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Pyknotic objects, I. Basic notionsApr 22 2019Pyknotic objects in are sheaves on the site of compacta. These provide a convenient way to do algebra and homotopy theory with additional topological information present. This appears, for example, when trying to contemplate the derived category of a ... More
K-theoretic crystals for set-valued tableaux of rectangular shapesApr 21 2019In earlier work with C. Monical (2018), we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of semistandard ... More
Triangular curves and cyclotomic Zariski tuplesApr 19 2019The purpose of this paper is to exhibit infinite families of conjugate projective curves in a number field whose complement have the same abelian fundamental group, but are non-homeomorphic. In particular, for any $d>3$ we find Zariski tuples parametrized ... More
The Monsky--Washnitzer toposApr 19 2019Mimicking Ogus's construction, we define a site, for varieties over a field of char. p > 0, using Monsky--Washnitzer algebras/weak formal schemes. We prove a comparison theorem between the MW cohomology and a certain analytic cohomology.
Isotropic motivesApr 19 2019In this article we introduce the local versions of the Voevodsky category of motives with Z/p-coefficients over a field k, parameterized by finitely-generated extensions of k. We introduce the, so-called, flexible fields, passage to which is conservative ... More
A survey on maximal green sequencesApr 19 2019Maximal green sequences appear in the study of Fomin-Zelevinsky's cluster algebras. They are useful for computing refined Donaldson-Thomas invariants, constructing twist automorphisms and proving the existence of theta bases and generic bases. We survey ... More
Triangular decomposition of character varietiesApr 18 2019We introduce an original definition of character varieties associated to a punctured compact oriented surface with at least one puncture per boundary component. These character varieties are affine Poisson varieties which coincide with the Culler-Shalen ... More
Rationally elliptic toric varietiesApr 18 2019We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.
On singular moduli that are S-unitsApr 18 2019Recently Yu. Bilu, P. Habegger and L. K\"uhne proved that no singular modulus can be a unit in the ring of algebraic integers. In this paper we study for which sets S of prime numbers there is no singular modulus that is an S-units. Here we prove that ... More
Cohomology of the moduli space of cubic threefolds and its smooth modelsApr 18 2019We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily-Borel and toroidal compactifications of the ... More
Weighted Lifted Codes: Local Correctabilities and Application to Robust Private Information RetrievalApr 18 2019Low degree Reed-Muller codes are known to satisfy local decoding properties which find applications in private information retrieval (PIR) protocols, for instance. However, their practical instantiation encounters a first barrier due to their poor information ... More
A proof of the Hasse-Weil inequality for genus 2 à la ManinApr 18 2019We prove the Hasse-Weil inequality for genus 2 curves given by an equation of the form y^2 = f(x) with f a polynomial of degree 5, using arguments that mimic the elementary proof of the genus 1 case obtained by Yu. I. Manin in 1956.
Relative $\mathbb{A}^1$-homology and its applicationsApr 18 2019In this paper, we prove an $\mathbb{A}^1$-homology version of the Whitehead theorem with dimension bound. We also prove an excision theorem for $\mathbb{A}^1$-homology, Suslin homology, and $\mathbb{A}^1$-homotopy sheaves. In order to prove these results, ... More
The perfect cone compactification of quotients of type IV domainsApr 18 2019The perfect cone compactification is a toroidal compactification which can be defined for locally symmetric varieties. We show that the perfect cone compactification of quotients of type IV domains by the action of the stable orthogonal group has canonical ... More
Conjugacy of Levi subgroups of reductive groups and a generalization to linear algebraic groupsApr 18 2019We investigate Levi subgroups of a connected reductive algebraic group G, over a ground field K. We parametrize their conjugacy classes in terms of sets of simple roots and we prove that two Levi K-subgroups of G are rationally conjugate if and only if ... More
Coincidence of two Swan conductors of abelian charactersApr 18 2019There are two ways to define the Swan conductor of an abelian character of the absolute Galois group of a complete discrete valuation field. We prove that these two Swan conductors coincide.
Monotonic invariants under blowupsApr 18 2019We prove that the numerical invariant $3\mu-4\tau$ increases when we perform a blowup for a reduced irreducible plane curve singularity. This provides a new perspective to understand the question posed by A. Dimca and G.-M. Greuel. Moreover, our work ... More
On the space of theta functions for a prime levelApr 18 2019Hecke expected that an explicit set of theta series obtained from maximal orders of the definite quaternion algebra over Q which is ramified at a prime N will be a basis of the space of holomorphic modular forms of weight 2 and level N. However, later ... More
The chow cohomology of affine toric varietiesApr 17 2019We study the Fulton-Macpherson Chow cohomology of affine toric varieties. In particular, we prove that the Chow cohomology vanishes in positive degree. We prove an analogous result for the operational $K$-theory defined by Anderson and Payne.
Moderately ramified actions in positive characteristicApr 17 2019In characteristic two and dimension two, wild Z/2Z-actions on k[[u,v]] ramified precisely at the origin were classified by Artin, who showed in particular that they induce hypersurface singularities. We introduce in this article a new class of wild quotient ... More
Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flowApr 17 2019We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an $n$-dimensional projective manifold $X$ with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded for all time ... More
A sufficient condition for a toric weak Fano 4-fold to be deformed to a Fano manifoldApr 17 2019In this paper, we introduce the notion of toric special weak Fano manifolds, which have only special primitive crepant contractions. We study the structure of them, and in particular completely classify smooth toric special weak Fano 4-folds. As a result, ... More
Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genusApr 17 2019We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that ... More
On the K-theoretic fundamental classes of Deligne-Lusztig varietiesApr 17 2019In this paper we express the class of the structure sheaves of the closures of Deligne--Lusztig varieties as explicit double Grothendieck polynomials in the first Chern classes of appropriate line bundles on the ambient flag variety. This is achieved ... More
Autonomous first order differential equationsApr 17 2019The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on differentially ... More
A simple character formulaApr 17 2019In this paper we prove a character formula expressing the classes of simple representations in the principal block of a simply-connected semisimple algebraic group G in terms of baby Verma modules, under the assumption that the characteristic of the base ... More
The Integral Chow Ring of $\bar{M}_2$Apr 17 2019In this paper we compute the Chow ring of the moduli stack $\bar{M}_2$ of stable curves of genus 2 with integral coefficients.
Moduli of $\ell$-adic pro-étale local systems for smooth non-proper schemesApr 16 2019Let $X$ be a smooth scheme over an algebraically closed field. When $X$ is proper, it was proved in \cite{me1} that the moduli of $\ell$-adic continuous representations of $\pi_1^\et(X)$, $\LocSys(X)$, is representable by a (derived) $\Ql$-analytic space. ... More
Algebraic stability of meromorphic maps descended from Thurston's pullback mapsApr 16 2019Let $\phi:S^2 \to S^2$ be a post-critically finite branched overing. Under certain conditions, by work of Koch, $\phi$ induces a meromorphic self-map $R_{\phi}$ on the moduli space $\mathcal{M}_{0,n}$; $R_{\phi}$ descends from Thurston's pullback map ... More
Cluster Structures on Double Bott-Samelson CellsApr 16 2019We introduce double Bott-Samelson cells defined by a symmetrizable generalized Cartan matrix $C$ and a pair of positive braids $(b,d)$ in the braid group associated to $C$. We prove that the coordinate rings of double Bott-Samelson cells are upper cluster ... More
Graded Quivers, Generalized Dimer Models and Toric GeometryApr 16 2019The open string sector of the topological B-model model on CY $(m+2)$-folds is described by $m$-graded quivers with superpotentials. This correspondence extends to general $m$ the well known connection between CY $(m+2)$-folds and gauge theories on the ... More
On isolation of singular zeros of multivariate analytic systemsApr 16 2019We give a separation bound for an isolated multiple root $x$ of a square multivariate analytic system $f$ satisfying that the deflation process applied on $f$ and $x$ terminates after only one iteration. When $x$ is only given approximately, we give a ... More
Computing the Lie algebra of the differential Galois group: the reducible caseApr 16 2019In this paper, we explain how to compute the Lie algebra of the differential Galois group of a reducible linear differential system. We achieve this by showing how to transform a block-triangular linear differential system into a Kolchin-Kovacic reduced ... More
The de Rham functor for logarithmic D-modulesApr 16 2019In the first part we deepen the six-functor theory of (holonomic) logarithmic D-modules, in particular with respect to duality and pushforward along projective morphisms. Then, inspired by work of Ogus, we define a logarithmic analogue of the de Rham ... More
The de Rham functor for logarithmic D-modulesApr 16 2019Apr 18 2019In the first part we deepen the six-functor theory of (holonomic) logarithmic D-modules, in particular with respect to duality and pushforward along projective morphisms. Then, inspired by work of Ogus, we define a logarithmic analogue of the de Rham ... More
The 0-th Fitting ideal of the Jacobian module of a plane curveApr 16 2019We describe the 0-th Fitting ideal of the Jacobian module of a plane curve in terms of determinants involving the Jacobian syzygies of this curve. This leads to new characterizations of maximal Tjurina curves, that is of non free plane curves, whose global ... More
A transformation rule for natural multiplicitiesApr 16 2019Apr 17 2019For multiplicities arising from a family of ideals we provide a general approach to transformation rules for a ring extension \'etale in codimension one. Our result can be applied to bound the size of the local \'etale fundamental group of a singularity ... More
A transformation rule for natural multiplicitiesApr 16 2019For multiplicities arising from a family of ideals we provide a general approach to transformation rules for a ring extension \'etale in codimension one. Our result can be applied to bound the size of the local \'etale fundamental group of a singularity ... More
p-adic equidistribution of CM pointsApr 16 2019Let $X$ be a modular curve and consider a sequence of Galois orbits of CM points in $X$, whose $p$-conductors tend to infinity. Its equidistribution properties in $X({\bf C})$ and in the reductions of $X$ modulo primes different from $p$ are well understood. ... More
The stable category of Gorenstein flat sheaves on a noetherian schemeApr 16 2019For a semi-separated noetherian scheme, we show that the category of cotorsion Gorenstein flat quasi-coherent sheaves is Frobenius and a natural non-affine analogue of the category of Gorenstein projective modules over a noetherian ring. We show that ... More
Depth functions of powers of homogeneous idealsApr 16 2019We settle a conjecture of Herzog and Hibi, which states that the function depth $S/Q^n$, $n \ge 1$, where $Q$ is a homogeneous ideal in a polynomial ring $S$, can be any convergent numerical function. We also give a positive answer to a long-standing ... More
Gevrey expansions of hypergeometric integrals IIApr 16 2019We study integral representations of the Gevrey series solutions of irregular hypergeometric systems under certain assumptions. We prove that, for such systems, any Gevrey series solution, along a coordinate hyperplane of its singular support, is the ... More
Images of analytic map germs and singular fibrationsApr 16 2019For a map germ $G$ with target $(\bC^{p}, 0)$ or $(\bR^{p}, 0)$ with $p\ge 2$, we focus on two phenomena which do not occur when $p=1$: the image of $G$ may be not well-defined as a set germ, and a local fibration near the origin may not exist. We show ... More
Uniform matrix product states from an algebraic geometer's point of viewApr 16 2019We apply methods from algebraic geometry to study uniform matrix product states. Our main results concern the topology of the locus of tensors expressed as uMPS, their defining equations and identifiability. By an interplay of theorems from algebra, geometry ... More
An action of the Polishchuk differential operator via punctured surfacesApr 16 2019For a family of Jacobians of smooth pointed curves there is a notion of tautological algebra. There is an action of $\mathfrak{sl}_2$ on this algebra. We define and study a lifting of the Polishchuk operator, corresponding to $f\in \mathfrak{sl}_2$, on ... More
Bounded volume denominators and bounded negativityApr 16 2019In this paper we study the question of whether on smooth projective surfaces the denominators in the volumes of big line bundles are bounded. In particular we investigate how this condition is related to bounded negativity (i.e., the boundedness of self-intersections ... More
Free Proalgebraic GroupsApr 16 2019Replacing finite groups by linear algebraic groups, we study an algebraic-geometric counterpart of the theory of free profinite groups. In particular, we introduce free proalgebraic groups and characterize them in terms of embedding problems. The main ... More
Deformation Theory and Partition Lie AlgebrasApr 15 2019A theorem of Lurie and Pridham establishes a correspondence between formal moduli problems and differential graded Lie algebras in characteristic zero, thereby formalising a well-known principle in deformation theory. We introduce a variant of differential ... More
Uniform bound for the number of rational points on a pencil of curvesApr 15 2019Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number ... More
On the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ of relatively high degreeApr 15 2019We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r$. We let $\mathcal{H}^\mathcal{L}_{d,g,r}$ ... More
Nombre de composantes connexes d'une variété réelle et R-placesApr 15 2019The purpose of this paper is to present results and open problems related to R-places. The first section recalls basic facts, the second introduces R-places and their relationship with orderings and valuations. The third part involves Real Algebraic Geometry ... More
Le théorème de réduction stable de Deligne et MumfordApr 15 2019The stable reduction theorem of Deligne and Mumford --- The moduli space of smooth projective curves of genus $g$ is a quasi-projective algebraic variety, but is not projective. To understand its geometry, it may be crucial to consider compactifications ... More
Derived category of Finite Spaces and Grothendieck DualityApr 15 2019We obtain some fundamental results, as Bokstedt-Neeman Theorem and Grothendieck duality, about the derived category of modules on a finite ringed space. Then we see how these results are transfered to schemes in a simple way and generalized to other ringed ... More
Integral points on twisted Markoff surfacesApr 15 2019We study the integral Hasse principle for affine varieties of the form ax^2+y^2+z^2-xyz=m ,using Brauer-Manin obstruction, and we produce examples whose Brauer groups include 4-torsion elements .We will construct their explicit representatives and compute ... More
Multidimensional topological Galois theoryApr 14 2019In this preprint we present an outline of the multidimensional version of topological Galois theory. The theory studies topological obstruction to solvability of equations "in finite terms" (i.e. to their solvability by radicals, by elementary functions, ... More
Horizontal strips and spaces of quadratic differentialsApr 14 2019A (meromorphic) quadratic differential is a (meromorphic) section of the tensor square of the canonical bundle of a Riemann surface. They arose in the study of quasiconformal mappings in the works of Oswald Teichm\"uller, and have played a mayor role ... More
Infinitesimal Bloch regulatorApr 14 2019In this paper, we continue our project of defining and studying the infinitesimal versions of the classical, real analytic, invariants of motives. Here, we construct an infinitesimal analog of Bloch's regulator. Let $X/k$ be a scheme of finite type over ... More
A non-commutative differential module approach to Alexander modulesApr 14 2019The theory of R. Crowell on derived modules is approached within the theory of non-commutative differential modules. We also seek analogies to the theory of cotangent complex from differentials in the commutative ring setting. Finally we give examples ... More
On the decomposition theorem for intersection de Rham complexesApr 14 2019We establish a positive characteristic analogue of intersection cohomology for polarized variations of Hodge structure. This includes: a) the decomposition theorem for the intersection de Rham complex; b) the $E_1$-degeneration theorem for the intersection ... More
Stable bases of the Springer resolution and representation theoryApr 14 2019In this note, we collect basic facts about Maulik and Okounkov's stable bases for the Springer resolution, focusing on their relations to representations of Lie algebras over complex numbers and algebraically closed positive characteristic fields, and ... More
The Massey vanishing conjecture for number fieldsApr 13 2019A conjecture of Min\'a\v{c} and T\^an predicts that for any n>2, any prime p and any field k, the Massey product of n Galois cohomology classes in H^1(k,Z/pZ) must vanish if it is defined. We establish this conjecture when k is a number field.
The S_n-equivariant top weight Euler characteristic of M_{g,n}Apr 12 2019We prove a formula, conjectured by Zagier, for the S_n-equivariant Euler characteristic of the top weight cohomology of M_{g,n}.
Varieties of planes on intersections of three quadricsApr 12 2019We study the geometry of spaces of planes on smooth complete intersections of three quadrics, with a view toward rationality questions.
Sylvester-Gallai type theorems for quadratic polynomialsApr 12 2019We prove Sylvester-Gallai type theorems for quadratic polynomials. Specifically, we prove that if a finite collection $\mathcal Q$, of irreducible polynomials of degree at most $2$, satisfy that for every two polynomials $Q_1,Q_2\in {\mathcal Q}$ there ... More
On the Mumford-Tate conjecture for hyperkähler varietiesApr 12 2019We study the Mumford-Tate conjecture for hyperk\"{a}hler varieties. Building on work of Markman, we show that it holds in arbitrary codimension for all varieties of $\mathrm{K}3^{[m]}$-type. For an arbitrary hyperk\"{a}hler variety satisfying $b_2(X)>3$ ... More
Semi-conformal structure on certain vertex superalgebras associated to vertex superalgebroidsApr 12 2019In this paper, we first give the definiton of a vertex superalgebroid. Then we construct a family of vertex superalgebras associated to vertex superalgebroids. As a main result, we find a sufficient and necessary condition that this vertex superalgebras ... More
Rational real algebraic models of compact differential surfaces with circle actionsApr 12 2019We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group $\mathbb{S}^1$ up to equivariant isomorphisms. As an application, we show that every compact differentiable ... More
Logarithmic abelian varieties, Part VI: Local moduli and GAGFApr 12 2019This is Part VI of our series of papers on log abelian varieties. In this part, we study local moduli and GAGF of log abelian varieties.
Tropical Lagrangians and Homological Mirror SymmetryApr 12 2019We produce for each tropical hypersurface $V(\phi)\subset Q=\mathbb{R}^n$ a Lagrangian $L(\phi)\subset (\mathbb{C}^*)^n$ whose moment map projection is a tropical amoeba of $V(\phi)$. When these Lagrangians are admissible in the Fukaya-Seidel category, ... More
What Makes a Complex VirtualApr 12 2019Virtual resolutions are homological representations of finitely generated $\text{Pic}(X)$-graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex ... More
Klein coverings of genus 2 curvesApr 11 2019We investigate the geometry of \'etale $4:1$ coverings of smooth complex genus 2 curves with the monodromy group isomorphic to the Klein four-group. There are two cases, isotropic and non-isotropic depending on the values of the Weil pairing restricted ... More
Fano varieties of K3 type and IHS manifoldsApr 11 2019We construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure and we link some of them to projective families of irreducible holomorphic symplectic manifolds.
O-minimal de Rham cohomologyApr 11 2019In the present paper we elaborate an o-minimal de Rham cohomology theory for abstract-definable $\mathcal{C}^p$ manifolds with $1\leq p\leq \infty$ in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential ... More
A survey of the additive dilogarithmApr 10 2019Borel's construction of the regulator gives an injective map from the algebraic $K$-groups of a number field to its Deligne-Beilinson cohomology groups. This has many interesting arithmetic and geometric consequences. The formula for the regulator is ... More
The sequence of mixed Łojasiewicz exponents associated to pairs of idealsApr 10 2019We analyze the sequence $\mathcal L^*_J(I)$ of mixed \L ojasiewicz exponents attached to any pair $I,J$ of monomial ideals of finite colength of the ring of analytic function germs $(\mathbb C^n,0)\to \mathbb C$. In particular, we obtain a combinatorial ... More
Lehn's formula in Chow and Conjectures of Beauville and VoisinApr 10 2019The Beauville-Voisin conjecture for a hyperk\"ahler manifold X states that the subring of the Chow ring A^*(X) generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of X. We prove a weak version of this ... More
Obstructions to weak approximation for reductive groups over $p$-adic function fieldsApr 10 2019We establish arithmetic duality theorems for short complexes associated to reductive groups over $p$-adic function fields. Using dualities, we deduce obstructions to weak approximation for quasi-split reductive groups. Finally, we give an application ... More
Algorithm for studying polynomial maps and reductions modulo prime numberApr 10 2019In our previous paper an effective algorithm for inverting polynomial automorphisms was proposed. Also the class of Pascal finite polynomial automorphisms was introduced. Pascal finite polynomial maps constitute a generalization of exponential automorphisms ... More
Equivariant splitting of the Hodge--de Rham exact sequenceApr 10 2019Let $X$ be an algebraic curve with an action of a finite group $G$ over a field $k$. We show that if the Hodge-de Rham short exact sequence of $X$ splits $G$-equivariantly then the action of $G$ on $X$ is weakly ramified. In particular, this generalizes ... More
Derived categories of Thaddeus pair moduli spaces via d-critical flipsApr 09 2019We show that the moduli spaces of Thaddeus pairs on smooth projective curves and those of dual pairs are related by d-critical flips, which are virtual birational transformations introduced by the second author. We then prove the existence of fully-faithful ... More
Asymptotic Syzygies in the Setting of Semi-Ample GrowthApr 09 2019We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld \cite{einErmanLazarsfeld16} we give an explicit range in which the graded Betti numbers of $\mathbb{P}^{n_1}\times ... More
Dynamically affine maps in positive characteristicApr 09 2019We study fixed points of iterates of dynamically affine maps (a generalisation of Latt\`es maps) over algebraically closed fields of positive characteristic $p$. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function ... More
Simple embeddings of rational homology balls and antiflipsApr 09 2019Let $V$ be a regular neighborhood of a negative chain of $2$-spheres (i.e. exceptional divisor of a cyclic quotient singularity), and let $B_{p,q}$ be a rational homology ball which is smoothly embedded in $V$. Assume that the embedding is simple, i.e. ... More
Constructing Separable Arnold Snakes of Morse PolynomialsApr 09 2019We give a new and constructive proof of the existence of a special class of univariate polynomials whose graphs have preassigned shapes. By definition, all the critical points of a Morse polynomial function are real and distinct and all its critical values ... More
Mirror curve of orbifold Hurwitz numbersApr 09 2019Edge-contraction operations form an effective tool in various graph enumeration problems, such as counting Grothendieck's dessins d'enfants and simple and double Hurwitz numbers. These counting problems can be solved by a mechanism known as topological ... More
Perfect points on genus one curves and consequences for supersingular K3 surfacesApr 09 2019Apr 10 2019We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus one curves over function fields admit no points over the perfect closure of the base field) and use it to show that any ... More
Perfect points on genus one curves and consequences for supersingular K3 surfacesApr 09 2019We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus one curves over function fields admit no points over the perfect closure of the base field) and use it to show that any ... More
F-algebra--Rinehart Pairs and Super F-algebroidsApr 09 2019In this note we define F-algebra--Rinehart pairs and super F-algebroids and study the connection between them.
Quadratic Chabauty for (bi)elliptic curves and Kim's conjectureApr 09 2019We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets $\mathcal{X}(\mathbb{Z}_p)_2$ containing ... More
The bottleneck degree of algebraic varietiesApr 09 2019A bottleneck of a smooth algebraic variety $X \subset \mathbb{C}^n$ is a pair of distinct points $(x,y) \in X$ such that the Euclidean normal spaces at $x$ and $y$ contain the line spanned by $x$ and $y$. The narrowness of bottlenecks is a fundamental ... More
A surface birational to an Enriques surface with non-finitely generated automorphism groupApr 09 2019We will show that there is a smooth complex projective surface, birational to some Enriques surface, such that the automorphism group is discrete but not finitely generated.
More barriers for rank methods, via a "numeric to symbolic" transferApr 08 2019We prove new barrier results in arithmetic complexity theory, showing severe limitations of natural lifting (aka escalation) techniques. For example, we prove that even optimal rank lower bounds on $k$-tensors cannot yield non-trivial lower bounds on ... More
Arithmetic occult period mapsApr 08 2019Several natural complex configuration spaces admit surprising uniformizations as arithmetic ball quotients, by identifying each parametrized object with the periods of some auxiliary object. In each case, the theory of canonical models of Shimura varieties ... More
WDVV-Type Relations for Disk Gromov-Witten Invariants in Dimension 6Apr 08 2019The first author's previous work established Solomon's WDVV-type relations for Welschinger's invariant curve counts in real symplectic fourfolds by lifting geometric relations over possibly unorientable morphisms. We apply her framework to obtain WDVV-style ... More
Loose edges and factorization theoremsApr 08 2019Let $ R $ be a regular local ring with maximal ideal $ \mathfrak{m} $. We consider elements $ f \in R $ such that their Newton polyhedron has a loose edge. We show that if the symbolic restriction of $f$ to such an edge is a product of two coprime polynomials, ... More
On the moduli space of holomorphic G-connections on a compact Riemann surfaceApr 08 2019Let $X$ be a compact connected Riemann surface of genus at least two and $G$ a connected reductive complex affine algebraic group. The Riemann--Hilbert correspondence produces a biholomorphism between the moduli space ${\mathcal M}_X(G)$ parametrizing ... More
A monodromy criterion for existence of Neron models of abelian schemes in characteristic zeroApr 08 2019We consider the problem of existence of Neron models for a family of abelian varieties over a base of dimension greater than 1. We show that when S is of equicharacteristic zero, the condition of toric additivity introduced in [Ore18] is sufficient for ... More
Wall-crossings and a categorification of $K$-theory stable bases of the Springer resolutionApr 07 2019We compare the $K$-theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure-Lusztig operators in the affine Hecke algebra. It ... More
Moduli of certain wild covers of curvesApr 07 2019A fine moduli space is constructed, for cyclic-by-$\mathsf{p}$ covers of an affine curve over an algebraically closed field $k$ of characteristic $\mathsf{p}>0$. An intersection of finitely many fine moduli spaces for cyclic-by-$\mathsf{p}$ covers of ... More
Unexpected complex caustics of the complex elliptic billardApr 07 2019The article studies a genralization of the elliptic billard to the complex domain. We show that the billard orbits also have caustics, and that the number of such caustics is bigger than for the real case. For example, for a given ellipse E, there exists ... More