Papers

Let $U$ be an affine log Calabi–Yau variety. Although special and finitely generated valuations are defined using a log CY-Fano compactification of $U$, we prove that the corresponding skeleta in the dual complex are independent of this choice, and hence invariant under $\mathrm{Aut}(U)$. We also show that a finitely generated valuation is special if and only if it is maximal with respect to a natural partial order induced by regular functions on $U$. We then consider the affine log Calabi–Yau threefold obtained as the complement of the Markov cubic surface in $\mathbb{A}^3$. We describe the action of the three Vieta involutions on the special skeleton and relate it to the $(\infty,\infty,\infty)$-triangle reflection group on the hyperbolic plane. We also show that, for every finite triangulation of the dual complex, the special skeleton fails to be locally closed on some simplex. Via the cone construction, this provides a counterexample to a conjecture of the author and Xu.

We construct explicit families of maximal singular curves over finite fields. A maximal singular curve is a curve that attains the Aubry-Perret bound, which is the natural extension of the Hasse-Weil-Serre bound on the maximum number of rational points on a smooth curve over a finite field. Starting from a smooth elliptic or hyperelliptic curve $\tilde{C}$ over $\mathbb{F}_{q}$ ($q$ odd), we generate singular curves $C$ with non-split nodes or cusps. In our approach we use linear projections of geometric embeddings, and we apply Stöhr's embedding of hyperelliptic Gorenstein curves and the Rosa-Stöhr theory of trigonal Gorenstein curves. The construction is applied to the Tafazolian and Tafazolian-Torres smooth maximal curves. Finally, we also determine the gonality of all obtained curves.

We study genus two curves whose Jacobians have complex multiplication by a biquadratic CM field. In this setting the Jacobian decomposes as a product of two elliptic curves with complex multiplication by orders in the same imaginary quadratic field. We derive upper bounds for the primes of stable bad reduction of such curves in terms of the discriminants of the CM orders. To do so, we give an explicit description of principally polarized abelian surfaces with CM by a biquadratic field via small CM cycles on an orthogonal Shimura variety of signature (3,2). We use it to prove an arithmetic intersection formula between such CM cycles with Humbert surfaces in terms of coefficients of incoherent Eisenstein series and representation numbers of ternary positive definite quadratic forms. Finally, employing the arithmetic properties of higher Green functions, we show that primes of stable bad reduction always exist.

We prove a criteria for the Behrend function values on the Hilbert scheme of $n$-points on $\C^3$ from local monodromy and the Denef–Loeser's Lefschetz trace formula of jet spaces. If the power two, three, or four of the monodromy is unipotent, it reduces respectively to a Hessian parity formula, a cubic-complement formula, or a quartic formula. This gives a Milnor fiber and monodromy based proof of nonconstancy of the Behrend function on $\Hilb^n(\C^3)$ after Jelisiejew-Kool-Schmiermann. We also calculate two ideals in $\Hilb^9(\C^3)$ and $\Hilb^{12}(\C^3)$ so that the Behrend function is not $\pm 1$, answering a question of Ricolfi.

We study the Bourbaki degree of line arrangements in the projective plane and its interaction with the intersection lattice and the syzygies of the gradient ideal. We show that the Bourbaki degree is obtained by evaluating the reduced characteristic polynomial at the initial degree of the first syzygy module. This leads to a sharp upper bound and to refinements involving the two largest intersection multiplicities, with consequences for the global Tjurina number, the freeness defect, and Terao's conjecture, including an improvement of Dimca's numerical criterion when the smaller exponent is at least five. We also establish addition–deletion formulas and show that the defect from the sharp bound is monotone under line addition. Finally, we prove that the Bourbaki degree is determined by the intersection lattice for arrangements with at most eight lines, while examples with isomorphic intersection lattices and different Bourbaki degrees exist for every number of lines at least nine.

We study the canonical formal solution space of a homogeneous $A$-hypergeometric system in a generic direction. We present as a finite-dimensional cokernel the quotient of that space by the span of the series obtained by intrinsic perturbation, taken over all exponents occurring in that space and all ordered negative support families, and we compute the codimension of that span. We show that the presentation and the codimension transfer to the holomorphic solutions on a common nonsingular domain. We apply these results, for every integer $q\geq 1$, to a homogeneous $A$-hypergeometric system of lattice rank 2 with $5q$ columns. We show that the canonical formal solution space of this system has dimension $12q^2$ and that the span of the series obtained by intrinsic perturbation has codimension exactly $\lceil 3q^2/4\rceil$ in that space.

We prove there are $1442$ four-bar coupler curves through nine generic points in the plane, and thus resolve Alt's problem. We obtain this proof in three steps. First, we identify the space of coupler curves with a Zariski open subset of $\textrm{Gr}(3,6)$. Next, we formulate the polynomial system representing the nine-point path synthesis problem in these coordinates and modify it to obtain the mixed volume $5538$. Finally, we prove that $4096$ of the branches of the generic sparse polynomial system with that support escape the torus in the sparse limit. Thus, we obtain an upper bound of $5538-4096=1442$ for the generic solution count. A lower bound of $1442$ is achieved via numerical certification on one instance.

We develop a root-theoretic localization formalism for genus-zero Gromov–Witten invariants of flag varieties and of smooth zero loci of globally generated homogeneous vector bundles. The resulting invariants are expressed as finite sums over decorated trees whose contributions are determined by the root system of the ambient flag variety and by the torus weights of the defining bundle. We connect these computations with the Theory of Hodge Atoms of Katzarkov–Kontsevich–Pantev–Yu via the matrix of small quantum multiplication by first Chern class. Let $X$ be a Fano fourfold with $h^{3,1}(X)=1$ arising as a hyperplane section of a suitable Fano fivefold. Its cohomology decomposes into a monodromy-fixed part and the middle vanishing cohomology. This decomposition is preserved by quantum multiplication by the first Chern class. Moreover, it acts by a scalar on the vanishing summand. We show, for the monodromy-fixed block, that (a) if every eigenvalue has algebraic multiplicity at most two and $X$ is Hodge general, then $X$ is irrational; (b) if every eigenvalue has Jordan defect at most one and $X$ is rational, then every weak factorization contains a smooth surface center whose minimal model is a projective K3 surface. Many applications are presented. Using the recently introduced quantum connection approach to irrationality, we obtain from the computations here presented the irrationality of the very general smooth complete intersection of a hyperplane and a quadric in $\operatorname{LG}(3,6)$; the very general smooth quadric section in $G_2/P_2$; and the very general smooth complete intersection of five hyperplanes and a quadric in the spinor 10-fold.

Brane tilings are bipartite graphs on a 2-torus that encode the Lagrangians of 4d N=1 supersymmetric gauge theories arising on D3-branes probing toric Calabi-Yau 3-folds. Among these bipartite graphs, only those that satisfy geometric consistency conditions correspond to well-behaved quantum field theories. We train a convolutional neural network (CNN) to distinguish geometrically consistent from inconsistent brane tilings directly from their Kasteleyn matrices. We study a family of 4d N=1 theories obtained by adding diagonal edges to the hexagonal faces of the brane tiling for the abelian orbifold C^3/Z_3 x Z_3, and find that the CNN identifies geometric inconsistency with high accuracy. For inconsistent brane tilings that can be rendered consistent by Higgsing a single bifundamental chiral field, we show that gradient-based saliency analysis can be used to localize the responsible chiral fields with accuracy well above a matched random baseline. These results demonstrate that explainable AI (XAI) can be used to identify local defects responsible for inconsistencies in supersymmetric gauge theories.

We prove that a smooth projective surface over $\mathbb{C}$ admitting a full exceptional collection of line bundles is rational.

Let $G$ be a finite abelian group, and let $k$ be an infinite perfect field containing a primitive root of unity of order the exponent of $G$. We prove that the $G$-equivariant version of Gabber's presentation lemma proposed by Bachmann holds over $k$ if and only if $G$ is cyclic of $p$-power order, for some prime $p$.

This paper studies the Floer-theoretic aspects of homological mirror symmetry inspired by proposals of Perutz and Siebert and the Gross–Siebert intrinsic mirror symmetry program. Given a maximally unipotent degeneration of smooth projective Calabi–Yau manifolds over the punctured disk, we construct a ring using the fixed point Floer cohomology groups of the iterates of the monodromy of the degeneration equipped with the pair of pants product. Under the assumption that this ring is commutative, we can consider a candidate mirror family defined by the relative Proj construction. Further assuming that a smooth fiber $X_t$ contains a so-called tropical Lagrangian section, we construct a fully faithful embedding from the derived category of perfect complexes on our candidate mirror family into the Fukaya category of $X_t$. We verify both of these assumptions for certain Batyrev–Borisov toric degenerations, as well as some toric degenerations of Calabi–Yau threefolds coming from the Gross–Siebert reconstruction algorithm. These two geometric hypotheses are both phrased to support the general study of mirror symmetry for maximally unipotent degenerations of Calabi–Yau manifolds, largely reducing the symplectic inputs for proving homological mirror symmetry to the problem of constructing tropical Lagrangian sections.

We show that for every smooth complex cubic threefold $Y$, the fourfold $Y\times\mathbb{P}^1$ is irrational. As an application, we construct a smooth stably rational complex Fano variety $X$ which is irrational, while $X^m$ is rational for every $m\geq2$ and $\mathrm{Sym}^2 X$ is rational.

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $S$ be a minimal smooth projective surface of general type over $k$. If the second Chern class satisfies $c_2(S)<0$, then the order of the automorphism group satisfies $|\mathrm{Aut}_k(S)|<2598(K_S^2)^4$, and the order of every abelian subgroup $G\subset \mathrm{Aut}_k(S)$ satisfies $|G|<81(K_S^2)^3$.

In this paper we consider the general $n-2$ dimensional linear section $X$ of the Grassmannian $\mathbb G(1,n)$ of lines in $\mathbb{P}^n$, that is a Fano variety of index 1. We first prove that for $n$ odd $X$ is birational to a hypersurface defined by a pfaffian determinant of order $n+1$ of linear forms in $\mathbb{P}^{n-1}$, and we study this birational transformation in some detail. Then for $n$ even, we prove that $X$ is unirational and birational to a hypersurface defined by a pfaffian determinant of order $n$ of linear forms in $\mathbb{P}^{n-1}$. Moreover we prove that the hypersurface of $\mathbb{P}^n$ described by the lines corresponding to the points in $X$ has degree $n-1$ and is rational, and we find a rational parametrization of it.

Let $f \colon S \rightarrow B$ be a fibred surface over an algebraically closed field $k$ of characteristic $p \geq 5$, where $S$ is a minimal smooth projective surface of general type and $B$ is a smooth projective curve of genus $b\geq 2$. We prove that the group of fibration-preserving automorphisms of $f$ has order at most $15658(K_S^2)^{4}$. Furthermore, we provide an example to show that the exponent $4$ of the polynomial bound is sharp.

We study the quantum $K$-theory of the flag variety $G/B$. For each minuscule fundamental weight $\varpi$, we construct an explicit relation in the torus-equivariant quantum $K$-theory $QK_T(G/B)$. The relation can be regarded as a quantum deformation of the character of the irreducible representation with highest weight $\varpi$.

We interpret the convolution constructions of [arXiv:2604.26357] in Hodge-theoretic terms, extending the Hodge defect calculation of [arXiv:2609.19341]. For Hodge–Tate modules on $\mathbb C^*$ smooth outside $1$, the defect is the action of an element of the Deligne–Terasoma transport algebra expressed through the Drinfeld associator. For convolutions of iterated Kummer extensions, comparison of mixed Hodge structures produces a beta factor and yields the regularized double shuffle relations.

For a vector bundle $E$ on a smooth projective curve $C$, one defines the Quot scheme $Q:=Q_d(E,C)$ parametrizing subsheaves of $E$ having length $d$ torsion quotients. We describe the indecomposable components of the tangent bundle $T_Q$ of $Q$, thus characterizing when $T_Q$ is indecomposable. We also characterize when $T_Q$ is simple; in fact, we completely describe the endomorphism rings of the indecomposable components of $T_Q$. As an application, we determine when two products of such Quot schemes can be isomorphic.

We study the asymptotic density of $\mathbb{F}_q$-point-free linear sections of geometrically irreducible projective varieties over finite fields. We then apply these results to rank-metric codes via determinantal varieties. Our approach recovers the known cases in which the density tends to $0$ or $1$ and determines the limit in the cases where it was previously unknown. To compute these previously unknown limits, we extend the notion of quasireflexivity to higher-dimensional varieties and show that determinantal varieties satisfy this property. This allows us to invoke the Chebotarev density theorem for varieties over finite fields to obtain the desired estimate.

Let $q=3q_0^2$, where $q_0=3^s$ and $s\ge3$. We prove that a smooth, projective, geometrically irreducible curve over $\F_q$ with $q^3+1$ rational points and genus \[ \frac32q_0(q-1)(q+q_0+1) \] is $\F_q$-isomorphic to the Ree curve.

We study $K$-regularity for complex varieties with finite quotient singularities. We prove that for such varieties, $K_1$-regularity is equivalent to smoothness whenever the singular locus is contained in an affine closed subvariety (e.g., affine varieties or those with isolated singularities). In contrast, we construct projective varieties with finite quotient singularities which are $K_m$-regular for every $m$ but are not local complete intersections. Our smoothness criterion is obtained by affirmatively answering Fogarty's 1988 question concerning differential forms on finite quotients.

Given a complex algebraic variety, we relate the product structure on its absolute Hodge cohomology with mixed Hodge formality, a notion of formality in the sense of rational homotopy, that takes into account the mixed Hodge structures present in complex algebraic geometry. Namely, we prove that mixed Hodge formality implies the splitting of absolute Hodge cohomology as a square-zero extension. We define and study certain products, akin to Massey products, which live in the absolute Hodge cohomology and obstruct its splitting. We also relate mixed Hodge formality to the existence of multiplicative Chow-Künneth decompositions.

We consider $V_{g,n}(\{b_i\})$, the Weil-Petersson volumes of the moduli space of Riemann surfaces of genus $g$ with $n$ geodesic boundaries of lengths $b_i$, $(i=1,...,n)$, and various ${ N}{\geq}1$ supersymmetric generalizations of them, using a framework within which they are specializations of a larger family of quantities naturally governed by the integrable KdV hierarchy. An exact boundary-addition identity is derived that directly relates $V_{g,n+1}(\{b_1,b_2,...,b_n,b\})$, for any $b$, to the corresponding $n$-boundary KdV data underlying $V_{g,n}(\{b_1,b_2,...,b_n\})$. For ordinary Weil-Petersson volumes, expansion about the special removable-cone value $b=2\pi i$ reproduces three known results of Do and Norbury, organizing them as the first levels of a complete hierarchy. The higher levels are naturally built from additional KdV data, which we interpret geometrically in intersection theory as a tower of $\kappa$-decorated volumes. Several other recursive relations in the literature are illuminated by the identity, and various new ones are derived. The cases with ${ N}=1,2$ and (small) ${N}=4$ supersymmetry are also covered by the identity, and we exhibit some of the striking special features and results arising in each case. Among many applications presented, we use the identity to uncover the intersection theory descriptions of ${N}=2$ and small ${ N}=4$ Weil-Petersson volumes.

We prove that every germ of an abelian relation of a codimension-one web on a complex projective manifold is defined over a Liouvillian extension of the function field.

For $r> 2$, we study the moduli space parameterising fixed degree morphisms $\mathbb P^r\to X$ where $X$ is a smooth hypersurface of low degree. More precisely, we prove the following result: let $n\geq 2, e\geq 1$ and $X\subset \mathbb P^{n-1}$ a smooth degree $d\geq 2$ hypersurface over an algebraically closed field of characteristic zero or greater than $d$, then $\mathrm{Mor}_e(\mathbb P^r, X)$ is irreducible of the expected dimension if \[ n> 2^d(d-1)\binom{de+r-1}{r-1}. \] Our result extends the result of Browning-Yamagishi for $r=2$ via multiblock Weyl differencing.

A theorem of Kollár, Miyaoka, and Mori states that on a rationally connected variety, any finite collection of points lies on a rational curve. Motivated by this, it is natural to ask what one can say about families of curves passing through many general points of an arbitrary smooth projective variety X. When X has nonnegative Kodaira dimension, we establish a sharp linear lower bound for the genus of such curves in terms of the number of points and the dimension of X, generalizing a theorem of Arapura and Archava. By contrast, the least possible gonality of a connecting family eventually stabilizes as the number of points grows. We characterize its limiting value in terms of varieties dominating X that are generically finite covers of rationally connected varieties. As an illustration, we study these invariants for hypersurfaces of large degree, determining in particular the joint asymptotic behavior of the minimal connecting genus as the number of points and the degree vary. Finally, we briefly consider higher-dimensional connecting subvarieties, proving linear bounds for their canonical volumes and computing asymptotic results for hypersurfaces.

We consider a certain pair of families of Picard rank 1 Calabi–Yau threefolds, that have appeared earlier in mathematical literature in unrelated contexts: the family $\mathcal{X}$ of degree 33 threefolds in $G(2,6)$ (constructed by Miura) and the family $\mathcal{Y}$ of arithmetically Gorenstein degree 21 threefolds in $\mathbb{P}^8$ (constructed by Schenck–Stillman–Yuan). After establishing a natural geometric correspondence between their general members, we go on to show that any pair of corresponding threefolds in these families satisfy certain classical dualities. We moreover discover that their geometries may be related by a mathematical gauged linear sigma model, using which we prove that they are derived equivalent. This settles a conjecture of Miura, who predicted the existence of non-trivial Fourier–Mukai partners to members of $\mathcal{X}$, based on a study of its mirror moduli. This conjecture was also formulated later by Gerhardus–Jockers, in a physical context. In fact, starting from the other family $\mathcal{Y}$, we conjecturally arrive at the same mirror moduli. Thus, such a pair is a new, and quite possibly the last, addition to the small list of deformation families of non-birational, double-mirror Calabi–Yau threefolds having Picard rank 1.

We develop a complex-analytic approach to the model theory of the (real) unrestricted exponential. As a consequence we derive sharp forms of many of the foundational results for the structure ${\mathbb R}^\text{RE}_{\exp}$ (and more general structures). In particular we establish sharp o-minimality, a sharp form of Wilkie's theorem of the complement, a sharp form of Wilkie's conjecture and a sharp form of piecewise definability by terms. Our approach is based on a complexification of the LE-preparation theorem of Lion–Rolin. We also develop a parallel complex theory for ${\mathbb R}_\text{an,exp}$, proving for example that the rational points of height $H$ on a nowhere-dense definable set can be interpolated by an algebraic hypersurface of degree $\text{poly}(\log H)$. This generalizes a theorem of Cluckers–Pila–Wilkie who proved the same statement for ${\mathbb R}_\text{an}^\text{pow}$.

Let $K$ be a Henselian discretely valued field with excellent ring of integers $\mathcal{O}_K$ and algebraically closed residue field $k$. Let $X\to\mathbb{P}^1_K$ be a cyclic cover of degree $n$ prime to the characteristic of $k$. In joint work with Obus and Srinivasan (arXiv:2609.20553), we define an integer called the conductor-discriminant contribution $\text{cdc}(y)$ associated to a multiplicity $2$ point $y$ of the branch divisor of the normalization in $K(X)$ of a regular $\mathcal{O}_K$-model $\mathcal{Y}$ of $\mathbb{P}^1_{K}$; modulo several key results about $\text{cdc}(y)$, we prove a conductor-discriminant inequality for $X$, extending previous work of Ogg, Saito, Liu, Srinivasan, and Obus$-$Srinivasan. In this companion paper, we supply the necessary technical results for $\text{cdc}(y)$. In particular, we show $\text{cdc}(y)$ is non-negative except under highly restrictive conditions on $n$ and the structure of the branch divisor at $y$. Moreover, if $\text{cdc}(y)$ is negative, we show the spectrum of the complete local ring of any point lying over $y$ under the normalization of $\mathcal{Y}$ in $K(X)$ is a rational double point. Along the way, we show the non-negativity of a related quantity, the conductor exponent-discriminant contribution $\text{cedc}(y)$, which is used in our joint work with Obus and Srinivasan to give a new proof of a result of Kohls.