We report some work in progress on the relationship between étale and quasicoherent cohomological dimensions.

We study Ziegler pairs of line arrangements from both numerical and homological perspectives. First, we show that for arrangements of $d<9$ lines the intersection lattice determines the exponent data considered here. Then we list six distinct Ziegler pair with $d=10$. In particular, we construct higher-degree examples with the same intersection lattice, the same minimal degree of a Jacobian relation, and the same Hilbert function of the Milnor algebra, but with different minimal graded free resolutions.

In this note we investigate the Fitting ideals associated to the Tjurina ideal of a non quasi-homogeneous plane curve singularity. Special properties occur when the difference between Milnor number and Tjurina number is at most 2.

In 1929, O. Zariski found a pair of complex plane algebraic curves of the same degree and with the same collection of singularities, but embedded into the plane in a topologically different way. Accordingly, such curves belong to different components of the equisingular family. This phenomenon has been intensively studied till now. In this note, we propose a different insight on this subject: Two curves $C',C''\subset\PP^2$ form an {\it anti-Zariski pair}, if $(\PP^2,C')$ and $(\PP^2,C'')$ are homeomorhic, but $C'$ and $C''$ belong to different components of the equisingular family. We exhibit examples of anti-Zariski pairs and discuss related issues.

Inspired by the model theory of difference fields in characteristic zero, a class of automorphisms of an algebraic variety, here called compound fundamental isotrivial, is introduced. These are algebraic dynamical systems that are built up via a finite sequence of equivariant fibrations from (possibly nonautonomous) algebraic dynamics which trivialise after base extension over themselves. Every wild automorphism of an abelian variety is compound fundamental isotrivial. Conversely, it is shown that the only irreducible projective varieties admitting a wild automorphism that is compound fundamental isotrivial are the abelian varieties. That is, the wild automorphism conjecture of Reichstein, Rogalski, and Zhang is here proven for compound fundamental isotrivial dynamics. Along the way, a counterexample to the naive generalisation of the conjecture to the nonautonomous setting of $σ$-varieties is provided.

For a regular semisimple Higgs bundle with a smooth spectral curve, we prove that, over the \etale\ locus, the Atiyah class of the underlying bundle is induced by the Atiyah class of the spectral line bundle and takes values in the centralizer of the Higgs field. Further, when the discriminant is reduced, we construct a logarithmic refinement across the branch divisor: the Atiyah class extends as a class with logarithmic poles and values in a natural regularized centralizer sheaf.

We give a conceptual proof for Hochster's Theorem, which asserts that each spectral space is homeomorphic to the spectrum of a ring. Given a ground field and a spectral space, our ring is constructed as filtered direct limit of prime-finite ring, which are attached in a functorial way to finite Kolmogoroff spaces. The construction simplifies an argument of Ershov along these lines. Our crucial ingredient is an assembly of finite Kolmogoroff spaces in terms of coequalizers and pushouts of one-dimensional spaces, and Schwede's observation on prime ideals in cartesian squares of rings.

During the last several years, Spencer Bloch emphasized that Archimedean hight pairings between two homologically trivial cycles of complimentary arithmetic dimensions, or the closely related classes in the biextension, should lead to interesting analogs of the classical cross ratio. In this note I calculate such a generalized cross-ratio for planes in the projective space.

We show that for a quasi-compact quasi-separated scheme $X$ with an ample family of line bundles, the connective half $\text{QCoh}(X)_{\geq0}$ of the standard $t$-structure on the derived $\infty$-category of quasi-coherent sheaves is compactly generated by a connective perfect object if and only if $X$ is quasi-affine.

Any linear space of square matrices has an associated eigenvector variety. Its points are eigenvectors of matrices from that linear space. We present a systematic study of eigenvector varieties, with focus on Lie algebras and Hamiltonians of quantum systems.

We study the cohomology of a quasimap space that categorifies the constant term of the geometric Eisenstein series for the mirabolic parabolic subgroup of $GL$ over the function field $\mathbb{F}_q(C)$ of a smooth projective curve $C$. This cohomology carries a natural action of an algebra of correspondences whose commutative subalgebra is the ring of regular functions on the Coulomb branch, which here is the $A_{n}$-surface singularity. A choice of rank-one local system on $C$ induces an action of the étale fundamental group on the Coulomb branch; the scheme-theoretic fixed locus carries a natural vector bundle. Our main result identifies the cohomology of the quasimap space with the local cohomology of this vector bundle, for a generic range of parameters.

We provide counterexamples to the unimodal minimal filling architecture conjecture for polynomial neural networks (PNNs) with power activation functions. Fixing the input and output widths, the conjecture states that any minimal filling architecture has unimodal widths for the hidden layers. We found counterexamples via a frontier search, recursive dimension bounds on neurovarieties, and symbolic computation. Notably, several subarchitectures of our main example exhibit large defect, in contrast with the predominantly small-defect behavior observed in prior literature.

We show that the realization theorem of Fernández de Bobadilla, which identifies the Milnor fiber of a weighted-homogeneous polynomial with the complement of a germ of analytic set, can be combined with the systematic Massey product constructions of Grbić-Linton for moment-angle complexes $\mathcal{Z}_K = \mathcal{Z}_K(D^2, S^1)$ to produce weighted-homogeneous polynomials whose Milnor fibers are arbitrarily highly connected and non-formal. The original application of this strategy, due to Fernández de Bobadilla, used the Denham-Suciu classification of lowest-degree triple Massey products and yielded only 2-connected non-formal Milnor fibers. The Grbić-Linton framework, which constructs non-trivial $n$-fold Massey products in $H^*(\mathcal{Z}_K;\mathbb{Z})$ for arbitrary $n$ and in arbitrary cohomological degrees, removes this connectivity restriction entirely.

We develop a geometric approach to the regularized double shuffle relations for multiple zeta values, based on convolution of perverse sheaves on $\mathbb{C}^*$ and inspired by the approach of Deligne and Terasoma. We introduce semi-holonomy isomorphisms associated with pro-unipotent paths and show that their compatibility with multiplicative convolution is equivalent to a condition on the pro-unipotent fundamental group, the homological pentagon equation. We prove that this condition is equivalent to the regularized double shuffle relations, yielding a geometric proof that the pentagon equation implies these relations. The approach is purely topological and avoids Hodge-theoretic and Tannakian methods.

We give an explicit description of the BPS Lie algebra of any quiver with zero potential, by relating the perverse filtration on the cohomological Hall algebra with certain limit conditions on polynomials. Our results also give a partial description of the perverse filtration for arbitrary potential, which we conjecture is complete in the case of tripled quivers with canonical cubic potential.

We develop the theory of transfer and norm maps for finite group schemes, extending classical results from finite group theory to a context where induction and restriction are not necessarily bi-adjoint. In the additive setting, we construct a transfer map for both modules and $\rm Ext $ groups and prove that its surjectivity characterizes relative projectivity, establishing a generalization of Higman's criterion. In the multiplicative setting, we define a relative norm map for algebras with a group scheme action. We compare this norm with other versions in the literature, proving that it coincides with Mumford's norm for finite morphisms and on fields is a power of the classical field norm.

Let $E/\mathbb{Q}$ be an elliptic curve and let $ρ_E \colon \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{GL}(2, \widehat{\mathbb{Z}})$ be the adelic Galois representation attached to $E$. Much work has been done in recent years to study the image of $ρ_E$ (up to conjugation) as part of Mazur's so called ``Program B.'' In this paper, we describe and implement an efficient algorithm to compute the image of $ρ_E$ in $\operatorname{GL}(2, \widehat{\mathbb{Z}})$ (up to conjugation) for an elliptic curve $E/\mathbb{Q}$ with complex multiplication (CM) and $j$-invariant not $0$ or $1728$.

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero. An irreducible algebraic variety $X$ over $\mathbb{K}$ of dimension $n$ is called additive if it admits a regular action of the additive group $(\mathbb{K}^n, +)$ with an open orbit, and uniquely additive if this action is unique up to isomorphism. Huang and the second author have previously determined all additive smooth Fano toric threefolds. Here we determine all additive and uniquely additive smooth Fano toric varieties of dimension up to $6$ by computational means, and give a detailed classification for dimension up to $4$. To this effect, we introduce the AdditiveToricVarieties package for Macaulay2, a software system for algebraic geometry and commutative algebra, with methods for working with additive group actions on complete toric varieties. Our work relies on results by Arzhantsev, Dzhunusov and Romaskevich, who relate the existence and uniqueness of such actions to conditions on the Demazure roots of the fans corresponding to the toric varieties. We also prove that every smooth complete toric variety of Picard rank two is additive.

A fundamental property of Segre classes is their birational invariance. This invariance implies that the Segre class of a closed subscheme only depends on the integral closure of the defining ideal sheaf. In this paper, we show that, conversely, the Segre class of a closed subscheme encodes an integral dependence criterion for its defining ideal sheaf. As an application, we prove that Aluffi's Segre zeta function provides an integral dependence criterion for homogeneous ideals in polynomial rings.

We study energy minimization problems in quantum chemistry through the lens of computational algebraic geometry. We focus on minimizing the Rayleigh quotient of a Hamiltonian over a tensor train variety. The complex critical points of this problem approximate eigenstates of the quantum system, with the global minimum approximating the ground state. We call the number of critical points the Rayleigh-Ritz degree. We first study the Rayleigh-Ritz degree and introduce the Rayleigh-Ritz discriminant, which describes Hamiltonians that lead to a deficient number of critical points. We then specialize this framework to tensor train varieties: we identify instances when they are Segre products of projective spaces, report what we know about their defining ideals, and present a birational parametrization from products of Grassmannians. We use homotopy continuation to compute all critical points of this optimization problem over various tensor train and determinantal varieties. Finally, we use these results to benchmark state-of-the-art methods, the Alternating Linear Scheme and Density Matrix Renormalization Group.

In 2018, Kalck and Yang showed that the singularity categories associated with $3$-dimensional Gorenstein quotient singularities are triangle equivalent (up to direct summands) to small cluster categories associated with McKay quivers with potential. We introduce higher McKay quivers with potential and generalize Kalck and Yang's theorem to arbitrary dimensions. The singularity categories we consider occur as the stable categories of categories of Cohen-Macaulay modules. We refine our description of the singularity categories by showing that these categories of Cohen-Macaulay modules are equivalent to Higgs categories in the sense of Wu. Moreover, we describe the singularity categories in the non-Gorenstein case.

We study the large-$N$ asymptotics of the central trace of the heat kernel on compact classical groups. For every classical family $G_N\subset \mathrm{GL}_N(\C)$, we prove a full large-$N$ asymptotic expansion, using a highest weights/partitions correspondence adapted to the large-rank regime, under which the eigenvalues of the Laplace--Beltrami operator stabilize as observables in the algebra of shifted symmetric functions. Then, we prove a random surface representation of the trace in terms of ramified coverings of the torus. We provide two independent applications: an explicit large-rank counting law for the Casimir spectrum, with exponential Hardy--Ramanujan-type growth in contrast with the polynomial behavior of Weyl's law at fixed rank, and a rigorous probabilistic formulation of the Yang--Mills/Hurwitz duality on a two-dimensional torus initiated by Gross and Taylor, completing a previous work of the authors. We also extend this duality to a Yang--Mills/Gromov--Witten duality by expressing the coefficients of the central heat trace as explicit functionals of the generating function of Gromov--Witten invariants.

Building sets were introduced in the study of wonderful compactifications of hyperplane arrangement complements and were later generalized to finite meet-semilattices. Convex geometries, the duals of antimatroids, offer a robust combinatorial abstraction of convexity. Supersolvable convex geometries and antimatroids appear in the study of poset closure operators, Coxeter groups, and matroid activities. We prove that the building sets on a finite meet-semilattice form a supersolvable convex geometry. As an application, we demonstrate that building sets and nested set complexes respect certain restrictions of finite meet-semilattices unifying and extending results of several authors.

In this paper we develop an algebraic theory to study the problem of finding the minimum distance point from an algebraic variety with respect to the Hermitian distance function. The theory generalizes the Euclidean Distance degree introduced in arXiv:1309.0049, replacing a positive symmetric bilinear form by a Hermitian form. Various examples are presented to show the robustness of the machinery.

A complex hyperplane arrangement $\mathcal{A}$ is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group $G$ decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the $I$-adic completion of the Alexander invariant $B(G)$ also decomposes as a direct sum of "local" invariants and the Chen ranks of $G$ are the sums of the local contributions. Moreover, if $B(G)$ is separated, then the degree 1 cohomology jump loci of the complement of $\mathcal{A}$ have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.

We prove a recent conjecture of the fourth named author with P. Norbury that states a system of universal polynomial relations among the kappa classes on the moduli spaces of algebraic curves. The proof involves localization and materialization analysis of the spin Gromov-Witten theory of the projective line and is dictated by $\mathbb{Z}_2$-equivariant topological recursion.

We prove Demailly's Conjecture concerning the lower bound for the Waldschmidt constant in terms of the initial degree of the second symbolic powers for any set of generic points or very general points in $\mathbb{P}^N$. We also discuss the Harbourne-Huneke Containment and the aforementioned Demailly's Conjecture for general points and show the results for sufficiently many general points and general points in projective spaces with low dimensions.

Let X be a proper smooth variety over an algebraically closed field k of characteristic p>0. This thesis studies the p-primary component of the Brauer group of X.

In our series of papers, we prove that smooth Fano threefolds in positive characteristic lift to the ring of Witt vectors. Moreover, we show that they satisfy Akizuki-Nakano vanishing, $E_1$-degeneration of the Hodge to de Rham spectral sequence, and torsion-freeness of Crystalline cohomologies. In this paper, we establish these results for the case when $|-K_X|$ is very ample and the Picard group is generated by $ω_X$.

The main result of the paper is a description of conormal Lie algebras of Feigin-Odesskii Poisson structures. In order to obtain it we introduce a new variant of a definition of a Feigin-Odesskii Poisson structure: we define it using a differential on the second page of a certain spectral sequence. In the general case this spectral sequence computes morphisms and higher Ext's between filtered objects in an abelian category. Moreover, we use our definition to give another proof of the description of symplectic leaves of Feigin-Odesskii Poisson structures.