Papers

In this article, using the theory of $\delta$-geometry, we construct a canonical $z$-isocrystal $(\mathbf{H}_\delta(E), \mathfrak{f}^*)$ admitting a Hodge-Pink structure for any abelian Anderson module $E$. The Hodge-Pink structure on $\mathbf{H}_\delta(E)$ induces a natural filtration $(\mathbf{H}_\delta(E) \supset \mathbf{X}_{\mathrm{prim}}(E) \supset \{0\})$. The elements of $\mathbf{X}_{\mathrm{prim}}(E)$ are represented by primitive delta characters associated to $E$. We establish a natural morphism from $\mathbf{H}_\delta(E)$ to the associated de Rham cohomology module $\mathbf{H}^*_{\mathrm{dR}}(E)$, which is strictly compatible with the aforementioned filtration and the classical Hodge filtration $(\mathbf{H}^{*}_{\mathrm{dR}}(E)\supset {\mathrm{Lie}(E)^{*}}\supset \{0\})$ on $\mathbf{H}^*_{\mathrm{dR}}(E)$. Moreover, we show that the map induces an isomorphism between $\mathbf{X}_{\mathrm{prim}}(E)$ and $\mathrm{Lie}(E)^*$. Hence our isomorphism provides an interesting interpretation of the invariant differentials of $E$ as primitive delta characters of $E$. Furthermore, when $E$ is a Drinfeld module, we show that the constructed $z$-isocrystal $\mathbf{H}_\delta(E)$ is weakly admissible. Consequently, the positive equal characteristic analogue of the Fontaine functor associates a crystalline $z$-adic Galois representation to the $\delta$-geometric object $\mathbf{H}_\delta(E)$. In the case, when $E$ is the Carlitz module, we show that the Galois representation associated to $\mathbf{H}_\delta(E)$ is indeed the usual one coming from the Tate module.

Given a smooth projective variety $X$ of dimension $n$ and a closed subscheme $Z \subseteq X$, it is in general a difficult problem to determine whether $X \setminus Z$ is isomorphic to $\mathbf A^n$. In the case where $X$ is a projective bundle over $\mathbf P^1$ and the restriction of every irreducible component of $Z$ to every fibre is a hyperplane, we give a complete geometric characterisation in terms of the irreducible components of $Z$ and their intersections. In the appendix, we use this to obtain a coordinate-free explanation for the recent first counterexample to the Jacobian conjecture.

In this paper, over the field of complex numbers, we prove that a numerical Godeaux surface contains at most six pairwise disjoint $(-2)$-curves, and that this bound is sharp. As an application, we refine the classification of involutions on smooth minimal surfaces of general type with $p_g=0$ and $K^2=7$: the divisorial fixed part $R$ satisfies $R^2=-1$, the involution acts trivially on $H^*(S,\mathbb{Q})$, and, if the minimal resolution of the quotient is of general type, it is a numerical Campedelli surface containing five pairwise disjoint $(-2)$-curves. Another application concerns commuting involutions on smooth minimal surfaces of general type with $p_g=0$ and $K^2=8$.

In the study of Nash equilibria of finite-player games, one often seeks equilibria that are compatible with predetermined constraints, either determined by the players or by an external agent. We discuss the algebraic loci, called Nash loci, of games whose Nash equilibrium scheme intersects a fixed algebraic variety in a product of projective spaces. We determine their dimensions and multidegrees in multiprojective space, and their equations for two-player games and for small multiple-player games. The multilinear equations defining the Nash equilibrium scheme allow us to describe Nash loci in the language of Grassmannians and Plücker coordinates. Motivated by this fact, we relate Nash loci to multigraded associated varieties, which are subvarieties in products of Grassmannians that generalize the multigraded Cayley-Chow hypersurfaces of Osserman and Trager.

For smooth hypersurfaces over rational function fields of characteristic greater than the degree and with sufficiently large constant field, we improve the number of variables required in Birch's theorem from an exponential function of the degree to a quadratic one. This agrees, up to constants, with the sharp quadratic threshold for the unconditional existence of rational points on smooth hypersurfaces. A circle method argument reduces the required cancellation to lower bounds for the codimensions of certain singular loci associated with complete exponential sums over finite fields. Our main innovation is a new method for proving these bounds: we introduce the notion of multiplication rank for the linear functionals indexing these exponential sums and combine the resulting rank stratification with a weighted degeneration of the Jacobian equations to obtain a codimension estimate that grows linearly with multiplication rank.

In this article, we construct noncommutative cluster varieties, $\mathcal{A}_{R,S}$, for each reduced root system $R$ and marked surface $S$ simultaneously generalizing the cluster varieties of Fock-Goncharov, Li, Goncharov-Shen, Berenstein-Retakh, Goncharov-Kontsevich, and our previously introduced polygonal cluster algebras. Additionally, we define a large class of algebraic groups, we call Jordan split groups. Given a reduced root system $R$ and a family of Jordan algebras, the Lie algebra for $G$ is constructed by unifying the Tits-Kantor-Koecher construction for a single Jordan algebra with the construction of a split Lie algebra. The notion of Jordan split groups is closely related to a grading of its Lie algebra by the root system $R$. We show that these gradings are usually induced by a choice of standard parabolic subalgebra $\mathfrak{p}_\Theta$ and we classify $R$-graded pairs $(G,\Theta)$ via a condition depending only on the subset $\Theta\subset \Delta$ of the set of simple roots. Next, we define Jordan algebra points of $\mathcal{A}_{R,S}$ which parameterize $G$-local systems on $S$ with boundary decoration related to cosets $G/U_\Theta$ when $G$ is Jordan split of type $R$. When $S$ is a disk, points of $\mathcal{A}_{R,S}$ parameterize configurations of decorated flags. We use this to give noncommutative cluster structures on the double $R$-Bruhat cells of $G$, generalizing the cluster algebras of Berenstein-Fomin-Zelevinsky. When each Jordan algebra is formally real, we say that $G$ has a positive structure with respect to $\Theta$. This defines a positive semigroup in $G$. For real algebraic groups, the pairs $(G,\Theta)$ which have positive structures are exactly those which admit a positive structure as defined by Guichard-Wienhard and we give algebraic proofs of many of the properties of positive configurations of flags and of positive representations.

Let $k=\mathbb F_q$. We classify, up to two-sided $k$-Möbius equivalence, all separable indecomposable tame exceptional maps $\mathbf P^1_k\to\mathbf P^1_k$ whose geometric Galois closure has genus one. The resulting arithmetic fixed-field forms are encoded by Frobenius-stable affine elliptic quotient data recovered from the cover; we prove a converse and an exact equivalence criterion. The same data determine branch arithmetic, geometric and arithmetic monodromy, the constant field, and behavior over every finite extension. In particular, over each $\mathbb F_{q^r}$, permutation is equivalent to exceptionality. We obtain explicit rank-two support formulas, necessary and sufficient occurrence criteria, and exact two-sided class counts in every tame signature, including the surviving cases in characteristics $2$ and $3$. In characteristic greater than $3$, tameness is automatic. A sharp $3$-adic obstruction arises only for a stronger cubic self-endomorphism realization problem, not for the fixed-field classification itself.

In this paper we study the problem of realizing finite group schemes as fundamental group schemes of smooth projective varieties. We establish Bertini-type results and prove the triviality of fundamental group schemes of mildly singular complete intersections in projective space. With this in hand, we are able to go through a Godeaux–Serre construction and prove that, over an infinite perfect field $k$, any finite group scheme $G/k$ can be realized as the $S$-, extended Nori and Nori fundamental group schemes of a connected smooth projective variety of any dimension at least $\dim \text{Lie}(G)+2$.

We study nearest and farthest Voronoi diagrams of lines in $\mathbb{R}^3$ under the Euclidean metric when all $n$ lines belong to one ruling of a smooth doubly ruled real quadric. For arbitrary line sites, the combinatorial complexity of the nearest Voronoi diagram is known only to lie between $\Omega(n^2)$ and $O(n^{3+\varepsilon})$. Under general-position assumptions, we prove that both diagrams in the ruling class have at most $4n(n-3)$ vertices and $O(n^2)$ total combinatorial complexity. Conversely, for every $n \ge 4$, one ruling of a fixed non-rotational one-sheeted hyperboloid contains a general-position set of $n$ lines with at least $(n-2)(n-3)/2$ distinct regular nearest vertices, where regular means that exactly four lines support the vertex and their three defining bisectors meet transversely. Thus the worst-case complexity of the nearest Voronoi diagram in this class is $\Theta(n^2)$, while the farthest diagram has $\Theta(n^2)$ complexity for every general-position input, since it has exactly $n(n-1)$ three-dimensional cells. Under the Plücker embedding, the ruling is a conic, and the condition for a line to be tangent to a Euclidean sphere restricts to a binary quartic. At a regular vertex, the four supporting parameters exhaust its roots, and sign alternation forces two arcs of the parameter circle to be site-free. This leaves only $n(n-3)/2$ possible cyclic support types, while Bézout's theorem bounds the number of centers for each type by eight. The same reduction yields an exact $O(n^2)$-time algorithm that, after cyclically sorting the site parameters, enumerates all finite nearest and farthest vertices as constant-degree real univariate representations.

Motivated by theoretical problems in deep learning, we conjecture that post-composing a fixed number of pairwise distinct nonconstant polynomials with a generic polynomial of sufficiently large degree yields linearly independent polynomials. This generalizes Newman–Slater's theorem on powers of polynomials. We establish several cases of this conjecture and its origin-passing variant: We prove the result for two polynomials, and for an arbitrary number of polynomials when their degrees are bounded. Furthermore, we show how the conjecture implies a complete understanding of the identifiability (i.e., parameter symmetries) of deep fully connected neural network architectures with generic polynomial activation functions. In particular, for network architectures with layer-specific activations of increasing degree, our established versions of the conjecture fully characterize the set of parameters yielding the same end-to-end network function. As a special case, we fully resolve the identifiability of shallow polynomial networks.

In this paper, we study the structures of single connected $k$-leaky $(r+1)$-completed Hurwitz numbers. We first prove that, for fixed genus, after extracting an explicit product of Pochhammer type factors, the stable connected leaky Hurwitz numbers for partition $\mu$ are polynomials in the quotients $[\mu_i]$ modulo $k+r$. We then give a closed formula for the generating function of single connected leaky Hurwitz numbers, which can be used to study the genus dependence of leaky Hurwitz numbers.

We provide necessary and sufficient conditions for the specializations of a vertically parametrized polynomial system to attain the maximal number of complex nonzero solutions. To each vertical system, we attach a pair consisting of a projective simplicial toric variety and a toric vector bundle. The toric vector bundle allows for a homogenization of the polynomials in the Cox ring of the toric variety, providing a homogeneous ideal with the same zeros over the torus as the original system. We show that the maximal number of isolated solutions is attained if and only if this ideal has no solutions in the faces of the toric variety and prove that this happens for generic values of the parameters. In addition, we provide a novel formula for this generic number of zeros as an alternating sum of mixed volumes over a family of polytopes attached to the toric vector bundle.

In this paper we prove that the Generalised Cone Conjecture holds for generic Sakai surfaces, which completes the proof of the Generalised Cone Conjecture in dimension two. Furthermore, we show that there exists a non-generic Sakai surface for which no meaningful variant of the conjecture holds.

We obtain a complete classification of rank two semistable sheaves with maximal third Chern class on the Fano threefold $X_5$ of index 2 and degree 5. Also, we describe moduli spaces of such sheaves in the general case of big second Chern class. These moduli spaces are smooth rational varieties. The study uses the theory of tilt-stability and Bridgeland stability conditions.

A separating curve is a real curve whose real part disconnects its complex part. It also admits totally real morphisms to the projective line, and the separating gonality is the minimal possible degree for such morphisms. The separating gonality is bounded from below by the number of connected components of the real part of the curve. In this paper, we build on a strategy of Manzaroli to improve this lower bound for some curves embedded in the projective plane or in a Hirzebruch surface.

Let $S$ be a smooth projective surface with $q(S)=0$. We give numerical criteria for the nef cones of $S^{[n,n+1]}$ and $S^{[1,n]}$ to decompose as sums of pullbacks of nef cones under their natural morphisms. These criteria recover the known decompositions for the projective plane, Hirzebruch surfaces, and Picard rank one K3 surfaces, and apply to del Pezzo surfaces with $2\le K_S^2\le7$. For a del Pezzo surface of degree one, we show that the corresponding pullback decompositions fail for both $S^{[n,n+1]}$ and $S^{[1,n]}$.

We study properties of the transcendental numerical dimension on compact Kähler manifolds. In particular, we establish a connection between this invariant and degenerate divisors in fibrations, extending few results from the projective setting. We also study fibrations between compact Kähler manifolds with rationally connected general fibre. We prove that every pseudo-effective line bundle contained in a tensor power of the cotangent bundle comes from a pseudo-effective line bundle contained in the corresponding tensor power on the base, up to an explicit relation involving degenerate divisors. Finally, combining these results with a theorem of Cao–Păun, we answer a question they posed on rational quotients. More precisely, let \(q : X\dashrightarrow Q\) be the rational quotient of a compact Kähler manifold \(X\), and let \(L\) be a pseudo-effective line bundle on \(X\) admitting an injection \(L\to(\Omega_X^1)^{\otimes m},~m \geq 1\). We prove that \[ \nu(L,X)\leq\nu(K_Q,Q). \] To the best of our knowledge, both the descent theorem and this inequality are new even in the projective setting.

We investigate the form of the Saito metric on quotients of the reflection representation of an extended affine Weyl group, restricted to an arbitrary stratum of its discriminant. We compute the Saito determinant for all strata and Dynkin types and show that it is divisible by a product of q-analogues of restricted roots for the associated hyperplane arrangement. Our results simultaneously generalise the factorisation in the Weyl denominator formula to higher codimension strata, and provide q-analogue refinements of results of Antoniou-Feigin-Strachan for Coxeter groups.

Motivated by the Ekedahl-Shepherd-Barron-Taylor conjecture, we study arithmetic criteria for the global and local integrability of foliations in characteristic zero. Our main result is a finite-holonomy theorem for the regular parts of invariant prime divisors of corank one foliations which are \(p\)-closed for almost all primes. As a consequence, we give a complete proof for corank one foliations of the statement proposed by Ekedahl-Shepherd-Barron-Taylor: a foliation on a smooth projective variety which is \(p\)-closed for almost all primes is algebraically integrable whenever it admits a compact leaf. We also introduce a local analytic notion of \(p\)-closedness for foliation germs on normal complex varieties and formulate local conjectures relating it to meromorphic and holomorphic first integrals. We establish holomorphic integrability for analytically \(p\)-closed foliations with canonical singularities on smooth surfaces and on klt surface germs. Combining these local results with the finite-holonomy theorem, we obtain holomorphic first integrals for certain analytically \(p\)-closed non-dicritical surface germs and for analytically \(p\)-closed simple singularities in arbitrary dimension. Finally, we formulate a connection-theoretic counterpart to local holomorphic integrability in terms of flat meromorphic extensions of the Bott partial connection, and verify it in dimension two.

We study the GIT quotients of the Peterson variety $\mathrm{Pet}_n\subset \mathrm{GL}(n,\mathbb C)/B$ under a one-parameter subgroup $\lambda:\mathbb G_m \to T$ with respect to the linearization $\mathcal L(\chi)$ given by a regular dominant character $\chi$ in the root lattice. Using the Richardson stratification, we describe the semistable and stable loci explicitly in terms of subsets of simple roots. This determines the GIT chamber decomposition and the corresponding wall-crossing morphisms. In the deep chamber, the quotient is shown to be isomorphic to the weighted projective space $\mathbb P(1,2,\ldots,n-1)$. We obtain a complete chamber-theoretic characterization of normality and describe how normality varies with the choice of linearization. We also prove that the quotient is smooth if and only if $n\le3$, independently of the regular dominant linearization. These results describe how the singular geometry of the Peterson variety is reflected in the variation of its GIT quotients.

Let $\mathcal{Z}_w^{(\mathbb{F}_q)}$ be the $\mathbb{F}_q$-linear subspace of $\mathbb{F}_q(\!(\theta^{-1})\!)$ spanned by Thakur's multiple zeta values $\zeta_A(\mathfrak{s})$ of weight $w$. We prove that $\sum_{w=1}^{\infty} \left(\dim_{\mathbb{F}_q} \mathcal{Z}_w^{(\mathbb{F}_q)}\right) x^w = \frac{x(1-x^q)(1-2x+x^q)}{(1-2x+x^{q+1})^2}$. Moreover, we construct an explicit $\mathbb{F}_q$-basis of $\mathcal{Z}_w^{(\mathbb{F}_q)}$, and prove that any $\mathbb{F}_q$-linear relation among Carlitz multiple polylogarithm values $\operatorname{Li}_A(\mathfrak{s})$ is an $\mathbb{F}_q$-linear combination of quadruple-carry relations. This result can be regarded as an $\mathbb{F}_q$-analogue of the corresponding $\mathbb{F}_q(\theta)$-theorem proved by Chang–Chen–Mishiba and independently by Im–Kim–Le–Ngo Dac–Pham. Our discovery of the quadruple-carry relations is inspired by the recent work of Im–Kim–Ngo Dac. These relations may be viewed as $\mathbb{F}_q$-analogues of the double-shuffle relations among classical multiple zeta values $\zeta(\mathfrak{s})$.

Let $X$ be an $\epsilon$-lc projective variety of dimension $n\geq 2$ such that $-K_X$ is nef and $0<\epsilon \leq 1$ is a real number. Then for any nef divisors $D_1,\dots,D_{n-2}$, \[ c_1(X)^{2}\cdot D_1\cdots D_{n-2}\leq \frac{2(1+\epsilon)}{\epsilon}\,\hat c_2(X)\cdot D_1\cdots D_{n-2}. \] In particular, there exists a Kawamata–Miyaoka type inequality \[ c_1(X)^n\leq \frac{2(1+\epsilon)}{\epsilon}\,\hat c_2(X)\cdot c_1(X)^{n-2}. \]

We record lower bounds for permanent circuits with valid division over characteristic zero, counting nonscalar multiplications and divisions while additions and scalar operations are free. Chapter 5 of OpenAI's Ten Advances supplies the block construction and critical-locus estimate; these, with the parameter choice made here and the classical bounds of Strassen and Baur-Strassen, give liminf as n tends to infinity of L_div(per_n)/(n^2 log_2 log_2 n) >= 1/12. Our finite-parameter refinement for matching-minor polynomials observes that the coefficient vector at deletion size h lies in a space of dimension at most min{binom(t,h), binom(t,d-h)}. Twelve machine-checked declarations of the Lean development accompanying OpenAI's manuscript on border determinantal complexity of the permanent (September 24, 2026) imply, by a short written argument, the existence of a geometric slice; a further written, unformalized application of the same criterion gives L_div(per_n) >= (n^2/119790) log_2(n/44) for n >= 1936. Assuming Proposition 7.3 of that manuscript gives the bound (n^2/86400) log_2(n/48) for n >= 1408. These are order-of-growth results; the elementary Hessian bound n^2/2 is larger at practical orders.

We study the exceptional component of the space $\mathbb{F}(2,\mathbb{P}^3)$, of codimension-one foliations of degree two on $\mathbb{P}^3$. We describe the geometry of its boundary and prove that it has four irreducible components, all of dimension $12$. Three of these components contain a dense subset given by the orbit of a logarithmic foliation of type $(1,1,2)$, while the fourth contains a family of pull-back type foliations from $\mathbb{P}^2$ whose orbits have dimension $11$.

We prove that for a connected, geometrically unibranch variety $X$ over a perfect field of positive characteristic and a dense open subset $U \subset X$, the canonical homomorphism between the Tannaka duals of the categories of overconvergent isocrystals is a quotient map. We also show that the analogous statement for convergent isocrystals, and for all isocrystals, fails. The counterexample is based on Crew's construction of a unit-root $F$-isocrystal that is convergent but not overconvergent.

Motivated by the principle that positivity of the Beauville–Bogomolov–Fujiki (BBF) form controls the geometry of hyperkähler manifolds, we study BBF positivity from the perspective of bimeromorphic geometry, focusing on its relation to the Moishezonness of hyperfujiki manifolds (natural bimeromorphic analogues of hyperkähler manifolds). We prove that any Moishezon hyperfujiki manifold carries a big line bundle with positive BBF square. We also prove that the existence of a BBF-positive integral $(1,1)$-class implies Moishezonness for hyperfujiki $4$-folds and, more generally, for hyperfujiki $2n$-folds under a weak Kähler minimal-model condition. The proof mainly uses the theory of primitive symplectic varieties developed by B. Bakker–C. Lehn, a construction of small bimeromorphic models for certain $K$-trivial manifolds by I. Biswas–J. Cao–S. Dumitrescu–H. Guenancia, and the theory of pull-backs of reflexive differential forms established by S. Kebekus–C. Schnell. As an application, we give a Hodge-theoretic description of the Moishezon locus in certain smooth families by combining this criterion with the period theory developed by B. Anthes–A. Cattaneo–S. Rollenske–A. Tomassini.

Let X be a smooth, complex Fano variety, and delta(X) its Lefschetz defect. It is known that if delta(X) is at least 4, then X is isomorphic to a product SxT where S is a surface, whereas if delta(X)=3, X admits a del Pezzo fibration with a prescribed structure. In this paper, we study the case where delta(X)=2. Our first result is that there exists a sequence of flips X–>X' such that X' is smooth and admits a conic bundle structure X'->Y, with rho(X)-rho(Y)=2. The conic bundle factors as X'->X”->Y, where X'->X” is a smooth blow-up, and f: X”->Y is again a conic bundle. Then we show that either f is smooth, or there exists a del Pezzo fibration g: X->T with rho(X)-rho(T)=3 and T smooth, admitting two possible generic fibers, and we describe the relative cone NE(g). Finally for one of the possible generic fibers we give a structure theorem for the del Pezzo fibration g; we show that T is Fano and that X can be reconstructed from T and a suitable divisor class on T. As an application, we construct a family of Fano 4-folds X exhibiting this behaviour.

We prove the multivariable Strong Monodromy Conjecture for the local topological zeta function of arbitrary tuples of nonzero nonunit plane curve germs. The entries may be nonreduced and may have common irreducible factors. We also prove that the non-equivariant local motivic and topological zeta functions have the same polar locus and generic pole orders, using a multivariable extension of the Nicaise–Xu definition of motivic pole order. Our arguments combine twisted Poincaré residues with resolution formulas and specialization to one variable. In arbitrary dimension, we prove that a nonzero iterated residue class on a stratum of a log resolution implies the vanishing of the Bernstein–Sato ideal at the corresponding parameter. In particular, every topological polar hyperplane whose order equals the ambient dimension is contained in this zero locus. For algebraic germs, the topological pole order is at most the motivic pole order, and the two attain maximal order simultaneously.

Let $X$ be a connected smooth complex projective surface. We prove an effective-divisor version of the Arcara–Miles conjecture, together with its strict analogue. For every divisorial Bridgeland stability condition, failure of stability, respectively semistability, of a line bundle or its relevant shift is detected by a natural subobject associated with a non-zero effective Cartier divisor $C$ satisfying $C^2<0$. The proof combines minimal-rank destabilizers, slope Harder–Narasimhan filtrations, the Bogomolov–Gieseker inequality, and an ordered Lorentzian partial-sum estimate that forces the minimal rank to be one. Consequently, strict semistability of a line bundle or its relevant shift is detected by a negative effective divisor, and a numerical semistability condition arising from the twisted deformed Hermitian–Yang–Mills equation is equivalent to stability under all integral scalings.

This is the first of two papers in which we prove the Generalised Cone Conjecture on surfaces.