Papers

We study the growth of the bidegree of an ample irreducible curve in $\mathbb{P}^1 \times \mathbb{P}^1$ under a product polynomial endomorphism $\varphi=(f,g)$, where at least one of $f$ and $g$ is non-exceptional. We prove that, if the curve $C$ is not preperiodic under $(f^a,g^b)$ for any $a,b\geq 1$, then the bidegree of $\varphi^n(C)$ is asymptotic to $(\text{deg}(g)^n,\text{deg}(f)^n)$. As an application of this exponential growth, we prove a geometric analogue of Silverman's theorem on the finiteness of $S$-integral points in orbits. Namely if $C$ is not $(f^a,g^b)$-preperiodic and $C'$ is not totally invariant for $\varphi$, then for any infinite sequence ${n_i}$ of positive integers, the union of the intersections $$ \bigcup_{i \geq 1} \left( \varphi^{n_i}(C)\cap C' \right) $$ is Zariski dense in $C'$.

Let $p$ be a prime. We prove that every compatible branch datum of degree $p$ over the sphere is realizable by a connected branched cover. The three-point case is constructed in residue characteristic $p$. Henrio's moment theorem supplies the distinct-point moment solutions from which we construct a special primitive tail for each prescribed partition; a second application underlies the new tail required by a positive source genus. These tails are joined by a logarithmic deformation datum and embedded in one subgroup of $S_p$ containing a common regular subgroup of order $p$. Wewers's lifting theorem produces a three-point Galois cover in characteristic zero. The quotient by a point stabilizer has degree $p$ and the prescribed three ramification profiles. The fusion and realization results of Edmonds–Kulkarni–Stong then give the assertion for an arbitrary number of branch values. As consequences, the connected prime-degree Hurwitz potential has full support on the Riemann–Hurwitz locus, every corresponding connected relative Gromov–Witten invariant of $\mathbf P^1$ is nonzero, the two-relative-point disconnected sector with even completed-cycle orders at most $p$ is strictly positive subject to the dimension constraint, and the connected transposition sector is strictly positive.

We prove the conjecture of Huang, Jiang, and Oblomkov (HJO) giving a geometric extension of the Rogers–Ramanujan and Andrews–Gordon identities for every torus-knot singularity $X^a=Y^b$ with coprime $1<a<b.$ For a prime power $q$, let $\mathcal{NC}_n^{a,b}(\mathbb F_q)$ denote the set of pairs of commuting nilpotent $n\times n$ matrices $(A,B)$ over $\mathbb F_q$ satisfying $A^a=B^b$. We establish the threefold equality between their normalized counts, the HJO $q$-series $Z_{a,b}$, and the explicit infinite product $P_{a,b}$: \[ \underbrace{\vphantom{\Bigg|} \prod_{m\geq1}(1-q^{-m}) \Biggl(\sum_{n=0}^{\infty} \frac{\lvert\mathcal{NC}_n^{a,b}(\mathbb F_q)\rvert} {\lvert\operatorname{GL}_n(\mathbb F_q)\rvert}\Biggr) }_{\text{point count}} = \underbrace{\vphantom{\Bigg|}Z_{a,b}(q^{-1}) }_{\text{\(q\)-series}} = \underbrace{\vphantom{\Bigg|}P_{a,b}(q^{-1}) }_{\text{infinite product}}. \] Our main result is a stronger finite identity: the rank $N$ HJO sum equals $(q;q)_N$ times the generating function for balanced cylindric partitions with entries bounded by $N$. Taking $N\to\infty$ yields the HJO conjecture. The proof combines the compositional rational shuffle theorem of Bergeron–Garsia–Leven–Xin and Mellit with a multiplicativity theorem for slope operators and a determinantal model for bounded cylindric partitions, linked by a common $q$-difference equation. The finite identity and the HJO conjecture have been formalized unconditionally in Lean by AxiomProver.

We prove conductor-discriminant inequalities for all $\mathbb{Z}/n$-covers of $\mathbb{P}^1$ defined over discretely valued fields $K$ with excellent valuation ring $\mathcal{O}_K$ and perfect residue field of characteristic not dividing $n$, modulo some calculations appearing in work of the third author (arXiv:2609.20585). Specifically, when such a curve $X$ is given by $y^n = f(x)$ with $f(x) \in\mathcal{O}_K[x]$ and $n\mid\text{deg}(f)$, and if $\mathcal{X}$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $\mathcal{X}$ is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$. This is a direct generalization of previous work of the first two authors on hyperelliptic curves, which in turn generalized work of Ogg, Saito, Liu, and the second author. When $f$ is monic, this strengthens a result of Kohls stating that the conductor exponent of the Jacobian of such a curve is bounded above by $(n-1)v_K(\text{disc}(\text{rad}(f)))$.

We show that the ACC conjecture for minimal log discrepancies holds for both bounded generalized sub-pairs in arbitrary dimension and bounded threefolds with arbitrary boundaries.

We classify the automorphism groups of smooth cubic threefolds and fourfolds. We also show that there are $156$ (respectively, $40$) connected families of smooth cubic fourfolds (respectively, threefolds) with specified automorphism group action. For each family, the group action and defining equations are also given. The classification combines both representation theory (based on GAP) and lattice theory (based on SageMath).

We prove a generalization of the version, due to A. Gabrielov, of the Łojasiewicz inequality for not necessarily restricted semi-Pfaffian sets. Unlike the most variants of the Łojasiewicz inequality, it describes the rate of growth of a Pfaffian function on a semi-Pfaffian set $X$ not only in a neighbourhood of its zero set in the domain of $X$ but also in a neighbourhood of zeroes on the boundary of the domain. Gabrielov's version is essentially one-dimensional, we generalize it to a multidimensional case.

This is a survey on Pham-Brieskorn varieties and their various properties, targeted for graduate students.

We study logarithmic Poisson and de Rham cohomologies associated with a class of inhomogeneous free divisors in the affine plane. These divisors arise as inhomogeneous deformations of reduced normal crossing divisors and admit explicit bases for their modules of logarithmic vector fields in the sense of Saito. Using these bases, we construct the corresponding logarithmic Poisson structures and describe explicitly the induced Koszul bracket on logarithmic differential 1-forms within the framework of Lie-Rinehart algebras. We then determine the logarithmic Poisson cochain complex and compute its cohomology for the class under consideration. Furthermore, by means of the logarithmic Spencer complex, we identify the corresponding logarithmic de Rham complex and compute its cohomology. These computations provide explicit cohomological invariants for the class of inhomogeneous divisors considered and show how logarithmic Poisson and de Rham theories extend the corresponding constructions for reduced normal crossing divisors. In addition, we establish an explicit comparison between the logarithmic cohomological theories: they are naturally isomorphic in any degree $k \neq 1$, whereas in degree 1 the logarithmic de Rham cohomology group is a split one-dimensional extension of the logarithmic Poisson cohomology group.

We prove that a generically trivial torsor under a reductive group scheme over an unramified regular local ring is trivial.

Let $X$ be a smooth complex affine variety of dimension $n$, and let $F=(f_1,\ldots,f_r)$ be a tuple of nonzero regular functions on $X$ such that $f:=\prod_{i=1}^r f_i$ is not invertible. We study the zero loci of the Bernstein-Sato ideals $B_F^{\mathbf a}$ for nonnegative integral shifts $\mathbf a$. For a fixed log resolution, every codimension-one irreducible component of $Z(B_F^{\mathbf a})$ is a hyperplane of the form $L_E(\mathbf s)+k_E+c=0$ with $c$ a positive integer. We give a new proof of this result using localized maximal and minimal extensions of relative D-modules. We also prove that $c\leq L_E(\mathbf a)+(n-1-\delta_f)L_E(\mathbf 1)-k_E$, where $\delta_f=\min\{n-1,\alpha_f\}$ and $\alpha_f$ is the minimal exponent of $f$. The problem of obtaining such an upper bound for arbitrary tuples (in particular, for $r>1$) was raised by Budur, van der Veer, and Van Werde, and the above inequality resolves it. To obtain the upper bound, we compare the diagonal slice of $Z(B_F^{\mathbf 1})$ with the root set of $b_f$. A finite covering by translates, combined with diagonal specialization and the log-resolution description, shows that these sets have the same least and greatest points. Saito's root estimate at their common least point then yields the upper bound. We further establish a divisor-valued formulation of the local index comparison, recovering the detection of monodromy support via monodromy zeta functions and the multivariable A'Campo formula.

Let $f_m(x_0,\ldots,x_{2m-2}) =\det(x_{i+j})_{0\leq i,j\leq m-1}$ be the determinant of the generic symmetric Hankel matrix of size $m$. We determine its global Bernstein–Sato polynomial. The proof combines the nearby cycles of the corresponding secant hypersurface with an explicit Archimedean zeta integral over the cone of positive definite Hankel matrices. A Gauss–Radau parametrization reduces this integral to a Morris integral and hence to a quotient of Gamma functions. Its primitive poles are detected by test functions supported arbitrarily close to the vertex of the cone. Lichtin's theorem, applied to a secant-stratum log resolution, fixes the integer shift between these poles and the roots of the Bernstein–Sato polynomial. A relative $D[s]$-lattice realization of nearby cycles, together with the simplicity of the corresponding intersection complexes, gives uniqueness and multiplicity one in every monodromy class. The induction in the matrix size follows from the local product structure of secant varieties.

We prove that the integral Chow groups $CH^i$ and mixed Hodge degree $H^{2i, (i, i)}$ cohomology groups of really full rank (RFR) sink-recurrent cluster varieties vanish for $i > 0$. In particular this applies to braid varieties and open Richardson varieties in any Lie type. Our main tool is the construction of a stratification of any RFR sink-recurrent cluster variety $\mathcal{A}(\Sigma)$ into (affine spaces times) RFR sink-recurrent cluster varieties of seeds with fewer mutable vertices than $\Sigma$. We employ the theory of Voevodsky motives, and towards this end we prove that the cycle class maps are isomorphisms onto the lowest-weight part of rational Borel-Moore homology for any mixed Tate variety over a number field. We then show that RFR sink-recurrent cluster varieties have mixed Tate and, in fact, split motives. Finally, we use our results to deduce vanishing theorems about the Khovanov-Rozansky homology groups of closures of positive braids and generation properties of the cohomology of closed Richardson, projected Richardson, and brick varieties.

Perverse schobers are a categorification of perverse sheaves, originally proposed by Kapranov and Schechtman. The purpose of this paper is to initiate a microlocal study of perverse schobers. We first categorify $\operatorname{Perv}(\mathbb{C},R)/\operatorname{Loc}(\mathbb{C})$, the category of perverse sheaves on a complex line with singular points at $R$, modulo local systems. We use this to propose a general definition for microlocal perverse schobers supported on the open conormal to a germ of a hypersurface, and we conjecture that this is invariant under Radon transform. Here we make the key observation that while the analogous quotient of perverse schobers $\mathsf{2Perv}(\mathbb{C},R)/\mathsf{2Loc}(\mathbb{C})$ is a reasonable categorification, the resulting theory fails to be invariant under Radon transform. In fact, our proposed categorification can be recovered by correcting an instance of this failure in a universal manner. We prove Radon invariance when our hypersurface is the curve $y^m=x^n$ in $\mathbb{C}^2_{x,y}$. Along the way, we explain how our theory relates to Fourier transforms of perverse schobers, periodic SODs, and spherical monads.

Conifold boundary conditions constrain higher genus topological string free energies, but the cancellation of subleading poles is not implied by their leading singular behavior. We prove a regularity theorem for connected elliptic completed cycle traces and use it to establish the Nekrasov-Shatashvili conifold gap for the maximal contact relative Gromov-Witten theory of the projective plane with a smooth cubic boundary and a top Hodge class insertion. The higher genus local/relative correspondence expresses the ordinary potential in terms of the relative potential and differentiated lower genus contributions. We identify its conifold polarized correction with an elliptic trace. A central difference identity resums each non-vacuum partition weight into a content product multiplied by a regular exponential. For arbitrary regular parts of the input potentials and any nome with a simple zero, the normalized non-vacuum connected contribution is divisible by the conifold coordinate and by the square of the coupling. The entire polar part and finite part are consequently determined by the vacuum constants. A triangular induction transfers the known ordinary local projective plane gap to the relative theory. We also obtain finite part identities, finite formulas for regular Taylor coefficients, and reconstruction of the holomorphic ambiguities. The argument is formal in the coupling and gives holomorphic remainders at every fixed genus.

This paper continues the study of the $K$-theory of monoid schemes, using it to give a useful definition of the higher $K$-theory of a matroid via its Bergman fan.

Let $V$ be a complex symplectic vector space and let $\mathbb{P}(V)$ carry the induced contact structure and symplectic polarity. We study the loci of hypersurfaces containing codimension-two cones supported on hyperplanes. The starting point is the following characterization: a projective subvariety $X\subset\mathbb{P}(V)$ which is a hypersurface of degree at least two in a hyperplane $H$ is involutive if and only if it is a cone whose vertex contains the polar point $\sigma(H)$. We give a short proof, construct the corresponding incidence spaces, compute their dimensions and express the incidence degrees as Segre-class integrals. In $\mathbb{P}^3$ the incidence map is birational for every $d\ge m\ge 2$ except $(m,d)=(2,2)$, and the same holds in $\mathbb{P}^5$; the quadratic case is exceptional in every dimension, the incidence having generic degree $2n$ in $\mathbb{P}^{2n-1}$. We also give a sufficient criterion for birationality in higher dimension, closed formulas in $\mathbb{P}^3$ and coefficient formulas in $\mathbb{P}^5$. For quadric cones in $\mathbb{P}^5$ the degree is the product of shifted binomial factors and an irreducible polynomial of degree 36, and we explain where the shifted factors come from.

We define a quadratically enriched pushforward for proper morphisms between smooth, proper Deligne–Mumford stacks over an arbitrary field using Grothendieck–Serre duality. We apply this pushforward to define quadratically enriched Euler classes and Euler numbers of relatively oriented vector bundles on smooth, proper Deligne–Mumford stacks. Finally, we define quadratically enriched fundamental classes for smooth, proper schemes over an arbitrary field, and we use these to propose a definition of quadratically enriched quantum $K$-invariants when the moduli space of stable maps is smooth, tame, and relatively oriented.

In projective space $\mathbb{P}^r$, the normal bundle of a general rational curve of any degree $d \geq r$ is balanced except over fields of characteristic 2. In arXiv:2404.08102, Coskun, Larson, and Vogt found counterexamples to the naive generalization of this statement to general rational curves in Grassmannians and proposed a conjecture for when the normal bundle is balanced. In this paper, we prove their conjecture for the Grassmannians $G(2, 4)$, $G(2, 5)$, and $G(2, 6)$.

We compare two mixed Hodge structures attached to a unipotent variation $\mathbb V$ of mixed Hodge–Tate structures on $X=\mathbb P^1\setminus{0,1,\infty}$: the cohomology of its extension by $*$ at $0$ and by $!$ at $\infty$, and the vanishing cycles at $1$. The semi-holonomy isomorphism constructed in [arXiv:2604.26357] identifies these two structures on their underlying rational structures, compatibly with their weight filtrations, but need not respect their Hodge filtrations. We compute the resulting Hodge defect. Its universal noncommutative series is given by a specific component of the inverse Drinfeld associator multiplied by an explicit local monodromy factor. We conclude with a formal $A_\infty$ interpretation of the coefficients of the Hodge defect in terms of weight-framed Tate extensions.

Let $X$ be a normal variety in arbitrary characteristic, and let $\tilde X$ be the successive blowup along subvarieties $Z_1,\dotsc,Z_n\subseteq X$ of codimension at least two that have simple normal crossings and satisfy $Z_h\not\supseteq Z_i$ whenever $h<i$. We prove that the relative cone of curves $\overline{\operatorname{NE}}(\tilde X/X)$ is generated by the classes of finitely many elementary curves, and that every face admits a contraction over $X$. Moreover, we prove that every small face contraction admits a $D$-flip for every $\mathbb R$-Cartier divisor $D$ that is relatively anti-ample. We also compute the relative Cox sheaf and prove its finite generation.

We prove the André–Oort conjecture for hypersurfaces $V \subset Y(1)^n \cong \mathbb{A}^n_\mathbb{C}$ defined by an equation $a_1 x_1^m + \ldots + a_n x_n^m = b$, where $a_1, \ldots, a_n, b \in \overline{\mathbb{Q}}$ and $m \in \mathbb{Z}_{>0}$. Unlike previous proofs, our result is both effective and uniform in the height of the coefficients $a_1, \ldots, a_n, b$. This is the first effective proof of a uniform André–Oort statement for a class of subvarieties with arbitrary dimension and non-empty special locus. We also prove an analogous result for hypersurfaces $V \subset Y(1)^n \times \mathbb{G}_m^l$.

Let $k=\overline{\mathbb{F}_2}$. We construct a nine-dimensional $F$-rational affine variety $X$ admitting a locally closed embedding $X\hookrightarrow \mathbb{P}_k^{19}$ together with a dense open set $U\subseteq (\mathbb{P}^{19})^\vee$ such that $X\cap H$ is not $F$-rational (or even $F$-injective) for all $H\in U$. We also construct a nine-dimensional projective $F$-rational variety $\mathfrak{X}$ and a closed embedding $\mathfrak{X}\hookrightarrow\mathbb{P}^{29}_k$ under which the same conclusion holds.

Browning and Sawin conjectured that random hypersurfaces with sign coefficients are smooth with probability tending to one as the degree grows. We prove this conjecture and obtain a quantitative bound. For each $n\geq1$, a degree $d$ form in $n+1$ variables, with independent uniform coefficients in $\{-1,1\}$, defines a singular complex hypersurface with probability $O_n(d^{-1/2})$. The positive-dimensional singular loci occur with exponentially small probability. For $n\geq3$, the same exponential bound holds for failure of absolute irreducibility. These results have been formalized in Lean by AxiomProver assuming existing literature.

We study the geography of surfaces of general type with extremal cotangent dimension, in other words surfaces which have either no global holomorphic symmetric differentials at all or the maximal asymptotic growth of their number, i.e. big cotangent bundle. We are mainly interested in surfaces with low slope K^2 /$\chi$, which, as far as maximal cotangent dimension is concerned, were out of reach of previous methods. We prove vanishing theorems for symmetric logarithmic differentials on minimal rational surfaces and for differentials on their double covers and extend a bigness criterion of Sakai to fibrations of general type in the sense of Campana. As a consequence, we prove, on the one hand that generic Horikawa surfaces have no nontrivial symmetric differentials and on the other hand that there exist Horikawa surfaces with big cotangent bundle.

Applying the BBDJS minimal model to the derived symplectification of a $-1$-shifted contact derived Artin stack and descending algebraically along the structural free $\mathbb{G}_m$-action, we construct an $\ell$-adic perverse sheaf on any oriented such stack, and use Verdier's specialization equivalence for monodromic sheaves to equip it with a tame twisted monodromy operator $\theta$. We show that the local fundamental class of a Legendrian $L$ satisfies $\theta \circ \mu_L = (-1)^{\mathrm{vdim} L}\mu_L$, so that on Legendrians of odd virtual dimension, the class vanishes due to $\theta$-invariance. Moreover, both parities occur already on the $A_1$ chart while an odd Legendrian can carry a nonzero local class. We further formulate a contact analogue of Joyce's conjecture for a graded orientation, in which the orientation datum is twisted by the parity of the virtual dimension. Under the assumption of a monodromic refinement of the symplectic conjecture, we construct the categorified Legendrian 2-categories $\mathfrak{L}\mathcal{F}_c(X)$ and $LLeg_0$ via $\ell$-adic pull-push functors. Finally, we show that the contact Behrend function is identically $1$, so that the associated Donaldson-Thomas invariant is the compactly supported étale Euler characteristic of the classical truncation, and that the higher traces of $\theta$ recover the singularity type that the first trace discards. As an application, we show that the symplectic invariant of a derived intersection of conic Lagrangians in a cotangent bundle vanishes identically, while the contact invariant computes the Euler characteristic of the projectivized intersection, with an explicit formula for conormal bundles.

Let K be an algebraically closed field of characteristic zero. Many of the geometric properties of the plane curves in K^2 with one place at infinity can be obtained from the arithmetic of numerical semigroups associated with these curves. In this survey, we recall the main properties of these semigroups as well as the geometric results obtained from these properties. We also address some problems.

Let $(G,X)$ be a Shimura datum of abelian type satisfying the hypotheses of the main theorem, and suppose that the associated Shimura varieties have hyperspecial good reduction at $p$. Their special fibres are stratified by the $\sigma$-conjugacy classes $b\in B(G_{\mathbb{Q}_p},\mu_h^{-1})$. For an individual Newton stratum, this paper constructs a stabilized formula for the alternating traces of Frobenius–Hecke correspondences on its compactly supported $\ell$-adic cohomology, with coefficients in an $\ell$-adic local system. The formula, obtained by modifying the Langlands–Kottwitz method, expresses these Lefschetz numbers as elliptic stable geometric distributions on endoscopic groups of $G$, placing the cohomological traces in a form suitable for comparison with automorphic spectral data. We also give a group-theoretic criterion determining which elliptic endoscopic groups can contribute to a given Newton stratum. For certain unitary Shimura varieties, we exhibit an intermediate Newton stratum with no non-trivial endoscopic contribution, although there are non-trivial endoscopic contributions in the cohomology of the ambient Shimura variety.

We explore pedal curves (a special case of a geometric locus) of a the plane curve called a bicorn. This curve has two cusps, which induce the existence of singularities of the pedal. We study the singularities of the pedals in each case. For this various presentations are used for the curves, parametric and symbolic presentations. Their properties and influence on the quality of the output is briefly discussed. In particular, the determination of points of self?intersection requires a parametric presentation; in this case a rational parametrization is the most efficient. Work is performed in an environment involving Dynamic Geometry, automated commands and a Computer Algebra System. Finally we describe a connection between our plane curves and some art, making the topic here suitable for STEAM education.

Let \pi: \mathcal X \rightarrow \Delta be a smooth proper family of compact complex manifolds such that the central fiber \mathcal X_0 is Kähler. Then all the fibers close to 0 are Kähler and {\pi} is a weakly Kähler morphism, even if \mathcal X is not a Kähler space. In this paper we show that if dim \mathcal X_0 = 4 and K_{\mathcal X_0} is nef, then K_{\mathcal X_t} is nef for all t in a neighborhood of the origin.