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On determinantal curves and complexified phase-lock areas in a model of Josephson junctionAug 22 2019The paper deals with a three-parameter family of special double confluent Heun equations that was introduced and studied by V.M.Buchstaber and S.I.Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. ... More
Remarks on generating series for special cyclesAug 22 2019In this note, we consider special algebraic cycles on the Shimura variety S associated to a quadratic space V over a totally real field F, |F:\Q|=d, of signature ((m,2)^{d_+},(m+2,0)^{d-d_+}), 1\le d_+<d. For each n, 1\le n\le m, there are special cycles ... More
Hyperbolicity of coarse moduli spaces and isotriviality for certain familiesAug 22 2019In this paper, we prove the Kobayashi hyperbolicity of the coarse moduli spaces of canonically polarized or polarized Calabi-Yau manifolds in the sense of complex $V$-spaces (a generalization of complex $V$-manifolds in the sense of Satake). As an application, ... More
Non-abelian Hodge Theory and Related TopicsAug 22 2019This paper is a survey aimed on the introduction of non-abelian Hodge theory that gives the correspondence between flat bundles and Higgs bundles, and some topics arising from this theory, especially some recent developments on the study of the relevant ... More
Invariant Hypersurfaces and Nodal Components of FoliationsAug 22 2019It is known that there is at least an invariant analytic curve passing through each of the components in the complement of nodal singularities, after the reduction of singularities of a germ of singular foliation in ${\mathbb C}^2,0$}. Here, we state ... More
Torus orbit closures in flag varieties and retractions on Weyl groupsAug 22 2019In this manuscript, we define three kinds of retractions on Weyl groups or finite Coxeter groups and study the relations among them. The first is what we call a \textit{geometric retraction} $\mathcal{R}^g_Y$ associated to a torus orbit closure $Y$ in ... More
Étale cohomology of rank one $\ell$-adic local systems in positive characteristicAug 22 2019We show that in positive characteristic special loci of deformation spaces of rank one $\ell$-adic local systems are quasilinear. From this we deduce the Hard Lefschetz theorem for rank one $\ell$-adic local systems and a generic vanishing theorem.
Rouquier dimension of some blow-upsAug 22 2019Rapha\"{e}l Rouquier introduced an invariant of triangulated categories which is known as Rouquier dimension. Orlov conjectured that for any smooth quasi-projective variety $X$ the Rouquier dimension of $D^b_{\mathrm{coh}}(X)$ is equal to $\mathrm{dim}\, ... More
Motivic sheaves revisitedAug 22 2019In earlier work (arXiv:0801.0261), we gave a definition of an abelian category of motivic (constructible) sheaves over a base in characteristic zero using Nori's method. This category has Hodge and etale realizations, and is stable under inverse and direct ... More
Poisson structure on the moduli spaces of sheaves of pure dimension one on a surfaceAug 22 2019Let S be a smooth complex projective surface equipped with a Poisson structure s and also a polarization H. The moduli space M_H(S,P) of stable sheaves on S having a fixed Hilbert polynomial P of degree one has a natural Poisson structure given by s, ... More
Classification of flat pencils of foliations on compact complex surfacesAug 22 2019Related to the classification of regular foliations in a complex algebraic surface, we address the problem of classifying the complex surfaces which admit a flat pencil of foliations. On this matter, a classification of flat pencils which admit foliations ... More
Moduli spaces of tropical curves are simply connectedAug 22 2019We develop basic techniques for studying fundamental groups and singular homology of generalized Delta-complexes. As an application, we show that the moduli spaces of tropical curves Delta_g and Delta_{g,n} are simply connected, for g at least 1. We also ... More
On the splitting principle for cohomological invariants of reflection groupsAug 21 2019Let $\mathrm{k}_{0}$ be a field and $W$ a finite orthogonal reflection group, which is subgroup of the orthogonal group of a regular symmetric bilinear space over $\mathrm{k}_{0}$. We prove Serre's splitting principle for cohomological invariants of $W$ ... More
The modularity of special cycles on orthogonal Shimura varieties over totally real fields under the Beilinson-Bloch conjectureAug 21 2019We study special cycles on a Shimura variety of orthogonal type over a totally real field of degree $d$ associated with a quadratic form in $n+2$ variables whose signature is $(n,2)$ at $e$ real places and $(n+2,0)$ at the remaining $d-e$ real places ... More
Algebraic integer totally in a compactAug 20 2019An algebraic integer is a complex number that is a root of some monic polynomial with integer coefficients. An algebraic integer is said to be totally in a compact of the complex plan if all its conjugates are in the same compact as well. Given a compact, ... More
P=W for Lagrangian fibrations and degenerations of hyper-Kähler manifoldsAug 20 2019We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-K\"ahler manifolds.
Gamma functions, monodromy and Apéry constantsAug 20 2019In their paper on the gamma conjecture in mirror symmetry, Golyshev and Zagier introduce what we refer to as an Ap\'ery series associated to an ordinary differential operator L with a point of maximal unipotent monodromy (MUM) at 0, a conifold singularity ... More
Torsors on loop groups and the Hitchin fibrationAug 20 2019In his proof of the fundamental lemma, Ng\^o established the product formula for the Hitchin fibration over the anisotropic locus. One expects this formula over the larger generically regular semisimple locus, and we confirm this by deducing the relevant ... More
The WYSIWYG compactificationAug 20 2019We show that the partial compactification of a stratum of Abelian differentials previously considered by Mirzakhani and Wright is not an algebraic variety. Despite this, we use a combination of algebro-geometric and other methods to provide a short, unconditional ... More
Characteristic classes of symmetric and skew-symmetric degeneracy lociAug 20 2019We give two formulas for the Chern-Schwartz-MacPherson class of symmetric and skew-symmetric degeneracy loci. We apply them in enumerative geometry, explore their algebraic combinatorics, and discuss K theory generalizations.
On automorphisms and the cone conjecture for Enriques surfaces in odd characteristicAug 20 2019We prove that, for an Enriques surface in odd characteristic, the automorphism group is finitely generated and it acts on the effective nef cone with a rational polyhedral fundamental domain. We also construct a smooth projective surface in odd characteristic ... More
Rings of differentiable semialgebraic functionsAug 20 2019In this work we analyze the main properties of the Zariski and maximal spectra of the ring ${\mathcal S}^r(M)$ of differentiable semialgebraic functions of class ${\mathcal C}^r$ on a semialgebraic set $M\subset\mathbb{R}^m$. Denote ${\mathcal S}^0(M)$ ... More
Tropical geometryAug 19 2019Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build with these new ... More
Abundance for uniruled pairs which are not rationally connectedAug 19 2019Let $(X,\Delta)$ be a projective log canonical pair of dimension $n$ such that $X$ is uniruled. If $X$ is not rationally connected, then $(X,\Delta)$ has a good model, assuming the Minimal Model Program in dimension $n-1$. If $X$ is rationally connected, ... More
Dominant rational maps from a very general hypersurface in the projective spaceAug 19 2019In this paper we study dominant rational maps from a very general hypersurface $X$ of degree at least $n+3$ in the projective $(n+1)$-space ${\mathbb P}^{n+1}$ to smooth projective $n$-folds $Y$. Based on Lefschetz theory, Hodge theory, and Cayley-Bacharach ... More
Addition-deletion results for the minimal degree of logarithmic derivations of arrangementsAug 19 2019We study the change of the minimal degree of a logarithmic derivation of a hyperplane arrangement under the addition or the deletion of a hyperplane, and give a number of applications. In particular, starting with Ziegler's example of a pair of arrangements ... More
A technical remark on the Donaldson-Futaki invariant for Fano reductive group compactificationsAug 19 2019We present an elementary way of computing the Donaldson-Futaki invariant associated to a test-configuration of an anti-canonically polarized Fano reductive group compactification.
Real Lagrangians in Calabi-Yau ThreefoldsAug 19 2019We compute the mod $2$ cohomology groups of real Lagrangians in Calabi-Yau threefolds using well-behaved torus fibrations constructed by Gross. To do this we study a long exact sequence introduced by Casta\~{n}o-Bernard and Matessi, which relates the ... More
Irrationality and monodromy for cubic threefoldsAug 19 2019We show the cohomological monodromy for the universal family of smooth cubic threefolds does not factor through the genus five mapping class group. This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds.
Irrationality and monodromy for cubic threefoldsAug 19 2019Aug 21 2019We show the cohomological monodromy for the universal family of smooth cubic threefolds does not factor through the genus five mapping class group. This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds.
Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate ringsAug 18 2019The elliptic algebras in the title are connected graded $\mathbb{C}$-algebras, denoted $Q_{n,k}(E,\tau)$, depending on a pair of relatively prime integers $n>k\ge 1$, an elliptic curve $E$, and a point $\tau\in E$. For fixed $n$ and $k$, they form a flat ... More
Algebraic conditions for the positivity of sectional curvatureAug 18 2019We examine algebraic conditions for the sectional positivity of the Riemann curvature operator. We describe sufficient conditions for dimension $n=4$, and complete characterisation for a dense open subset of the space of operators in dimension $4$.
Algebraic conditions for the positivity of sectional curvatureAug 18 2019Aug 20 2019We examine algebraic conditions for the sectional positivity of the Riemann curvature operator. We describe sufficient conditions for dimension $n=4$, and complete characterization for a dense open subset of the space of operators in dimension $4$. We ... More
On the Chow group of the self-product of a CM elliptic curve defined over a number fieldAug 18 2019In this note we study the cokernel of the restriction map from the Chow group of codimension 2 cycles on the spread of the self product of a CM-elliptic curve over the ring of integers of a number field to the codimension 2 cycles on the self of product ... More
On projective manifolds with pseudo-effective tangent bundleAug 18 2019In this paper, we develop the theory of singular hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold $X$ with pseudo-effective tangent bundle: $X$ admits a smooth fibration $X \to Y$ to a flat projective ... More
The rational cuspidal divisor class group of $X_0(N)$Aug 18 2019For any positive integer $N$, we completely determine the structure of the rational cuspidal divisor class group $\mathcal{C}(N)$ of $X_0(N)$, which is conjecturally equal to the group of rational torsion points on $J_0(N)$. More specifically, let $\ell$ ... More
The crystalline comparison of Ainf-cohomology: the case of good reductionAug 18 2019We provide a simple approach for the crystalline comparison of Ainf-cohomology, and reprove the comparison between crystalline and p-adic etale cohomology for formal schemes in the case of good reduction.
Dolbeault cohomology of complex manifolds with torus actionAug 18 2019We describe the basic Dolbealut cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM and LVMB-manifolds and, in most generality, complex manifolds ... More
Uniform Bounds for Periods of Endomorphisms of VarietiesAug 17 2019Suppose $X$ is a projective variety defined over a finite extension $K$ of $\mathbb{Q}_p$ and suppose $X$ admits a model $\mathcal{X}$ defined over the ring of integers $R$ of $K$. Let $f:{X}\rightarrow {X}$ be an endomorphism of $X$ defined over $K$ ... More
An Octanomial Model for Cubic SurfacesAug 16 2019We present a new normal form for cubic surfaces that is well suited for p-adic geometry, as it reveals the intrinsic del Pezzo combinatorics of the 27 trees in the tropicalization. The new normal form is a polynomial with eight terms, written in moduli ... More
Higher Connectivity of TropicalizationsAug 16 2019We show that the tropicalization of an irreducible d-dimensional variety over a field of characteristic 0is (d-l)-connected through codimension one, where l is the dimension of the lineality space of the tropicalization. From this we obtain a higher connectivity ... More
Integral cohomology of quotients via toric geometryAug 16 2019We describe the integral cohomology of a compact complex manifold $X$ quotiented by a cyclic group $G$ of prime order with only isolated fixed points. As a preliminary step, we investigate the integral cohomology of toric blow-ups of quotients of $\mathbb{C}^n$. ... More
Yano's conjectureAug 16 2019We present a proof of a conjecture proposed by T. Yano about the generic $b$-exponents of irreducible plane curve singularities.
Geometric local $\varepsilon$-factors in higher dimensionsAug 16 2019We use former results on geometric local $\varepsilon$-factors over curves in order to prove a factorization result for the determinant of the cohomology of an $\ell$-adic sheaf over an arbitrary proper scheme over a perfect field of positive characteristic ... More
The Injective Spectrum of a Right Noetherian Ring II: Sheaves and Torsion TheoriesAug 16 2019This is the second of two papers on the injective spectrum of a right noetherian ring. In the prequel, we considered the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the ... More
The Injective Spectrum of a Right Noetherian Ring I: Injective Spectra and Krull DimensionAug 16 2019The injective spectrum is a topological space associated to a ring $R$, which agrees with the Zariski spectrum when $R$ is commutative noetherian. We consider injective spectra of right noetherian rings (and locally noetherian Grothendieck categories) ... More
Walls for $G$-Hilb via Reid's recipeAug 15 2019The three-dimensional McKay correspondence seeks to relate the geometry of crepant resolutions of Gorenstein 3-fold quotient singularities $\mathbb{A}^3/G$ with the representation theory of the group $G$. The first crepant resolution studied in depth ... More
The cubo-cubic transformation and K3 surfacesAug 15 2019In this note we observe that the Cremona transformation in Oguiso's example of Cremona isomorphic but not projectively equivalent quartic K3 surfaces in three-dimensional projective space is the classical cubo-cubic transformation.
The fundamental group of binoid varietiesAug 15 2019Binoid schemes generalise monoid schemes, which in turn enable us to generalise toric varieties. Let $X$ be a binoid scheme. The aim of this paper is to calculate the topological fundamental group of $KX$, where $K=\mathbb{C}$ or $\mathbb{R}$. For the ... More
Sign conditions for the existence of at least one positive solution of a sparse polynomial systemAug 15 2019We give sign conditions on the support and coefficients of a sparse system of d generalized polynomials in d variables that guarantee the existence of at least one positive real root, based on degree theory and Gale duality. In the case of integer exponents, ... More
Tensor Operations on Group SchemesAug 15 2019In this paper we study multilinear morphisms between commutative group schemes and the associated tensor constructions. We will also do some explicit calculations and give examples that show that this theory behaves in a way that one would naturally expect. ... More
On Certain Morphisms between Flag VarietiesAug 15 2019The aim of this paper is to construct certain closed embeddings of Grassmannian varieties, using tensor operations on vector bundles. These embeddings generalize Segre and Pl\"ucker morphisms.
A toy model for the Drinfeld-Lafforgue shtuka constructionAug 15 2019The goal of this paper is to provide a categorical framework that leads to the definition of shtukas \`a la Drinfeld and of excursion operators a la V. Lafforgue. We take as the point of departure the Hecke action of Rep(G^L) on the category Shv(Bun_G) ... More
Probabilistic Saturations and Alt's ProblemAug 15 2019Alt's problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which ... More
15-nodal quartic surfaces. Part II: The automorphism groupAug 15 2019We describe a set of generators and defining relations for the group of birational automorphism group of a general 15-nodal quartic surface in the complex projective space.
Families of stable bundles on the fibres of the hyperkähler twistor projectionAug 14 2019Given a holomorphic vector bundle $E$ on the twistor space $\mathrm{Tw}(M)$ of a simple hyperk\"ahler manifold $M$, we view it as a family of bundles $\left\{E_I\right\}$ on the fibres $\pi^{-1}(I)$ of the twistor projection $\pi : \mathrm{Tw}(M) \to ... More
Deformations and BBF form on non-Kahler holomorphically symplectic manifoldsAug 14 2019In 1995, Dan Guan constructed examples of non-Kahler, simply-connected holomorphically symplectic manifolds. An alternative construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. We investigate topology and deformation ... More
An Automorphic Classification of Real Cubic CurvesAug 14 2019The action of ring automorphisms of the polynomial ring in two variables over the real numbers on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being selected.
Torus fibers and the weight filtrationAug 14 2019We show that if $(X,Y)$ is a simple normal crossings log Calabi--Yau pair, then there is a real torus of dimension equal to the codimension of the smallest stratum of $Y$ which can be used to construct $W_{2k-1}H^k(X \setminus Y;\mathbb{Q})$ for all $k$. ... More
Local and global applications of the Minimal Model Program for co-rank one foliations on threefoldsAug 14 2019We show that the Minimal Model Program for co-rank one foliations on threefolds terminates by proving foliation flips terminate. Moreover, we recover a full suite of powerful results on the birational structure of co-rank one foliations on threefolds ... More
Obstructed stable sheaves on elliptic surfaces -- Canonical singularities or notAug 14 2019Let $X$ be an elliptic surface over $P^1$ with $\kappa(X)=1$, and $M$ be the moduli scheme of rank-two stable sheaves on $X$ with $c_1=0$. We look into defining equations of $M$ at its singularity $E$. When the restriction of $E_{\eta}$ to the generic ... More
The Log Product FormulaAug 14 2019We prove a formula expressing the Log Gromov-Witten Invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of $V$ and $W$. This extends results of F. Qu and Y.P. Lee, who introduced this formula analogously to K. Behrend. ... More
Rigidity of Bott-Samelson-Demazure-Hansen variety for $F_4$ and $G_2$Aug 13 2019Let $G$ be a simple algebraic group of adjoint type over $\mathbb{C},$ whose root system is of type $F_{4}.$ Let $T$ be a maximal torus of $G$ and $B$ be a Borel subgroup of $G$ containing $T.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the ... More
Rigidity of Bott-Samelson-Demazure-Hansen variety for $PSO(2n+1, \mathbb{C})$Aug 13 2019Let $G=PSO(2n+1, \mathbb{C}) (n \ge 3)$ and $B$ be the Borel subgroup of $G$ containing maximal torus $T$ of $G.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ Let $Z(w, \underline{i})$ ... More
Hilbert-Kunz Multiplicity of Fibers and Bertini TheoremsAug 13 2019Let $k$ be an algebraically closed field of characteristic $p > 0$. We show that if $X\subseteq\mathbb{P}^n_k$ is an equidimensional subscheme with Hilbert--Kunz multiplicity less than $\lambda$ at all points $x\in X$, then for a general hyperplane $H\subseteq\mathbb{P}^n_k$, ... More
$p$-adic Integral GeometryAug 13 2019We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective set (reproving ... More
Parabolic subgroups and Automorphism groups of Schubert varietiesAug 13 2019Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $w$ be an element of the Weyl group $W$ and $X(w)$ be the Schubert variety in ... More
Gromov-Witten theory with maximal contactsAug 13 2019Let $X$ be a variety equipped with a normal crossings collection of hyperplane sections $D$. We prove that the genus zero, maximal contact logarithmic Gromov-Witten theory of the pair $(X,D)$ is equal up to a multiplicity to the local theory of the total ... More
A remark on algebraic cycles on cubic fourfoldsAug 13 2019In this short note we try to generalize the Clemens-Griffiths criterion of non-rationality for smooth cubic threefolds to the case of smooth cubic fourfolds.
Coarse density of subsets of $M_g$Aug 13 2019Let $\mathcal{M}_g$ be the moduli space of genus $g$ Riemann surfaces. We show that an algebraic subvariety of $\mathcal{M}_g$ is coarsely dense with respect to the Teichm\"uller metric (or Thurston metric) if and only if it is all of $\mathcal{M}_g$. ... More
On the log-local principle for the toric boundaryAug 12 2019Let $X$ be a complex projective variety and let $D=D_1+\cdots+D_l$ be a $\mathbb{Q}$-Cartier divisor with each $D_j$ irreducible and nef. The log-local principle of van Garrel-Graber-Ruddat conjectures that the genus 0 log Gromov-Witten theory of maximal ... More
RationalMaps, a package for Macaulay2Aug 12 2019This paper describes the RationalMaps package for Macaulay2. This package provides functionality for computing several aspects of rational maps such as whether a map is birational, or a closed embedding.
Tropically planar graphsAug 12 2019We study tropically planar graphs, which are the graphs that appear in smooth tropical plane curves. We develop necessary conditions for graphs to be tropically planar, and compute the number of tropically planar graphs up to genus $7$. We provide non-trivial ... More
The Dual Complex of a semi-log canonical SurfaceAug 12 2019Semi-log canonical varieties are a higher-dimensional analogue of stable curves. They are the varieties appearing as the boundary $\Delta$ of a log canonical pair $(X,\Delta)$, and also appear as limits of canonically polarized varieties in moduli theory. ... More
Integration-by-parts reductions of Feynman integrals using Singular and GPI-SpaceAug 12 2019We introduce an algebro-geometrically motived integration-by-parts (IBP) reduction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the computer ... More
Concomitants of Ternary Quartics and Vector-valued Siegel and Teichmüller Modular Forms of Genus ThreeAug 12 2019We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichm\"uller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double ... More
A class of perverse schobers in Geometric Invariant TheoryAug 12 2019Perverse schobers are categorifications of perverse sheaves. We construct a perverse schober on a partial compactification of the stringy K\"ahler moduli space (SKMS) associated by Halpern-Leistner and Sam to a quasi-symmetric representation X of a reductive ... More
Positivity Certificates via Integral RepresentationsAug 12 2019Complete monotonicity is a strong positivity property for real-valued functions on convex cones. It is certified by the kernel of the inverse Laplace transform. We study this for negative powers of hyperbolic polynomials. Here the certificate is the Riesz ... More
The elliptic Grothendieck-Springer resolution as a simultaneous log resolution of algebraic stacksAug 12 2019We construct an elliptic Grothendieck-Springer resolution as a simultaneous log resolution of algebraic stacks. Our construction extends a well-known simultaneous resolution of the coarse moduli space map for semistable principal bundles on an elliptic ... More
Special components of Noether-Lefschetz lociAug 12 2019We take a sum $C_1+r C_2,\ r\in\mathbb Q$ of a line $C_1$ and a complete intersection curve $C_2$ of type $(3,3)$ inside a smooth surface of degree $8$ and with $C_1\cap C_2=\emptyset$. We gather evidences to the fact that for all except a finite number ... More
Large automorphism groups of ordinary curves of even genus in odd characteristicAug 11 2019Let $\mathcal{X}$ be a (projective, non-singular, geometrically irreducible) curve of even genus $g(\mathcal{X}) \geq 2$ defined over an algebraically closed field $K$ of odd characteristic $p$. If the $p$-rank $\gamma(\mathcal{X})$ equals $g(\mathcal{X})$, ... More
Poisson--Kähler fibration I: curvature of the base manifoldAug 11 2019We start from a finite dimensional Higgs bundle description of a result of Burns on negative curvature property of the space of complex structures, then we apply the corresponding infinite dimensional Higgs bundle picture and obtain a precise curvature ... More
Schemes supported on the singular locus of a hyperplane arrangement in $\mathbb P^n$Aug 11 2019We introduce the use of liaison addition to the study of hyperplane arrangements. For an arrangement, $\mathcal A$, of hyperplanes in $\mathbb P^n$, $\mathcal A$ is free if $R/J$ is Cohen-Macaulay, where $J$ is the Jacobian ideal of $\mathcal A$. Terao's ... More
Log canonical thresholds of generic links of determinantal varietiesAug 11 2019We show that the log canonical threshold of a generic determinantal variety and its generic link are the same.
On the Universal ellipsitomic KZB connectionAug 11 2019We construct a twisted version of the genus one universal Knizhnik--Zamolodchikov--Bernard (KZB) connection introduced by Calaque--Enriquez--Etingof, that we call the ellipsitomic KZB connection. This is a flat connection on a principal bundle over the ... More
The unit map of the algebraic special linear cobordism spectrumAug 11 2019In joint work with Elmanto, Hoyois, Khan and Sosnilo, we computed infinite $\mathbb{P}^1$-loop spaces of motivic Thom spectra, using the technique of framed correspondences. This result allows us to express non-negative $\mathbb{G}_m$-homotopy groups ... More
Convex Algebraic Geometry of Curvature OperatorsAug 10 2019We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. More precisely, we determine in which dimensions $n$ this convex semialgebraic ... More
Finite generation and holomorphic anomaly equation for equivariant Gromov-Witten invariants of $K_{\mathbb{P}^1\times\mathbb{P}^1}$Aug 10 2019In this paper, we prove finite generation property and holomorphic anomaly equation for the equivariant Gromov-Witten theory of $K_{\mathbb{P}^1\times\mathbb{P}^1}$.
Higher signs for Coxeter groupsAug 10 2019We define and study cocycles on a Coxeter group in each degree generalizing the sign function. When the Coxeter group is a Weyl group, we explain how the degree three cocycle arises naturally from geometry representation theory.
Symmetry, Unimodality, and Lefschetz Properties for Graded ModulesAug 09 2019If $\mbk$ is algebraically closed of characteristic zero and $R = \mbk[x,y, z]$, we first investigate the Weak Lefschetz Property for the finite length $R$-module $M$ that is the cokernel of a map $\vp: \bds_{j=1}^{n+2} R(-b_j)\to\bds_{i=1}^n R(-a_i)$. ... More
On spin structures and orientations for gauge-theoretic moduli spacesAug 09 2019Let $X$ be a compact manifold, $G$ a Lie group, $P \to X$ a principal $G$-bundle, and $\mathcal{B}_P$ the infinite-dimensional moduli space of connections on $P$ modulo gauge. For a real elliptic operator $E_\bullet$ we previously studied orientations ... More
Computing zero-dimensional tropical varieties via projectionsAug 09 2019We present an algorithm for computing zero-dimensional tropical varieties using projections. Our main tools are fast unimodular transforms of lexicographical Gr\"obner bases. We prove that our algorithm requires only a polynomial number of arithmetic ... More
On Double Danielewski Surfaces and the Cancellation ProblemAug 09 2019We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms of these surfaces. ... More
Virtual intersection theoriesAug 09 2019We construct virtual fundamental classes in all intersection theories including Chow, K and algebraic cobordism for quasi-projective Deligne-Mumford stacks with perfect obstruction theories and prove the virtual pullback formula, the virtual torus localization ... More
Fourier-Mukai partners of abelian varietiesAug 09 2019We will discuss the Fourier-Mukai partners of a given abelian variety. The first part of the note is to give some basic theory of Fourier-Mukai partners and semi-homogenous vector bundles, then we will discuss the case when the kernel of an equivalence ... More
Fourier-Mukai partners of abelian varietiesAug 09 2019Aug 12 2019We will discuss the Fourier-Mukai partners of a given abelian variety. The first part of the note is to give some basic theory of Fourier-Mukai partners and semi-homogenous vector bundles, then we will discuss the case when the kernel of an equivalence ... More
On reconstructing subvarieties from their periodsAug 08 2019Let X be a smooth hypersurface of dimension n and Y a subvariety of dimension n/2. We give an algorithm which takes the periods of Y and returns an ideal. If Y is a complete intersection in the ambient space of X then we show that low degree equations ... More
Conics associated with totally degenerate curvesAug 08 2019Let $k$ be a field. Let $X/k$ be a stable curve whose geometric irreducible components are smooth rational curves. Taking Stein factorization of its normalization, we get a conic. We show the conic is non-split in certain cases. As an application, we ... More
Closed points on cubic hypersurfacesAug 08 2019We generalize some results of Coray on closed points on cubic hypersurfaces. We show certain symmetric products of cubic hypersurfaces are stably birational.
Some properties of a Brauer classAug 08 2019Let $X$ be a smooth proper curve defined over a field $k$. The representability of the relative Picard functor is obstructed by a class $\alpha\in\mathrm{Br}(\mathrm{Pic}_{X/k})$. We show the associated division algebra on $\mathrm{Pic}^0_{X/k}$ has natural ... More