Papers

Let $X$ be a quasi-projective scheme of dimension $d$ over an infinite field $k$, such that $X$ is reduced after removing all components of dimension $<d$, and let $X^*\subset X$ be a closed subset of dimension $<d$ such that $X\setminus X^*$ is regular of dimension $d$. Using a modification $\text{CH}^d(X, X^*)$ of the Chow group of 0-cycles on a singular variety defined by C. Weibel and the author in 1985, we construct a Chern class map $c_d:K_0(X)\to \text{CH}^d(X, X^*)$, a cycle class map from $\text{CH}^d(X, X^*)$ onto a subgroup $F^dK_0(X)$ of $K_0(X)$, and prove a Riemann-Roch theorem computing the two compositions of these two maps.

The theory of polyptych lattices seeks to incorporate, into a single combinatorial framework, the piecewise-linear bijections (mutations) that appear in many theories that generalize toric geometry, such as Newton-Okounkov bodies, toric degenerations, cluster varieties, and other areas. The link to algebraic geometry comes from the accompanying notion of detropicalizations of polyptych lattices, and their associated compactifications. The main result of this manuscript gives a concrete construction of a strict dual pair $(\mathcal{E}, \mathcal{F})$ of polyptych lattices, which we call a Gorenstein PL cone extension. To define our construction, we introduce the notion of Gorenstein PL cones, which are a polyptych analogue of a Gorenstein cone in the classical setting. If the original polyptych lattices are detropicalizable, then the new polyptych lattices $\mathcal{E},\mathcal{F}$ are also detropicalizable, via a simple explicit formula. Our Gorenstein PL cone extensions give a rich source of examples of strict dual pairs of polyptych lattices. In particular, any pair of Gorenstein-Fano polytopes $\Delta,\Delta'$ of dimension $n,n'$ respectively, gives rise to a strict dual pair $(\mathcal{E}(\Delta,\Delta'), \mathcal{F}(\Delta,\Delta'))$ of detropicalizable polyptych lattices of rank $n+n'+1$. Other examples can be built from cluster data.

We construct Laurent phenomenon structures on reductive branching algebras. For $A_{2n-1}\downarrow C_{n}$, $n\ge 2$, and the exceptional inclusions $D_4\downarrow G_2$, $E_6\downarrow F_4$, and $F_4\downarrow B_4$, we identify the branching algebras with upper Laurent phenomenon algebras, keeping the frozen coefficients polynomial. Degree-fibred categories of projective presentations give a common construction. It also gives cluster seeds of type $A_1$ for $B_3 \downarrow G_2$ and $G_2\downarrow A_2$. We give uniform sufficient conditions identifying a specialized Keel–Yu mirror algebra with an upper Laurent phenomenon algebra admitting a suitable binomial seed. The resulting theta basis is simultaneously adapted to the frozen boundary valuations. For $A_{2n-1}\downarrow C_{n}$ with $2\le n\le5$ and for the other exceptional inclusions, we obtain homogeneous theta bases parametrized by lattice points of explicit rational polyhedral cones, whose weight fibres compute every branching multiplicity. We also prove mutation invariance of finite upper bounds for LP patterns over a UFD.

Let $F=(f_1,\dots,f_r)$ be a family of nonzero polynomials on $\mathbb C^n$. More than ten years ago, Budur conjectured that $B_F$, the Bernstein–Sato ideal of $F$, is generated by products of linear polynomials with rational coefficients. With the aid of AI, we construct two counterexamples to Budur's conjecture. The examples indicate that main results in [Wu26] provide the optimal linearity properties that the zero locus of Bernstein-Sato ideals satisfies in general.

When studying the dynamical Mordell-Lang conjecture by using the $p$-adic method, experts have found that return sets of the $p$-adic version of the DML problem can be very sparse. In this article, we investigate the similar phenomenon for a complex analytic version of the DML problem and the DML problem for backward orbits.

We compare motivic nearby cycles at infinity with classes computed directly on rational compactifications. Let $f$ be a regular function on a smooth variety over a field of characteristic zero, and let $X$ be a smooth proper compactification. Suppose that a reduced divisor containing the boundary and the support of $\operatorname{div}_X(f)$ has simple normal crossings near the polar locus. We prove that the open-pair class defined by $1/f$ agrees with Raibaut's compactification-independent motivic nearby cycle at infinity. The key step is a local cancellation under blow-ups of transverse zero-pole intersections. For Laurent polynomials nondegenerate at infinity, this comparison yields a direct toric proof of Raibaut's essential-face formula. Equivariant Hodge realization and weighted Ehrhart theory give Newton formulas retaining cohomological degree and the limit-weight filtration, and determine irregular Hodge numbers through the limiting-irregular comparison. The weight-forgetting specialization computes the zero-potential stringy invariant of toric stacky fans with the same radial support, yielding a motivic-Hodge derivation of the corresponding Fano-Landau-Ginzburg mirror identity. In the full-dimensional weak Fano case, the refined formulas identify monodromy eigenvalues and Jordan-block sizes with inertia ages and anticanonical Lefschetz strings, giving a noncanonical comparison over $\mathbb C$.

Let D_N be the determinant of the generic symmetric N x N matrix, and let differential operators act by ordinary differentiation over a field of odd characteristic p. We determine the apolar ideal of D_N in every size. In addition to the classical quadratic relations, it is generated by one (p-1) x (p-1) determinant of differential variables for each set of 2p-2 indices. These additional generators are minimal modulo the quadratic ideal. An exterior-algebra realization identifies the remaining apolar relations with radicals of invariant pairings for SL_2. Semisimplification gives a Hilbert series in terms of Dyck paths of height at most p-2, and the Steinberg symmetrizer gives the explicit generators of the entire radical. We also compute the ranks of powers of a general linear form. The weak Lefschetz property holds exactly when N <= 2p-2, and the strong Lefschetz property holds exactly when N < p.

We propose a conjecture on how to recover an algebraic curve from the isogeny classes of its Prym varieties. Motivated by this conjecture, we analyze constructions of pairs of curves with isogenous Jacobians and several isogenous Pryms.

Let $K/E$ be a finite unramified extension of $p$-adic local fields, and let $H$ be a one-dimensional formal $\mathcal{O}_E$-module of finite height over $\mathcal{O}_K$. We introduce the exponential period map of $H$ and use it to construct a complete regular local ring $R_{H,K}$ with imperfect residue field, an endomorphism $\varphi_E$, and a commuting action of $\Gamma=\mathrm{Gal}(K(H[p^\infty](\overline{K}))/K)$. We prove an equivalence of categories between finitely generated $\mathcal{O}_E$-modules with a continuous $\mathrm{Gal}_K$-action and étale $(\varphi_E,\Gamma)$-modules over $R_{H,K}$. This recovers the classical cyclotomic and Lubin-Tate equivalences in the corresponding cases. In general, $R_{H,K}$ can have Krull dimension greater than one, and $\varphi_E$ need not lift the $q_E$-power Frobenius modulo $\pi_E$. The proof uses $F$-dynamical systems over $\mathcal{O}_E$, which combine contraction modulo $\pi_E$ with Frobenius on the residue field. For every flat $F$-dynamical system, we establish an equivalence of categories between étale $\varphi$-modules and continuous representations of the absolute Galois group of the residue field on finitely generated $\mathcal{O}_E$-modules.

We study Fourier-Mukai duality for a class of compactified Prym fibrations including moduli spaces of $\mathrm{SL}$ Higgs bundles over the elliptic locus. This leads to a shadow of the Hausel-Thaddeus conjecture, proof of the Corti-Hanamura motivic decomposition conjecture for these fibrations, and multiplicativity of the perverse filtration. Our approach can be described as a generalization of the Maulik-Shen-Yin package for compactified Jacobian fibrations to the case when the dual abelian fibration is a stack. This forces us to move beyond the case of full supports. Technical tools include a new pullback identity for the stacky Grothendieck-Riemann-Roch tau functor introduced by Toën, results on descent of (Arinkin-)Poincaré sheaves, and a comparison of Poincaré sheaves of the Prym varieties of families of smooth and nodal curves along the lines of Franco-Hanson-Horn-Oliveira.

Scott and Sokal asked whether every homogeneous polynomial with the half-plane property has a completely monotone negative power. We answer this question affirmatively and prove the Riesz-positivity conjecture of Michałek, Sturmfels, Uhler and Zwiernik, restated by Kozhasov, Michałek and Sturmfels: in $n$ variables, every exponent $\alpha\ge4096n^2$ is admissible, independently of the degree and coefficients. For complete hyperbolic polynomials the Riesz density is strictly log-concave, with relative Gaussian error at most $512n^2/\alpha$ and explicit curvature bounds. We characterize admissible exponents by a common spectral Dirichlet law, prove $n\le m+\alpha m(m-1)$ with its equality case, and obtain the sharp degree-dependent gap $0<\alpha<1/(2(m-1))$ whenever the degree-$m$ polynomial has a nonlinear irreducible factor. A nonnegative fourth-order defect of $-\log p$ vanishes at one point precisely for products of positive integer powers of Euclidean Jordan determinants. We classify the corresponding logarithmic Monge-Ampère equation and answer the question of Etingof, Kazhdan and Polishchuk about polynomial multiplicative Legendre transforms within irreducible complete hyperbolic polynomials; the general question has counterexamples, the Clifford quartics of Kogiso and Sato. The classification extends to reducible polynomials under a boundary-visibility hypothesis, and asymptotic common-power formulas for the Riesz densities force exact Jordan formulas.

We study the profinite non-rigidity of arithmetic lattices and its implications for the Kähler property. In the first part, we characterize the absolute Dynkin types that admit non-isomorphic real forms of higher-rank Lie groups containing torsion-free arithmetic lattices with isomorphic profinite completions. In the second part, as a geometric application, we answer a question asked independently by Arapura and Libgober, by showing that the Kählerness of finitely presented, residually finite groups is not determined by its profinite completion.

We develop several tools to study the local and global structure of the Teichmüller and moduli spaces for complex nilmanifolds. As a starting point, we prove a structure theorem for holomorphic maps between complex nilmanifolds, which allows for an explicit description of the Teichmüller and moduli spaces. We then introduce two period maps, and give general criteria for the existence of a complex structure on these spaces. To illustrate the theory, we prove a global Torelli theorem for various classes of nilmanifolds, including principal torus bundles over tori and almost abelian nilmanifolds. In particular, we show that there are nilmanifolds of arbitrary dimension and nilpotency index for which the Teichmüller space admits the structure of a complex manifold.

A smooth complete intersection $X\subset\mathbb{P}^{n+2}_{\mathbb{C}}$ of two quadrics admits an interpretation as a moduli space of bundles. For $n=2$, it parametrizes stable rank $2$ parabolic bundles on $\mathbb{P}^1$ with five marked points, while for $n=3$ it parametrizes stable rank $2$ bundles of odd degree with fixed determinant on a genus $2$ curve. For general $n$, it can be interpreted as a moduli space of semistable twisted $\mathrm{Spin}$-bundles. We study $X\setminus W$, where $W$ is the wobbly locus, consisting of bundles that admit a nonzero nilpotent Higgs field. We prove that $W$ is an irreducible divisor for $n\geq 3$ and compute $H_1(X\setminus W,\mathbb{Z})$. Using a root-stack description, we identify $\pi_1(X\setminus W)$ with the kernel of a monodromy homomorphism from an orbifold mixed braid group. The Reidemeister-Schreier method then yields an explicit presentation in every dimension. For $n=2$, where $X$ is a degree $4$ del Pezzo surface and $W$ is the union of its sixteen $(-1)$-curves, we obtain a presentation with $10$ generators and $25$ relators and prove that both numbers are minimal among all finite presentations. For $n=3$, we show that $\pi_1(X\setminus W)$ admits a presentation with $4$ generators and $78$ relators.

Let $X=\mathbb{P}^n$, where $p=n+1$ is prime. Consider the orbit of the standard numerical exceptional basis in $K_0(X)$ under mutations, sign changes, and isometries of the Euler form. We construct an invariant of this orbit: for every basis from the orbit, the square of rank of all basic vectors art congruent to $1$ modulo $p$. For $\mathbb{P}^4$ and $\mathbb{P}^6$, we present explicit numerical exceptional bases violating this congruence; hence the action is not transitive. We also establish a stronger obstruction modulo $p^2$ and use Polishchuk's theorem to show that no vector of the constructed bases is the class of an exceptional object. Finally, we prove that for every prime $p$, the category $D^b(\operatorname{Coh}\mathbb{P}^{p-1}_{\mathbb C})$ does not contain a pair of mutually orthogonal exceptional objects.

We revisit the correspondence between three-dimensional gauged linear sigma models and K-theoretic Gromov-Witten theory. We show that the K-theoretic Hori-Vafa formula for the grassmannian arises from the three-dimensional gauge theory upon including an appropriate anomaly-cancellation factor. Furthermore, we derive K-theoretic I-functions for isotropic grassmannians by incorporating symmetric and anti-symmetric tensor representation fields together with suitable boundary conditions.

We prove the (uniform) Yau-Tian-Donaldson Conjecture for transcendental Kähler classes in the case of trivial automorphisms. Indeed, we show the existence of Special Kähler Fujita Approximations of big cohomology classes associated to big test configurations, establishing the equivalence between the uniform $K$-stability and the strengthened version for models. More generally, we show that any big cohomology class on a compact Kähler manifold admits such Special Kähler Fujita Approximations: the volume and the analogue of the Riemann-Roch coefficient of the big class are both approximated by those of Kähler classes on higher compactifications.

Given a family of genus $g$ curves $f:\Psi\to S$ and a relative degree $d$ line bundle $L$ on $\Psi$ such that $f_*L$ is a vector bundle, the projectivization $\mathbb{P}(f_*L)$ is stratified by relative de Jonquières loci, which parameterize sections of $L$ with prescribed vanishing orders. We give universal formulas for the classes of these loci in terms of the relative $O(1)$ on $\mathbb{P}(f_*L)$ and tautological classes pulled back from the universal Picard stack $\mathrm{Pic}_g^d$. When $f$ is the family of lines in $\mathbb{P}^r$, we use the universal polynomials to compute $\mathrm{GL}_{r+1}$-equivariant classes of loci of hypersurfaces and complete intersections admitting lines with prescribed contact orders.

Let $(A,I)$ be a henselian pair such that $A$ is a domain of characteristic zero and $I\ne0$. We prove that any flat affine henselian scheme of finite presentation over $(A,I)$ has nonzero first structure-sheaf cohomology whenever one of its fibers over a point in $V(I)$ has positive dimension. This in particular gives a negative answer to the question by Devadas in the mixed-characteristic case. As an application, we also prove that the first structure-sheaf cohomology does not vanish for henselian affinoids of positive dimensions.

We study null geodesics in Reissner-Nordström-de Sitter spacetime and in its $\omega=-2/3$ Kiselev deformation through the elliptic curves defined by the radial equation. The relation of these curves to the non-critical $E_7$ Seiberg-Witten family yields differential equations for finite-distance light deflection, and explains the boundary source term through a suitable gauge of the Seiberg-Witten differential. This geometry separates the extremal point $Q^2=M^2$ from the Argyres-Douglas point $Q^2=9M^2/8$. We also determine the Kiselev critical and horizon loci, and show that, at fixed nonzero angular momentum, the Argyres-Douglas and ultracold loci meet only at zero Killing energy.

We prove that the support of every double Grothendieck polynomial is an $M^\natural$-convex set. Our main new tool is a rigidity result for the $K$-classes of multiprojective varieties with rational singularities.

We introduce the categories $\CM(X)$ and $\SCM(X)$ of reflexive sheaves on a minimal partial resolution $f\colon X \to \Spec R$ of a rational surface singularity $\Spec R$. The main result of this paper establishes that $\SCM(X)$ possesses a natural Frobenius structure, serving as a geometric counterpart to the algebraic Frobenius structure on special Cohen–Macaulay $R$-modules, introduced by Iyama–Wemyss and Iyama–Kalck–Wemyss–Yang. Utilizing this geometric framework, we establish an exact equivalence between $\SCM(X)$ and the category of special Cohen–Macaulay $R$-modules equipped with a specific exact structure, which induces a triangle equivalence between their stable categories. Consequently, this provides a direct, geometric proof of the Iyama–Kalck–Wemyss–Yang equivalence and yields a Buchweitz-type equivalence $\underline{\SCM}(X) \simeq D_{\sg}(X)$.

We associate to finite block size data \[ \bm n=(n_d)_{d\ge1},\qquad n_1>0, \] a positive definite cyclotomic form and a Householder reflection representation. Writing \[ N=\operatorname{lcm}\{d:n_d\ne0\},\qquad \wt=\sum_d d^2n_d, \] the form on the cyclic group $C_N$ is \[ B_{\bm n}(g,h)=\frac1{\wt}\sum_{(g^{-1}h)^d=1}d^2n_d. \] Its labelled Fourier spectrum reconstructs the complete multiplicity sequence by Möbius inversion. If the $n_d$ are the Wedderburn multiplicities of a split semisimple algebra $S=\prod_d M_d(k)^{n_d}$, the divisibility widths are dimensions of characteristic sums of Wedderburn blocks, and the form together with $\dim_kS$ recovers the isomorphism type of $S$. The main theorem is an arithmetic rigidity result. For $N\ge2$, the reflection group is finite if and only if it preserves a full rank lattice, equivalently \[ \supp(\bm n)=\{1,N\},\qquad n_1=N^2n_N. \] In this case $B_{\bm n}=\frac12(I+J)$ and the group is $W(A_N)\cong S_{N+1}$. In semisimple algebra terms, finiteness is equivalent to \[ S\cong k^{N^2m}\times M_N(k)^m \] for some $m\ge1$. For $N=2$, every nonfinite case is a dense copy of $D_\infty$ in the corresponding orthogonal group. For contraction algebras of threefold flops, Toda's width formulas identify the total scale and primitive Fourier mode with $\dim A_{\con}$ and $\dim A_{\con}^{\ab}$. This yields infinitude for all irreducible flops of length at least three and, in length two, \[ \Gamma_f\text{ finite}\quad\Longleftrightarrow\quad \dim A_{\con}=2\dim A_{\con}^{\ab}. \] The final part gives a complementary BPS/Riemann–Hilbert realization of the same reflection matrices.

We determine the dimensions of all derivative loci and all higher multiplicity loci of an elementary symmetric polynomial over an algebraically closed field of positive characteristic. For the degree-d polynomial in n variables, set Q = p^{v_p(n-d+1)}. The common zero locus of its order-q derivatives has dimension d - q - 1 + 1_{Q>d-q}, whereas the locus of zeros of multiplicity at least s has dimension d - s + 1_{Q>d}. The order-two case settles the dimension question in a conjecture of Orzel, with a necessary correction when d is a power of p. We classify all components attaining the larger dimension and prove that they are generically smooth. Along the coordinate components in the smaller-dimension cases, we determine the local rings and obtain generic multiplicity exactly Q. Finally, we identify the full translation stabilizer scheme and derive a Frobenius normal form. These results distinguish dimension jumps from infinitesimal thickening and give a uniform description of their dependence on the characteristic.

This paper considers the generic behaviour of sheaves supported on Schubert varieties in finite flag varieties. Our main results are that, for any fixed prime $p$, as the rank of the associated algebraic group grows, almost every intersection cohomology sheaf admits $p$-torsion in its stalks or costalks, and almost every indecomposable parity sheaf is not perverse. For certain families of partial flag varieties we also determine explicit asymptotics for this growth.

We prove the norm principle for the extended Clifford group $\Omega(q)$ of every nondegenerate quadratic form $q$ of even dimension at least $4$ over an arbitrary field $F$ of characteristic different from $2$, for all finite separable field extensions $L/F$. The main new ingredient is a factorization theorem for proper similitudes: for a quadratic extension $L/F$ and $f\in \mathrm{GO}^+(q)(L)$, there exist $u\in \mathrm{SO}(q)(L)$, a maximal $F$-torus $S\subset \mathrm{GO}^+(q)$, and $s\in S(L)$ such that $f=us$.

Let $G/P$ be a flag manifold with $G$ simple. We determine the rational homotopy groups $\pi _{\ast }(G/P)\otimes \mathbb{Q}$ and describe the rational cohomology ring $H^{\ast }(G/P;\mathbb{Q})$ solely in terms of the rational homotopy types of $G$ and $P$, respectively. The proof is based on the restrictions of basic Weyl invariants of $G$ to suitable simple factors of $P$, which also address a key nonvanishing issue not settled by earlier approaches of Meier, Shiga-Tezuka, Kotschick, and Terzic.

We prove that fiberwise-connected finite locally free commutative group schemes over a ring $R$ are equivalent to an explicit full subcategory of the derived category of modules over the Cartier ring of $R$.

A Demazure model of an algebraic $k$-torus $T$ split by a finite Galois extension of fields $L/k$ and splitting group $G=\text{Gal}(L/k)$ is a smooth projective $T_L$-toric variety $X_{\Sigma}$ such that the natural action of $G$ on $T_L$ extends to $X_{\Sigma}$. A smooth projective model of the algebraic $k$-torus $T$ can then be taken to be the quotient of a Demazure model by its splitting group. The existence of Demazure models of algebraic $k$-tori was determined by Brylinski and further refined by Colliot-Thélène, Harari and Skorobogatov. Voskresenskii and Kunyavskii's birational classifications of algebraic $k$-tori in dimensions 2 and 3 involved constructions of Demazure models of the algebraic $k$-tori corresponding to maximal finite subgroups of $\text{GL}(r,\mathbb{Z})$ for $r=2,3$. We discuss some explicit constructions of Demazure models of algebraic $k$-tori, making connections with the defining integral representations of their splitting groups. The constructions determine smooth projective Demazure models of the algebraic tori corresponding to maximal finite subgroups of $\text{GL}(4,\mathbb{Z})$. Since projective toric varieties are determined by lattice polytopes, our constructions focus on some highly symmetric families of polytopes, such as root polytopes and central transportation polytopes.

At the genesis of Galois theory at the end of the 19th century, enumerative geometers were concerned with the solvability of various problems: for instance whether one can express lines on a cubic surface in radicals in terms of the coefficients that define it. By work of Hermite and a celebrated theorem of Harris, this is equivalent to determining whether the monodromy group of a certain finite étale cover is solvable. This problem, as well as the related problem of computing bitangents to plane quartics, are both unsolvable. When one restricts to the locus of $G$-symmetric cubic surfaces, monodromy will drop and can become solvable. In this setting there are four flavors of monodromy one can consider: the one from restricting the classical cover, the one over the GIT quotient after modding out by projective transformations, and two related notions of "monodromy" coming from the stacky cover of $G$-symmetric lines on $G$-symmetric cubic surfaces - unlike in the non-symmetric setting these are all different. In this paper we characterize the relationships between all four notions of monodromy in the general setting of quotient stacks, and use these connections to compute all these monodromy groups for all 11 possible automorphism groups of smooth cubic surfaces, as well as for all 12 possible automorphism groups for smooth planar quartics. We demonstrate that, in the presence of any symmetry whatsoever, the problem of solving for lines on cubic surfaces or bitangents to plane quartics is solvable.