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Stratifications of the Singular Fibers of Mumford SystemsFeb 21 2019We study fibers of the Mumford system (even and odd) of order $g> 0$. These fibers are parametrized by hyperelliptic curves of genus $g$. In this article we write the fibers above the singular curves. To do this, we will use two methods, the first geometric ... More

A variety that cannot be dominated by one that liftsFeb 21 2019We prove a precise version of a theorem of Siu and Beauville on morphisms to higher genus curves, and use it to show that if a variety $X$ in characteristic $p$ lifts to characteristic $0$, then any morphism $X \to C$ to a curve of genus $g \geq 2$ can ... More

Differential Forms on Hyperelliptic Curves with Semistable ReductionFeb 20 2019Let $C$ be a hyperelliptic curve over a local field $K$ with odd residue characteristic, defined by some affine Weierstrass equation $y^2=f(x)$. We assume that $C$ has semistable reduction and denote by $\mathcal{X} \rightarrow \textrm{Spec}\, \mathcal{O}_K$ ... More

Finiteness properties of affine Deligne-Lusztig varietiesFeb 20 2019Affine Deligne-Lusztig varieties are closely related to the special fibre of Newton strata in the reduction of Shimura varieties or of moduli spaces of $G$-shtukas. In almost all cases, they are not quasi-compact. In this note we prove basic finiteness ... More

The Abel map for surface singularities III. Elliptic germsFeb 20 2019If $(\widetilde{X},E)\to (X,o)$ is the resolution of a complex normal surface singularity and $c_1:{\rm Pic}(\widetilde{X})\to H^2(\widetilde{X},{\mathbb Z})$ is the Chern class map, then ${\rm Pic}^{l'}(\widetilde{X}):= c_1^{-1}(l')$ has a (Brill--Noether ... More

A remark on Gromov-Witten-Welschinger invariants of $\mathbb{C} P^3\#\overline{\mathbb{C} P}^3$Feb 20 2019We generalize the formula of Gromov-Witten-Welschinger invariants of $\mathbb{C} P^3$ established by E. Brugall\'e and P. Georgieva in [BG16b] to $\mathbb{C} P^3\#\overline{\mathbb{C} P}^3$. Using pencils of quadrics, some real and complex enumerative ... More

A note on divisorial correspondences of semi-abelian varietiesFeb 20 2019Let S be a locally noetherian scheme and consider two extensions G_1 and G_2 of abelian S-schemes by S-tori. In this note we prove that the fppf-sheaf Corr _S(G_1,G_2) of divisorial correspondences between G_1 and G_2 is representable. Moreover, using ... More

Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curvesFeb 19 2019We define a symmetric monoidal category Trop which, roughly, has degrees of tropical curves as its objects and types of tropical curves as its morphisms. A symmetric monoidal functor with domain Trop is what we call a (2D) tropical quantum field theory ... More

Generating series for the Hodge-Euler polynomials of $GL(n,{\mathbb C})$-character varietiesFeb 18 2019With G=GL(n,C), let $\mathcal{X}_{\Gamma}G$ be the G-character variety of a given finitely presented group $\Gamma$, and let $\mathcal{X}^{irr}_{\Gamma}G \subset \mathcal{X}_{\Gamma}G$ be the locus of irreducible representation conjugacy classes. We provide ... More

Classification of Schubert Galois groups in Gr(4,9)Feb 18 2019We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, only 149 have Galois group that does not contain the alternating group. We identify ... More

Uniform Yomdin-Gromov parametrizations and points of bounded height in valued fieldsFeb 18 2019We prove a uniform version of non-Archimedean Yomdin-Gromov parametrizations in a definable context with algebraic Skolem functions in the residue field. The parametrization result allows us to bound the number of F_q[t]-points of bounded degrees of algebraic ... More

On Strassen's rank additivity for small three-way tensorsFeb 18 2019We address the problem of the additivity of the tensor rank. That is for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known as Strassen's ... More

Geometric local epsilon factorsFeb 18 2019Inspired by the work of Laumon on $\varepsilon$-factors and by Deligne's $1974$ letter to Serre, we give an explicit cohomological definition of $\varepsilon$-factors for $\ell$-adic Galois representations over henselian discrete valuation fields of positive ... More

Matroid connectivity and singularities of configuration hypersurfacesFeb 18 2019Consider a linear realization of a matroid over a field. One associates to it a configuration polynomial and bilinear form with polynomial coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first ... More

Moduli spaces of framed $G$--Higgs bundles and symplectic geometryFeb 18 2019Let $X$ be a compact connected Riemann surface, $D\, \subset\, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x\, \subsetneq\, G_x$ a Zariski closed subgroup for every $x\, \in\, D$. A framed principal ... More

Deformations of polystable sheaves on surfaces: quadraticity implies formalityFeb 18 2019We study relations between the quadraticity of the Kuranishi family of a coherent sheaf on a complex projective scheme and the formality of the DG-Lie algebra of its derived endomorphisms. In particular, we prove that for a polystable coherent sheaf of ... More

Isogenous components of Jacobian surfacesFeb 18 2019Let $\mathcal X$ be a genus 2 curve defined over a field $K$, $\mbox{char} K = p \geq 0$, and $\mbox{Jac} (\mathcal X, \iota)$ its Jacobian, where $\iota$ is the principal polarization of $\mbox{Jac} (\mathcal X)$ attached to $\mathcal X$. Assume that ... More

Isomorphisms between complements of projective plane curvesFeb 17 2019In this article, we study isomorphisms between complements of irreducible curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically closed field. Of particular interest are rational unicuspidal curves. We prove that if there exists ... More

On certain maximal hyperelliptic curves related to Chebyshev polynomialsFeb 17 2019We study hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs $(q,d)$ such that the hyperelliptic curve $\cC$ over a finite field $\FF_{q^2}$ corresponding to the equation $y^2 = \varphi_{d}(x)$ is ... More

Double coverings of arrangement complements and $2$-torsion in Milnor fiber homologyFeb 17 2019We prove that mod $2$ Betti numbers of the double covering of a complex hyperplane arrangement complement is combinatorially determined. The proof is based on a relation between mod $2$ Aomoto complex and the transfer long exact sequence. Applying the ... More

Physicists' $d=3+1$, $N=1$ superspace-time and supersymmetric QFTs from a tower construction in complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry and a purge-evaluation/index-contracting mapFeb 17 2019The complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry aspect of a superspace(-time) $\widehat{X}$ in Sec.\,1 of D(14.1) (arXiv:1808.05011 [math.DG]) together with the Spin-Statistics Theorem in Quantum Field Theory, which requires fermionic ... More

Complex tori, theta groups and their Jordan propertiesFeb 17 2019We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the group of bimeromorphic automorphisms of a product of the complex projective line and a complex torus of positive algebraic dimension.

Complex tori, theta groups and their Jordan propertiesFeb 17 2019Feb 19 2019We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the group of bimeromorphic automorphisms of a product of the complex projective line and a complex torus of positive algebraic dimension.

Higgs bundles, abelian gerbes and cameral dataFeb 16 2019We study the Hitchin map for $G_{\mathbb{R}}$-Higgs bundles on a smooth curve, where $G_{\mathbb{R}}$ is a quasi-split real form of a complex reductive algebraic group $G$. By looking at the moduli stack of regular $G_{\mathbb{R}}$-Higgs bundles, we prove ... More

On rigidity of trinomial hypersurfaces and factorial trinomial varietiesFeb 16 2019Trinomial varieties are affine varieties given by some special system of equations consisting of polynomials with three terms. Such varieties are total coordinate spaces of normal rational varieties with torus action of complexity one. For an affine variety ... More

Totally geodesic subvarieties in the moduli space of curvesFeb 16 2019In this paper we study totally geodesic subvarieties $Y \subset \mathsf{A}_g$ of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for $g\geq 4$. We prove that if $Y$ is generically contained in the Torelli ... More

Kawaguchi-Silverman conjecture for endomorphisms on several classes of varietiesFeb 16 2019We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC for group ... More

Int-amplified endomorphisms on normal projective surfacesFeb 16 2019We investigate int-amplified endomorphisms on normal projective surfaces. We prove that the output of the equivariant MMP is either a Q-abelian surface, a (equivariant) quasi-\'etale quotient of a smooth projective surface, a Mori dream space, or a projective ... More

Liftable derived equivalences and objective categoriesFeb 16 2019We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that is, isomorphic ... More

Palais-Smale values and stability of global Hölderian error bounds for polynomial functionsFeb 15 2019Let $f$ be a polynomial function of $n$ variables. In this paper, we study stability of global H\"{o}lderian error bound for a sublevel set $[f \le t]$ under a perturbation of $t$. Namely, we investigate the following questions: 1. Suppose that $[f \le ... More

The 2-dimensional Complex Jacobian Conjecture under the viewpoint of "pertinent variables"Feb 15 2019Let $F = (f,g): \C^2 \to \C^2$ be a polynomial map. The 2-dimensional Complex Jacobian Conjecture, which is still open, can be expressed as follows: "if $F$ satisfies the Non-Zero Condition $\det (JF(x,y)) = {\rm constant} \neq 0, \forall (x,y) \in \C^2$, ... More

Refined scattering diagrams and theta functions from asymptotic analysis of Maurer-Cartan equationsFeb 15 2019We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analysing the asymptotic behavior of Maurer-Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded ... More

Nearby cycles and semipositivity in positive characteristicFeb 15 2019We study restriction of logarithmic Higgs bundles to the boundary divisor and we construct the corresponding nearby-cycles functor in positive characteristic. As applications we prove some strong semipositivity theorems for analogs of complex polarized ... More

On the existence of flat generators of quasicoherent sheavesFeb 15 2019We show that for a quasicompact quasiseparated scheme $X$, the following assertions are equivalent: (1) the category $\operatorname{QCoh}(X)$ of all quasicoherent sheaves on $X$ has a flat generator; (2) the scheme $X$ is semiseparated.

WDVV equation and its application to relative Gromov--Witten theoryFeb 15 2019We derive a recursive formula for certain relative Gromov--Witten invariants with maximal tangency condition via WDVV equation. For certain relative pairs, we get explicit formulae of invariants using the recursive formula.

Degree of irrationality of very general abelian surfacesFeb 15 2019The degree of irrationality of a projective variety $X$ is defined to be the smallest degree rational dominant map to a projective space of the same dimension. For abelian surfaces, Yoshihara computed this invariant in specific cases, while Stapleton ... More

3264 Conics in a SecondFeb 14 2019Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given instance. This article illustrates how these two fields complement each other. Our focus lies on ... More

Holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds with Lagrangian fibrations and special Kahler geometryFeb 14 2019Let $M$ be a holomorphic symplectic K\"ahler manifold equipped with a Lagrangian fibration $\pi$ with compact fibers. The base of this manifold is equipped with a special K\"ahler structure, that is, a K\"ahler structure $(I, g, \omega)$ and a symplectic ... More

Quasi-complete intersections in P2 and syzygiesFeb 14 2019Let C \in P2 be a reduced, singular curve of degree d and equation f = 0. Let \Sigma denote the jacobian subscheme of C. We have 0 -> E -> 3.O -> I_\Sigma(d-1) -> 0 (the surjection is given by the partials of f). We study the relationships between the ... More

From partitions to Hodge numbers of Hilbert Schemes of SurfacesFeb 14 2019We celebrate the 100th anniversary of Srinivasa Ramanujan's election as a Fellow of the Royal Society, which was largely based on his work with G. H. Hardy on the asymptotic properties of the partition function. After recalling this revolutionary work, ... More

From partitions to Hodge numbers of Hilbert Schemes of SurfacesFeb 14 2019Feb 15 2019We celebrate the 100th anniversary of Srinivasa Ramanujan's election as a Fellow of the Royal Society, which was largely based on his work with G. H. Hardy on the asymptotic properties of the partition function. After recalling this revolutionary work, ... More

Non-symplectic involutions on manifolds of $K3^{[n]}$-typeFeb 14 2019We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant forms of the invariant and anti-invariant lattice ... More

Donaldson-Thomas invariants from tropical disksFeb 14 2019We prove that the quantum DT invariants associated to quivers with genteel potential can be expressed in terms of certain refined counts of tropical disks. This is based on a quantum version of Bridgeland's description of cluster scattering diagrams in ... More

The rigidity on the second fundamental form of projective manifoldsFeb 14 2019Let $M$ be a complex $n$-dimensional projective manifold in $\mathbb{P}^{n+r}$ endowed with the Fubini-Study metric of constant holomorphic sectional $1$, $\sigma$ its second fundamental form, and $\underline{|\sigma|}^2$ the mean value of the squared ... More

Stability conditions and braid group actions on affine $A_n$ quiversFeb 14 2019We study stability conditions on the Calabi-Yau-$N$ categories associated to an affine type $A_n$ quiver which can be constructed from certain meromorphic quadratic differentials with zeroes of order $N-2$. We follow Ikeda's work to show that this moduli ... More

Un adjointFeb 14 2019Given a scheme S and a flat morphism T \to S of finite presentation we define a surjective S-morphism to an {\'e}tale and separated S-scheme, which is universal in an obvious sense. Properties of this morphism are deduced from a thorough uses of quotients ... More

Equimultiplicity of families of map germs from $\mathbb{C}^2$ to $\mathbb{C}^3$Feb 13 2019In 1971, Zariski proposed some questions in Theory of Singularities. One of such problems is the so-called, nowadays, Zariski's multiplicity conjecture. In this work, we consider the version of this conjecture for families. We answer positively Zariski's ... More

Counting lines on projective surfacesFeb 13 2019We prove a bound on the number of lines on a smooth degree-d surface in three-dimensional projective space for $d \geq 3$. This bound improves a bound due to Segre and renders some of his arguments rigorous. It is the best known bound for $d \geq 6$.

Overdetermined Systems of Equations on Toric, Spherical, and Other Algebraic VarietiesFeb 13 2019Let $E_1,\ldots,E_k$ be a collection of linear series on an algebraic variety $X$ over $\mathbb{C}$. That is, $E_i\subset H^0(X, \mathcal{L}_i)$ is a finite dimensional subspace of the space of regular sections of line bundles $ \mathcal{L}_i$. Such a ... More

Factorization of KdV Schrödinger operators using differential subresultantsFeb 13 2019We address the classical factorization problem of a one dimensional Schr\"odinger operator $-\partial^2+u-\lambda$, for a stationary potential $u$ of the KdV hierarchy but, in this occasion, a "parameter" $\lambda$. Inspired by the more effective approach ... More

Global Invariant Branches of Non-degenerate Foliations on Projective Toric SurfacesFeb 13 2019We show that the isolated invariant branches globalize to algebraic curves, when we consider weak toric type complex hyperbolic foliations on projective toric ambient surfaces. To do it, we pass through a characterization of weak toric type foliations ... More

On the generalisation of cohomology with compact support to non-finite type schemesFeb 13 2019In this article we extend Deligne's construction of Grothendieck's six operations on the derived category of torsion sheaves over the \'etale site of a scheme for morphisms of finite type to a larger class of morphisms. This class includes profinite \'etale ... More

Volume form on moduli spaces of d-differentialsFeb 13 2019Given $d\in \mathbb{N}$, $g\in \mathbb{N} \cup\{0\}$, and an integral vector $\kappa=(k_1,\dots,k_n)$ such that $k_i>-d$ and $k_1+\dots+k_n=d(2g-2)$, let $\Omega^d\mathcal{M}_{g,n}(\kappa)$ denote the moduli space of meromorphic $d$-differentials on Riemann ... More

Étale coverings in codimension 1 with applications to Mori Dream SpacesFeb 13 2019The present paper is devoted to developing relations between Galois \'etale coverings in codimension 1 and \'etale fundamental groups in codimension 1 of algebraic varieties, aimed to studying the topology of Mori dream spaces. In particular, the universal ... More

Log-decay $F$-isocrystals on higher dimensional varietiesFeb 13 2019Let $k$ be a perfect field of positive characteristic and let $X$ be a smooth irreducible quasi-compact scheme over $k$. The Drinfeld-Kedlaya theorem states that for an irreducible $F$-isocrystal on $X$, the gap between consecutive generic slopes is bounded ... More

The matroid structure of vectors of the Mordell-Weil lattice and the topology of plane quartics and bitangent linesFeb 13 2019In this paper, we introduce the terminology of matroids into the study of Zariski-pairs related to rational elliptic surfaces, aiming to simplify the presentation and arguments involved. As an application, we provide new examples of Zariski $N$-ples of ... More

Perverse sheaves on semi-abelian varieties -- a survey of properties and applicationsFeb 13 2019We survey recent developments in the study of perverse sheaves on semi-abelian varieties. As concrete applications, we discuss various obstructions on the homotopy type of complex algebraic manifolds (expressed in terms of their cohomology jump loci), ... More

A reconstruction theorem for varietiesFeb 12 2019We show that varieties of dimension at least 2 over infinite fields are determined as abstract schemes by their Zariski topological spaces together with the rational equivalence relation on the set of effective divisors. This gives a universal Torelli ... More

Categorical Saito theory, I: A comparison resultFeb 12 2019In this paper, we present an explicit cyclic minimal $A_\infty$ model for the category of matrix factorizations $\MF(W)$ of an isolated hypersurface singularity. The key observation is to use Kontsevich's deformation quantization technique. Pushing this ... More

Local model of Hilbert-Siegel moduli schemes in $Γ_1(p)$-levelFeb 12 2019We construct a local model for Hilbert-Siegel moduli schemes with $\Gamma_1(p)$-level bad reduction over $\text{Spec }\mathbb{Z}_{q}$, where $p$ is a prime unramified in the totally real field and $q$ is the residue cardinality over $p$. Our main tool ... More

Exterior powers of the adjoint representation and the Weyl ring of $E_8$Feb 12 2019I derive explicitly all polynomial relations in the character ring of $E_8$ of the form $\chi_{\wedge^k \mathfrak{e}_8} - \mathfrak{p}_{k} (\chi_{1}, \dots, \chi_{r})=0$, where $\wedge^k \mathfrak{e}_8$ is an arbitrary exterior power of the adjoint representation ... More

Torus quotient of Richardson varieties in Orthogonal and Symplectic GrassmanniansFeb 12 2019For any simple, simply connected algebraic group $G$ of type $B,C$ and $D$ and for any maximal parabolic subgroup $P$ of $G$, we provide a criterion for a Richardson variety in $G/P$ to admit semistable points for the action of a maximal torus $T$ with ... More

The double point formula with isolated singularities and canonical embeddingsFeb 12 2019Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with isolated singularities. ... More

Characteristic cycles and Gevrey series solutions of $A$-hypergeometric systemsFeb 12 2019We compute the $L$-characteristic cycle of an $A$-hypergeometric system and higher Euler-Koszul homology modules of the toric ring. We also prove upper semicontinuity results about the multiplicities in these cycles and apply our results to analyze the ... More

Categorification of Legendrian knotsFeb 12 2019Feb 17 2019Perverse schober defined by Kapranov--Schechtman is a categorification of the notion of perverse sheaf. In their definition, a key ingredient is certain purity property of perverse sheaves. In this short note, we attempt to describe a real analogue of ... More

Categorification of Legendrian knotsFeb 12 2019Perverse schober defined by Kapranov--Schechtman is a categorification of the notion of perverse sheaf. In their definition, a key ingredient is certain purity property of perverse sheaves. In this short note, we attempt to describe a real analogue of ... More

On automorphisms of moduli spaces of parabolic vector bundlesFeb 11 2019Fix $n\geq 5$ general points $p_1, \dots, p_n\in\mathbb{P}^1$, and a weight vector $\mathcal{A} = (a_{1}, \dots, a_{n})$ of real numbers $0 \leq a_{i} \leq 1$. Consider the moduli space $\mathcal{M}_{\mathcal{A}}$ parametrizing rank two parabolic vector ... More

Degeneracy loci, virtual cycles and nested Hilbert schemes IIFeb 11 2019We express nested Hilbert schemes of points and curves on a smooth projective surface as "virtual resolutions" of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the ... More

Moving Seshadri Constants, and Coverings of Varieties of Maximal Albanese DimensionFeb 11 2019Feb 18 2019Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L \to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian \'etale covers of $X$ are arbitrarily ... More

Moving Seshadri Constants, and Coverings of Varieties of Maximal Albanese DimensionFeb 11 2019Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L \to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian \'etale covers of $X$ are arbitrarily ... More

Polylogarithms, Bloch complexes, and quiver mutationsFeb 11 2019For an integer n>2 we define a holomorphic complex valued weight n polylogarithm defined modulo (2pi i)^n/(n-1)! on the universal abelian cover of C-{0,1}. We analyze its functional equations and give a method for producing functional relations from quivers. ... More

Rigid isotopy of maximally writhed linksFeb 11 2019This is a sequel to the paper \cite{MO-mw} which identified maximally writhed algebraic links in $\rp^3$ and classified them topologically. In this paper we prove that all maximally writhed links of the same topological type are rigidly isotopic, i.e. ... More

Degeneration of pole order spectral sequences for hyperplane arrangements of 4 variablesFeb 11 2019For essential reduced hyperplane arrangements of 4 variables, we show that the pole order spectral sequence degenerates almost at $E_2$, and completely at $E_3$, generalizing the 3 variable case where the complete $E_2$-degeneration is known. These degenerations ... More

The Cohomology of the Grassmannian is a $gl_n$-moduleFeb 11 2019The integral singular cohomology ring of the Grassmann variety parametrizing $r$-dimensional subspaces in the $n$-dimensional complex vector space is naturally an irreducible representation of the Lie algebra of all the $n\times n$ matrices with integral ... More

An Eisenbud-Goto-type Upper Bound for the Castelnuovo-Mumford Regularity of Fake Weighted Projective SpacesFeb 11 2019We will give an upper bound for the $k$-normality of very ample lattice simplices, and then give an Eisenbud-Goto-type bound for some special classes of projective toric varieties.

Lorentzian polynomialsFeb 11 2019We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a nonzero Lorentzian polynomial has exactly one positive ... More

Graph sums in the Remodeling ConjectureFeb 11 2019The BKMP Remodeling Conjecture \cite{Ma,BKMP09,BKMP10} predicts all genus open-closed Gromov-Witten invariants for a toric Calabi-Yau $3$-orbifold by Eynard-Orantin's topological recursion \cite{EO07} on its mirror curve. The proof of the Remodeling Conjecture ... More

Generators for the $C^m$-closures of IdealsFeb 11 2019Let $\mathscr{R}$ denote the ring of real polynomials on $\mathbb{R}^{n}$. Fix $m\geq 0$, and let $A_{1},\cdots ,A_{M}\in \mathscr{R}$. The $ C^{m}$-closure of $\left( A_{1},\cdots ,A_{M}\right) $, denoted here by $ \left[ A_{1},\cdots ,A_{M};C^{m}\right] ... More

3d Mirror Symmetry and Elliptic Stable EnvelopesFeb 10 2019We consider a pair of quiver varieties (X;X') related by 3d mirror symmetry, where X =T*Gr(k,n) is the cotangent bundle of the Grassmannian of k-planes of n-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an ... More

The $a$-number of Certain Hyperelliptic CurvesFeb 10 2019In this paper, we compute a formula for the $a$-number of certain hyperelliptic curves given by the equation $y^2= x^m+1$ for infinitely many values of $m$. The same question is studied for the curve corresponding to $y^2= x^m+x$.

Derivator Six-Functor-Formalisms - Construction IIFeb 10 2019Starting from very simple and obviously necessary axioms on a (derivator enhanced) four-functor-formalism, we construct derivator six-functor-formalisms using compactifications. This works, for example, for various contexts over topological spaces and ... More

A minimax principle to the injectivity of the Jacobian conjectureFeb 10 2019The main result of this paper is to prove some type of Real Jacobian Conjecture. It is proved by the Minimax Principle and asserts if the eigenvalues of $F'(x)$ are bounded from zero and all the eigenvalues of $F'(x)+F'(x)^T$ are strictly same sign, where ... More

Doubly periodic monopoles and $q$-difference modulesFeb 10 2019An interesting theme in complex differential geometry is to find a correspondence between algebraic objects and differential geometric objects. One of the most attractive is the non-abelian Hodge theory of Simpson. In this paper, pursuing an analogue ... More

Singularities of the Moduli Space of n Unordered Points on the Riemann SphereFeb 10 2019We classify the finite groups associated to the singularities of the moduli space of $n/ge5$ unordered points on the Riemann sphere. We also realize the classification by an algorithm.

An improvement of the duality formalism of the rational etale siteFeb 10 2019We improve the arithmetic duality formalism of the rational etale site. This improvement allows us to avoid some exotic approximation arguments on local fields with ind-rational base, thus simplifying the proofs of the previously established duality theorems ... More

On the multipliers at fixed points of self-maps of the projective planeFeb 09 2019This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz ... More

On the multipliers at fixed points of self-maps of the projective planeFeb 09 2019Feb 15 2019This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz ... More

Étale Homotopy Obstructions of Arithmetic SpheresFeb 09 2019Let $K$ be a field of characteristic $\ne 2$ and let $X$ be the affine variety over $K$ defined by the equation $$ X:\ a_0x_0^2 + \cdots + a_nx_n^2 = 1 $$ where $n\ge 0$ and $a_i\in K$. In this paper we compute the lowest mod 2 \'{e}tale homological obstruction ... More

Modular Nekrasov-Okounkov formulasFeb 09 2019Using Littlewood's map, which decomposes a partition into its $r$-core and $r$-quotient, Han and Ji have shown that many well-known hook-length formulas admit modular analogues. In this paper we present a variant of the Han-Ji `multiplication theorem' ... More

Abelianisation of Logarithmic $\mathfrak{sl}_2$-ConnectionsFeb 09 2019We prove a functorial correspondence between a category of logarithmic $\mathfrak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $\pi : \Sigma \to ... More

Characterization of polynomials whose large powers have fully positive coefficientsFeb 09 2019We give a criterion which characterizes a real multi-variate Laurent polynomial with full-dimensional smooth Newton polytope to have the property that all sufficiently large powers of the polynomial have fully positive coefficients. Here a Laurent polynomial ... More

$(g,k)$-Fermat curvesFeb 08 2019Let $G$ be a co-compact torsion free Fuchsian group of genus $g \geq2$ and, for each integer $k \geq 2$, $G_{k}$ be its normal subgroup generated by the $k$-powers of the elements of $G$ together its commutators. There is a natural holomorphic embedding ... More

The Cremona group and its subgroupsFeb 08 2019This survey deals with the Cremona group via its subgroups.

Infinitesimal Lipschitz conditions on family of analytic varietiesFeb 08 2019In this work, we extend the concept of the double of an ideal defined in \cite{G2}, to the context of modules. We also obtain the genericity of the infinitesimal Lipschitz condition A for an enlarged class of analytic spaces.

Buryak-Okounkov formula for the n-point function and a new proof of the Witten conjectureFeb 08 2019We identify the formulas of Buryak and Okounkov for the n-point functions of the intersection numbers of psi-classes on the moduli spaces of curves. This allows us to combine the earlier known results and this one into a principally new proof of the famous ... More

A family of threefolds of general type with canonical map of high degreeFeb 08 2019In this note we provide a two-dimensional family of smooth minimal threefolds of general type with canonical map of degree 96, improving the previous known bound of 72.

Adaptive Step Size Control for Polynomial Homotopy Continuation MethodsFeb 08 2019In this paper we develop an adaptive step size control for the numerical tracking of implicitly defined paths in the context of polynomial homotopy continuation methods. We focus on the case where the paths are tracked using a predictor-corrector scheme ... More

Arithmetic subspaces of moduli spaces of rank one local systemsFeb 08 2019We show that closed subsets of the character variety of a normal complex variety, which are $p$-adically integral and Galois invariant, are motivic.

Invariants, Bitangents and Matrix Representations of Plane Quartics with 3-Cyclic AutomorphismsFeb 08 2019In this work we compute the Dixmier invariants and bitangents of the plane quartics with 3,6 or 9-cyclic automorphisms, we find that a quartic curve with 6-cyclic automorphism will have 3 horizontal bitangents which form an asysgetic triple. We also discuss ... More

Mazur's Conjecture and An Unexpected Rational Curve on Kummer Surfaces and their Superelliptic GeneralisationsFeb 08 2019We prove the following special case of Mazur's conjecture on the topology of rational points. Let $E$ be an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$. For a class of elliptic pencils which are quadratic twists of $E$ by quartic polynomials, ... More