cohomology of arithmetic groups. We study two cases in detail: the special unitary case, and the case of level structures on the moduli space 𝒜_g of abelian varieties.

We compute the top-weight rational cohomology of $A_g$ for $g=5$, $6$, and $7$, and we give some vanishing results for the top-weight rational cohomology of $A_8, A_9,$ and $ A_{10}$. When $g=5$ and $g=7$, we exhibit nonzero cohomology groups of $A_g$ in odd degree, thus answering a question highlighted by Grushevsky. Our methods develop the relationship between the top-weight cohomology of $A_g$ and the homology of the link of the moduli space of principally polarized tropical abelian varieties of rank $g$. To compute the latter we use the Voronoi complexes used by Elbaz-Vincent-Gangl-Soul\'e. Our computations give natural candidates for compactly supported cohomology classes of $A_g$ in weight $0$ that produce the stable cohomology classes of the Satake compactification of $A_g$ in weight $0$, under the Gysin spectral sequence for the latter space.

We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld we give an explicit range in which the graded Betti numbers of Pn1×Pn2 embedded by OPn1×Pn2(d1,d2) are non-zero.

Inspired by Kontsevich's graphic orbifold Euler characteristic we define a virtual Euler characteristic for any finite set of isomorphism classes of matroids of rank $r$. Our main result provides a simple formula for the virtual Euler characteristic for the set of isomorphism classes of matroids of rank $r$ realizable over $\mathbb{F}_2$ (i.e., binary matroids). We prove this formula by relating the virtual Euler characteristic for binary matroids to the point counts of certain subsets of Grassmanians over finite fields. We conclude by providing several follow-up questions in relation to matroids realizable over other finite prime fields, matroid homology, and beta invariants.

The goal of this note is to quantitatively study the behavior of asymptotic syzygies for certain toric surfaces, including Hirzebruch surfaces. In particular, we show that the asymptotic linear syzygies of Hirzebruch surfaces embedded by $\mathcal{O}(d,2)$ conform to Ein, Erman, and Lazarsfeld's normality heuristic. We also show that the higher degree asymptotic syzygies are not asymptotically normally distributed.

We explore the asymptotic behavior of the multigraded Castelnuovo--Mumford regularity of powers of ideals. Specifically, if $I$ is an ideal in the total coordinate ring $S$ of a smooth projective toric variety $X$, we bound the region $\operatorname{reg}(I^n)\subset\operatorname{Pic} X$ by proving that it contains a translate of the regularity of $S$ and is contained in a translate of the nef cone of $X$. Each bound translates by a fixed vector as $n$ increases. Along the way we prove that the multigraded regularity of a finitely generated torsion-free module is contained in a translate of the nef cone determined by the degrees of the generators of $M$, and thus contains only finitely many minimal elements.

We provide a number of new conjectures and questions concerning the syzygies of P1×P1. The conjectures are based on computing the graded Betti tables and related data for large number of different embeddings of P1×P1. These computations utilize linear algebra over finite fields and high-performance computing.

This note introduces the Macaulay2 package SchurVeronese, which gathers together data about Veronese syzygies and makes it readily accessible in Macaulay2. In addition to standard Betti tables, the package includes information about the Schur decompositions of the various spaces of syzygies. The package also includes a number of functions useful for manipulating and studying this data.

We provide a number of new conjectures and questions concerning the syzygies of ℙ^1×ℙ^1. The conjectures are based on computing the graded Betti tables and related data for large number of different embeddings of ℙ^1×ℙ^1. These computations utilize linear algebra over finite fields and high-performance computing.

We explore the relationship between multigraded Castelnuovo-Mumford regularity, truncations, Betti numbers, and virtual resolutions. We prove that on a product of projective spaces $X$, the multigraded regularity region of a module $M$ is determined by the minimal graded free resolutions of the truncations $M_{\geq\mathbf{d}}$ for $\mathbf{d}\in\operatorname{Pic}X$. Further, by relating the minimal graded free resolutions of $M$ and $M_{\geq\mathbf{d}}$ we provide a new bound on multigraded regularity of $M$ in terms of its Betti numbers. Using this characterization of regularity and this bound we also compute the multigraded Castelnuovo-Mumford regularity for a wide class of complete intersections.

In the early morning hours of June 28, 1969, police violently targeted LGBTQ+ people at the Stonewall Inn, and that led to the beginning of what has become known as the Stonewall Uprising.These actions, led by trans women of color, marked a crucial turning point in the LGBTQ+ rights movement in the United States and internationally.In honor and remembrance of these events, as well as the struggles and successes of all LGBTQ+ people, June is commonly know as LGBTQ+ Pride Month in the United States, Canada, and other countries.In celebration of this year's Pride Month, we are happy to discuss a particular aspect of the growing visibility of LGBTQ+ mathematicians.

We formulate several conjectures which shed light on the structure of Veronese syzygies of projective spaces. These conjectures are motivated by experimental data that we derived from a high-speed high-throughput computation of multigraded Betti numbers based on numerical linear algebra.

We introduce the VirtualResolution package for the computer algebra system Macaulay2. This package has tools to construct, display, and study virtual resolutions for products of projective spaces. The package also has tools for generating curves in $\mathbb{P}^1\times\mathbb{P}^2$, providing sources for interesting virtual resolutions.

We show that any abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective and explicit version of Poonen’s Bertini theorem over finite fields, which allows us to show the existence of smooth curves arising as hypersurface sections of bounded degree and genus. Additionally, for simple abelian varieties we prove a better bound. As an application, we show that for any elliptic curve E E over a finite field and any n ∈ N n\in \mathbb {N} , there exist smooth curves of bounded genus whose Jacobians have a factor isogenous to E n E^n .

We study systems of parameters over finite fields from a probabilistic perspective, and use this to give the first effective Noether normalization result over a finite field. Our central technique is an adaptation of Poonen's closed point sieve, where we sieve over higher dimensional subvarieties, and we express the desired probabilities via a zeta function-like power series that enumerates higher dimensional varieties instead of closed points. This also yields a new proof of a recent result of Gabber-Liu-Lorenzini and Chinburg-Moret-Bailly-Pappas-Taylor on Noether normalizations of projective families over the integers.

We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld we give an explicit range in which the graded Betti numbers of ℙ^n_1×ℙ^n_2 embedded by 𝒪_ℙ^n_1×ℙ^n_2(d_1,d_2) are non-zero. These bounds provide the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.