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Lagrangian pairs of pantsFeb 08 2018We construct a Lagrangian submanifold, inside the cotangent bundle of a real torus, which we call a Lagrangian pair of pants. It is given as the graph of the differential of a smooth function defined on the real blow up of a Lagrangian coamoeba. Lagrangian ... More
Graphes associés au groupe de CremonaFeb 08 2018To reinforce the analogy between the mapping class group and the Cremona group of rank 2 over an algebraic closed field, we look for a graph analoguous to the curve graph and such that the Cremona group acts on it non-trivially. The first candidate is ... More
Local Picard Group of Pointed Monoids and Their AlgebrasFeb 08 2018The main goal of this paper is to give an explicit formula for the cohomology of the sheaf of units of the punctured spectrum of a Stanley-Reisner ring. In particular we compute the local Picard group of $K [\triangle]$. To achieve this we study the corresponding ... More
Degree bound of Pólya PositivstellenstazFeb 08 2018P\'olya's Positivstellensatz on the $1$-simplex says that if $P(x)$ is a real polynomial such that $P(x)>0$ whenever $x \ge 0$, then all the coefficients of $(1+x)^mP(x)$ are positive whenever $m$ is large. Powers-Reznick gave a complexity estimate for ... More
A new presentation of the plane Cremona groupFeb 08 2018We give a presentation of the plane Cremona group over an algebraically closed field with respect to the generators given by the Theorem of Noether and Castelnuovo. This presentation is particularly simple and can be used for explicit calculations.
Frobenius powersFeb 08 2018This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and multiplier ideals ... More
Nash multiplicity sequences and Hironaka's order functionFeb 07 2018When $X$ is a $d$-dimensional variety defined over a field $k$ of characteristic zero, a constructive resolution can be achieved by successively lowering the maximum multiplicity via blow ups at smooth equimultiple centers. This is done by strarifying ... More
The monodromy of meromorphic projective structuresFeb 07 2018We study projective structures on a surface having poles of prescribed orders. We obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed $\mathrm{PGL}_2(\mathbb{C})$ local systems on the associated marked ... More
Ordinary $GL_2(F)$-representations in characteristic two via affine Deligne-Lusztig constructionsFeb 07 2018The group $\GL_2$ over a local field with (residue) characteristic $2$ possesses much more smooth supercuspidal $\ell$-adic representations, than over a local field of residue characteristic $> 2$. One way to construct these representations is via the ... More
Field extensions, Derivations, and Matroids over Skew HyperfieldsFeb 07 2018We show that a field extension $K\subseteq L$ in positive characteristic $p$ and elements $x_e\in L$ for $e\in E$ gives rise to a matroid $M^\sigma$ on ground set $E$ with coefficients in a certain skew hyperfield $L^\sigma$. This skew hyperfield $L^\sigma$ ... More
Tropicalized quartics and canonical embeddings for tropical curves of genus 3Feb 07 2018Brodsky, Joswig, Morrison and Sturmfels showed that not all abstract tropical curves of genus $3$ can be realized as a tropicalization of a quartic in the euclidean plane. In this article, we focus on the interior of the maximal cones in the moduli space ... More
$\infty$-topoi and Natural Phenomena: GenerationFeb 07 2018We show that the Segal topos of derived stacks over simplicial commutative $k$-algebras, which can be used to model natural phenomena, has a subobject classifier, something we regard as being a source from which dynamics is generated. This is done by ... More
Monodromy and Log GeometryFeb 06 2018A now classical construction due to Kato and Nakayama attaches a topological space (the "Betti realization") to a log scheme over $\mathbf{C}$. We show that in the case of a log smooth degeneration over the standard log disc, this construction allows ... More
Stratifications of affine Deligne-Lusztig varietiesFeb 06 2018Affine Deligne-Lusztig varieties are analogues of Deligne-Lusztig varieties in the context of affine flag varieties and affine Grassmannians. They are closely related to moduli spaces of $p$-divisible groups in positive characteristic, and thus to arithmetic ... More
Local to global principle for the moduli space of K3 surfacesFeb 06 2018Recently S. Patrikis, J.F. Voloch and Y. Zarhin have proven, assuming several well known conjectures, that the finite descent obstruction holds on the moduli space of principally polarised abelian varieties. We show an analogous result for K3 surfaces, ... More
On some local cohomology spectral sequencesFeb 06 2018We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The first type of spectral sequences involves the left derived functors of ... More
Automorphism group of a moduli space of framed bundles over a curveFeb 06 2018Let $X$ be a smooth complex projective curve of genus $g > 2$, and let $x\in X$ be a point. We compute the automorphism group of the moduli space of framed vector bundles on $X$ with a framing over $x$. It is shown that this group is generated by pullbacks ... More
Isotropic Subspaces of Schur ModulesFeb 06 2018It is a well-known fact that over the complex numbers and for a fixed $k$ and $n$, a generic $s$ in $Sym^2V^*$ vanishes on some $k$-dimensional subspace of $V$ if and only if $n\geq 2k$. Tevelev found exact conditions for the extension of this statement ... More
On residue maps for affine curvesFeb 05 2018We establish several compatibility results between residue maps in \'etale and Galois cohomology that arise naturally in the analysis of smooth affine algebraic curves having good reduction over discretely valued fields. These results are needed, and ... More
Motivic Chern classes and K-theoretic stable envelopesFeb 05 2018We study a K-theoretic characteristic class of singular varieties, namely the equivariant motivic Chern class. We prove that the motivic Chern class satisfies the axiom system of "K-theoretic stable envelopes," recently defined by Okounkov and studied ... More
Fixed point theorems involving numerical invariantsFeb 05 2018We exhibit invariants of smooth projective algebraic varieties with integer values, whose nonvanishing modulo p prevents the existence of an action without fixed points of certain finite p-groups. The case of base fields of characteristic p is included. ... More
Cuspidal cohomology of stacks of shtukasFeb 05 2018Let $G$ be a connected split reductive group over a finite field ${\mathbb F}_q$. The $\ell$-adic cohomology of stacks of $G$-shtukas is a generalisation of the space of automorphic forms over a function field. In this paper, we construct a constant term ... More
Moduli of non-standard Nikulin surfaces in low genusFeb 04 2018Primitively polarized genus $g$ Nikulin surfaces $(S,M,H)$ are of two types, that we call standard and non-standard depending on whether the lattice embedding $\mathbb{Z}[H] \oplus_{\perp} \mathbf{N} \subset \mathrm{Pic} (S)$ is primitive. Here $H$ is ... More
Lifting problem for linearly reductive torsors of curvesFeb 04 2018Let $U$ be a smooth relative curve over a complete discrete valuation ring $R$ with algebraically closed residue field $k$ of characteristic $p>0$ which admits a smooth compactification $X$ with $D=X\setminus U$ \'etale over $R$. Let $U_0$ denote its ... More
On direct images of twisted pluricanonical sheaves on normal varietiesFeb 03 2018Let $f: Y\rightarrow X$ be a proper morphism from a smooth variety onto a normal variety and $d$ the relative dimension of the map. Suppose the preimage $E$ of the singular locus of $X$ is a divisor. We show that for any $m>0$, modulo the torsion part, ... More
New example of modified moduli space of special Bohr - Sommerfeld lagrangian submanifoldsFeb 02 2018We present an example of modified moduli space of special Bohr - Sommerfeld lagrangian submanifolds for the case when the given algebraic variety is the full flag $F^3$ for $\mathbb{C}^3$ and the very ample bundle is $K^{- \frac{1}{2}}_{F^3}$
The finiteness of the genus of a finite-dimensional division algebra, and some generalizationsFeb 01 2018We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the ... More
Moduli of Stokes Torsors and Singularities of Differential EquationsFeb 01 2018Let M be a meromorphic connection with poles along a smooth divisor D in a smooth algebraic variety. Let Sol M be the solution complex of M. We prove that the good formal decomposition locus of M coincides with the locus where the restrictions to D of ... More
Invariant Whitney FunctionsFeb 01 2018A theorem of Gerald Schwarz [24, Thm. 1] says that for a linear action of a compact Lie group $G$ on a finite dimensional real vector space $V$ any smooth $G$-invariant function on $V$ can be written as a composite with the Hilbert map. We prove a similar ... More
Optimal Configurations in Coverage Control with Polynomial CostsJan 31 2018We revisit the static coverage control problem for placement of vehicles with simple motion on the real line, under the assumption that the cost is a polynomial function of the locations of the vehicles. The main contribution of this paper is to demonstrate ... More
Motivic Hodge modulesJan 30 2018We construct a quasi-categorically enhanced Grothendieck six-functor formalism on schemes of finite type over the complex numbers. In addition to satisfying many of the same properties as M. Saito's derived categories of mixed Hodge modules, this new ... More
The role of Coulomb branches in 2D gauge theoryJan 30 2018I give a simple construction of certain Coulomb branches $C_{3,4}(G;E)$ of gauge theory in 3 and 4 dimensions defined by Nakajima et al. for a compact Lie group $G$ and a polarisable quaternionic representation $E$. The manifolds $C(G; 0)$ are abelian ... More
New characterizations of freeness for hyperplane arrangementsJan 30 2018In this article we describe two new characterizations of freeness for hyperplane arrangements via the study of the generic initial ideal and of the sectional matrix of the Jacobian ideal of arrangements.
The Kähler geometry of Bott manifoldsJan 29 2018We study the K\"ahler geometry of stage n Bott manifolds, which can be viewed as $n$-dimensional generalizations of Hirzebruch surfaces. We show, using a simple induction argument and the generalized Calabi construction from [ACGT04,ACGT11], that any ... More
Purity of Crystalline StrataJan 29 2018Jan 30 2018Let $p$ be a prime. Let $n\in\mathbb N-\{0\}$. Let $\mathcal C$ be an $F^n$-crystal over a locally noetherian $\mathbb F_p$-scheme $S$. Let $(a,b)\in\mathbb N^2$. We show that the reduced locally closed subscheme of $S$ whose points are exactly those ... More
Matrix product states and the quantum max-flow/min-cut conjecturesJan 27 2018In this note we discuss the geometry of matrix product states with periodic boundary conditions and provide three infinite sequences of examples where the quantum max-flow is strictly less than the quantum min-cut. In the first we fix the underlying graph ... More
Affine Schubert calculus and double coinvariantsJan 27 2018We first define an action of the double coinvariant algebra $DR_n$ on the homology of the affine flag variety $\widetilde{Fl}_n$ in type $A$, and use affine Schubert calculus to prove that it preserves the image of the homology of the rational $(n,m)$-affine ... More
The totally nonnegative part of G/P is a ballJan 26 2018We show that the totally nonnegative part of a partial flag variety (in the sense of Lusztig) is homeomorphic to a closed ball.
Solvable Leibniz algebras with quasi-filiform Lie algebras of maximum length nilradicalsJan 26 2018In this paper solvable Leibniz algebras whose nilradical is quasi-filiform Lie algebra of maximum length, are classified. The rigidity of such Leibniz algebras with two-dimensional complemented space to nilradical is proved.
Connectedness of The Moduli Space of Artin-Schreier Curves of Fixed GenusJan 25 2018We study the moduli space $\mathcal{AS}_{g}$ of Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of positive characteristic $p$. The moduli space is partitioned by irreducible strata, where each stratum parameterizes Artin-Schreier ... More
Exotic components of $\mathrm{SO}(p,q)$ surface group representations, and their Higgs bundle avatarsJan 25 2018For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin ... More
Affine Grassmannians in A^1-algebraic topologyJan 25 2018Let k be a field. Denote by Spc(k)_* the unstable, pointed motivic homotopy category and by Omega_Gm: Spc(k)_* \to Spc(k)_* the Gm-loops functor. For a k-group G, denote by Gr_G the affine Grassmannian of G. If k is infinite and G is split reductive, ... More
Chern classes of automorphic vector bundles, IIJan 24 2018We prove that the $\ell$-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over $\bar{ \mathbb{Q}}_p$, descend to classes in the $\ell$-adic cohomology of the minimal ... More
A remark on uniform boundedness for Brauer groupsJan 22 2018The Tate conjecture for divisors on varieties over number fields is equivalent to finiteness of $\ell$-primary torsion in the Brauer group. We show that this finiteness is actually uniform in one-dimensional families for varieties that satisfy the Tate ... More
Estimates on volumes of homogeneous polynomial spacesJan 22 2018Jan 26 2018In this paper we develop the "local part" of our local/global approach to globally valued fields (GVFs). The "global part", which relies on these results, is developed in a subsequent paper.We study virtual divisors on projective varieties defined over ... More
Schubert Decomposition for Milnor Fibers of the Varieties of Singular MatricesJan 21 2018We consider the varieties of singular $m \times m$ complex matrices which may be either general, symmetric or skew-symmetric (with $m$ even). For these varieties we have shown in another paper that they had compact "model submanifolds", for the homotopy ... More
Cohomology of $p$-adic Stein spacesJan 20 2018We compute the $p$-adic \'etale and the pro-\'etale cohomologies of the Drinfeld half-space of any dimension. The main input is a new comparison theorem for the $p$-adic pro-\'etale cohomology of $p$-adic Stein spaces.
A method for construction of rational points over elliptic curves II: Points over solvable extensionsJan 18 2018I provide a systematic construction of points, defined over finite radical extensions, on any Legendre curve over any field of characteristic not equal two. This includes as special case Douglas Ulmer's construction of rational points over a rational ... More
Wildly Compatible Systems and Six OperationsJan 18 2018Let $S$ be an excellent regular scheme and let $X$ be a scheme separated and of finite type over $S$. Let $K_c(X, \mathbb{F}_{\lambda})$ be the Grothendieck ring of $\mathbb{F}_{\lambda}$-constructible sheaves on $X$, where $\mathbb{F}_{\lambda}$ is the ... More
Real-analytic Eisenstein series via the Poincaré bundleJan 17 2018A classical construction of Katz gives a purely algebraic construction of real-analytic Eisenstein series using the Gau\ss--Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and $p$-adic properties ... More
Stably irrational hypersurfaces of small slopesJan 16 2018We show that a very general complex projective hypersurface of dimension N and degree at least $ \lceil \log_2N \rceil+2$ is not stably rational. The same statement holds over any uncountable field of characteristic p>>N. This significantly improves earlier ... More
Ranks and Symmetric Ranks of Cubic SurfacesJan 16 2018We study cubic surfaces as symmetric tensors of format $4 \times 4 \times 4$. We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The corresponding ... More
Triviality properties of principal bundles on singular curves-IIJan 15 2018For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show, by constructing ... More
The adic tame siteJan 15 2018Jan 19 2018For every adic space $Z$ we construct a site $Z_t$, the tame site of $Z$. For a scheme $X$ over a base scheme $S$ we obtain a tame site by associating with $X/S$ an adic space $\textit{Spa}(X,S)$ and considering the tame site $\textit{Spa}(X,S)_t$. We ... More
A case of the generalized vanishing conjectureJan 14 2018In this paper, we show that the GVC (generalized vanishing conjecture) holds for the differential operator $\Lambda=(\partial_x-\Phi(\partial_y))\partial_y$ and all polynomials $P(x,y)$, where $\Phi(t)$ is any polynomial over the base field. The GVC arose ... More
Stability in the homology of Deligne-Mumford compactificationsJan 11 2018Using the the theory of FS^op modules, we study the asymptotic behavior of the homology of $\overline M_{g,n}$, the Deligne--Mumford compactification of the moduli space of curves, for $n >> 0$. An FS^op module is a contravariant functor from the category ... More
Reducible characteristic cycles of Harish-Chandra modules for $\mathrm{U}(p,q)$ and the Kashiwara-Saito singularityJan 10 2018We give examples of reducible characteristic cycles for irreducible Harish-Chandra modules for $\mathrm{U}(p,q)$ by analyzing a four-dimensional singular subvariety of $\mathbb{C}^8$. We relate this singularity to the Kashiwara-Saito singularity arising ... More
On singularity properties of convolutions of algebraic morphismsJan 09 2018Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth $K$-varieties, and let $V$ be a finite dimensional $K$-vector space. For two algebraic morphisms $\varphi:X\rightarrow V$ and $\psi:Y\rightarrow V$ we define a convolution operation, $\varphi*\psi:X\times ... More
Arithmetic surfaces and adelic quotient groupsJan 08 2018We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fiber over the infinite point of the base is taken into account. The calculations are presented via a short exact sequence. We relate ... More
Algebraic curves $A^{\circ l}(x)-U(y)=0$ and arithmetic of orbits of rational functionsJan 06 2018We give a description of pairs of complex rational functions $A$ and $U$ of degree at least two such that for every $d\geq 1$ the algebraic curve $A^{\circ d}(x)-U(y)=0$ has a factor of genus zero or one. In particular, we show that if $A$ is not a "generalized ... More
On a connectedness principle of Shokurov-Kollár typeJan 05 2018Let $(X,\Delta)$ be a log pair over $S$, such that $-(K_X+\Delta)$ is nef over $S$. It is conjectured that the intersection of the nonklt locus of $(X,\Delta)$ with any fiber $X_s$ has at most two connected components. We prove this conjecture in dimension ... More
Betti numbers for numerical semigroup ringsDec 30 2017We survey results related to the magnitude of the Betti numbers of numerical semigroup rings and of their tangent cones.
Birational geometry of singular Fano hypersurfaces of index twoDec 23 2017For a Zariski general (regular) hypersurface $V$ of degree $M$ in the $(M+1)$-dimensional projective space, where $M$ is at least 16, with at most quadratic singularities of rank at least 13, we give a complete description of the structures of rationally ... More
Embedded desingularization for arithmetic surfaces -- toward a parallel implementationDec 21 2017We present an approach for algorithmic embedded desingularization of arithmetic surfaces bearing in mind implementability. Our algorithm is based on work by Cossart-Jannsen-Saito, though we present a variant using a refinement of the order instead of ... More
Mixed Hodge structures with modulusDec 20 2017We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of ... More
Real Space Sextics and their TritangentsDec 18 2017The intersection of a quadric and a cubic surface in 3-space is a canonical curve of genus 4. It has 120 complex tritangent planes. We present algorithms for computing real tritangents, and we study the associated discriminants. We focus on space sextics ... More
Constructible 1-motives and exactnessDec 04 2017We prove that arbitrary pullbacks, as well as Betti and \'etale realisation functors, are t-exact for the constructible motivic t-structure on the category of cohomological 1-motives over a base scheme.
Grothendieck's homotopy theory, polynomial monads and delooping of spaces of long knotsDec 04 2017We extend some classical results - such as Quillen's Theorem A, the Grothendieck construction, and the characterisation of homotopically cofinal functors - from the homotopy theory of small categories to polynomial monads and their algebras. As an application ... More
On the Voevodsky motive of the moduli stack of vector bundles on a curveNov 29 2017We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky's category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective ... More
A Saito criterion for holonomic divisorsNov 28 2017We prove a variant of Saito's freeness criterion for holonomic divisors. It applies in particular to complex hyperplane arrangements.
Alternating minimization, scaling algorithms, and the null-cone problem from invariant theoryNov 21 2017Alternating minimization heuristics seek to solve a (difficult) global optimization task through iteratively solving a sequence of (much easier) local optimization tasks on different parts (or blocks) of the input parameters. While popular and widely ... More
The indeterminacy locus of the Voisin mapNov 16 2017Feb 06 2018Beauville and Donagi proved that the variety of lines $F(Y)$ of a smooth cubic fourfold $Y$ is a hyperK\"ahler variety. Recently, C. Lehn, M.Lehn, Sorger and van Straten proved that one can naturally associate a hyperK\"ahler variety $Z(Y)$ to the variety ... More
Solving Polynomial Systems via a Stabilized Representation of Quotient AlgebrasNov 13 2017We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to compute the structure ... More
Homogeneous locally nilpotent derivations of non-factorial trinomial algebrasOct 29 2017We describe homogeneous locally nilpotent derivations of the algebra of regular functions for a class of affine trinomial hypersurfaces. This class comprises all non-factorial trinomial hypersurfaces.
Improved Complexity Bounds for Counting Points on Hyperelliptic CurvesOct 10 2017We present a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus $g$ defined over $\mathbb{F}_q$. It is based on the approaches by Schoof and Pila combined with a modeling of the $\ell$-torsion by ... More
The Kähler Quotient Resolution of $\mathbb{C}^3/Γ$ singularities, the McKay correspondence and D=3 $\mathcal{N}=2$ Chern-Simons gauge theoriesOct 03 2017Nov 27 2017We advocate that a generalized Kronheimer construction of the K\"ahler quotient crepant resolution $\mathcal{M}_\zeta \longrightarrow \mathbb{C}^3/\Gamma$ of an orbifold singularity where $\Gamma\subset \mathrm{SU(3)}$ is a finite subgroup naturally defines ... More
Decomposition of degenerate Gromov-Witten invariantsSep 28 2017Dec 22 2017We prove a decomposition formula of logarithmic Gromov-Witten invariants in a degeneration setting. A one-parameter log smooth family X->B with singular fibre over b_0 \in B yields a family M(X/B,\beta) -> B of moduli stacks of stable logarithmic maps. ... More
Automorphisms of Danielewski varietiesSep 26 2017In 2007, Dubouloz introduced Danielewski varieties. Such varieties generalize Danielewski surfaces and provide counterexamples to generalized Zariski cancellation problem in arbitrary dimension. In the present work we describe the automorphism group of ... More
A Stabilized Normal Form Algorithm for Generic Systems of Polynomial EquationsAug 25 2017We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring $\mathbb{C}[x]/I$ associated to a zero-dimensional ideal $I$ generated by $n$ $\mathbb{C}$-polynomials in $n$ variables. We assume that the polynomials ... More
When is the heart of a t-structure a Grothendieck category?Aug 24 2017Let $\mathcal D$ be a triangulated category endowed with a $t$-structure $\mathfrak t=(\mathcal U,\Sigma \mathcal V)$ and denote by $\mathcal H:=\mathcal U\cap \Sigma\mathcal V$ its heart. In this paper we study the following well-known problem: Under ... More
Prism tableaux for alternating sign matrix varietiesAug 24 2017A prism tableau is a set of reverse semistandard tableaux, each positioned within an ambient grid. Prism tableaux were introduced to provide a formula for the Schubert polynomials of A. Lascoux and M.P. Sch\"utzenberger. This formula directly generalizes ... More
On the stable dynamical spectrum of complex surfacesAug 21 2017Aug 23 2017We characterize Salem numbers which have some power arising as dynamical degree of an automorphism on a complex (projective) 2-Torus, K3 or Enriques surface.
The $K(π, 1)$-problem for restrictions of complex reflection arrangementsAug 17 2017In 1962, Fadell and Neuwirth showed that the configuration space of the braid arrangement is aspherical. Having generalized this to many real reflection groups, Brieskorn conjectured this for all Coxeter groups. This follows from Deligne's seminal work ... More
Partial local resolution by characteristic zero methodsAug 16 2017Jan 19 2018We discuss to what extent the local techniques of resolution of singularities over fields of characteristic zero can be applied to improve singularities in general. For certain interesting classes of singularities, this leads to an embedded resolution ... More
On the dynamics of the Pappus-Steiner mapAug 14 2017We extract a two-dimensional dynamical system from the theorems of Pappus and Steiner in classical projective geometry. We calculate an explicit formula for this system, and study its elementary geometric properties. Then we use Artin reciprocity to characterise ... More
Schottky Algorithms: Classical meets TropicalJul 26 2017We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding. We offer solutions ... More
Hitchin and Calabi-Yau integrable systems via variations of Hodge structuresJul 19 2017A complex integrable system determines a family of complex tori over a Zariski-open and dense subset in its base. This family in turn yields an integral variation of Hodge structures of weight $\pm 1$. In this paper, we study the converse of this procedure. ... More
Changing Views on Curves and SurfacesJul 06 2017Nov 11 2017Visual events in computer vision are studied from the perspective of algebraic geometry. Given a sufficiently general curve or surface in 3-space, we consider the image or contour curve that arises by projecting from a viewpoint. Qualitative changes in ... More
Singularities and Semistable Degenerations for Symplectic TopologyJul 05 2017We overview our recent work defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case ... More
Quasi-coherent sheaves in differential geometryJul 04 2017Jul 28 2017It is proved that the category of simplicial complete bornological spaces over $\mathbb R$ carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal ... More
Branched holomorphic Cartan geometries and Calabi-Yau manifoldsJun 14 2017Jan 15 2018We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible ... More
Contractibility of the stability manifold for silting-discrete algebrasMay 30 2017We show that any bounded t-structure in the bounded derived category of a silting-discrete algebra is algebraic, i.e. has a length heart with finitely many simple objects. As a corollary, we obtain that the space of Bridgeland stability conditions for ... More
Localization of Bott-Chern classes and Hermitian residuesMay 26 2017We develop a theory of Cech-Bott-Chern cohomology and in this context we naturally come up with the relative Bott-Chern cohomology. In fact Bott-Chern cohomology has two relatives and they all arise from a single complex. Thus we study these three cohomologies ... More
Model topoi and motivic homotopy theoryApr 27 2017Given a small simplicial category $\mathcal{C}$ whose underlying ordinary category is equipped with a Grothendieck topology $\tau$, we construct a model structure on the category of simplicially enriched presheaves on $\mathcal{C}$ where the weak equivalences ... More
A software package to compute automorphisms of graded algebrasApr 17 2017We present a library autgradalg.lib for the free computer algebra system Singular to compute automorphisms of integral, finitely generated $\mathbb{C}$-algebras that are graded pointedly by a finitely generated abelian group. It implements the algorithms ... More
On the Arithmetic Dynamics of Monomial MapsApr 09 2017We generalized several results for the arithmetic dynamics of monomial maps, including Silverman's conjectures on height growth, dynamical Mordell-Lang conjecture, and dynamical Manin-Mumford conjecture. These results are originally known for monomial ... More
On the difference between permutation polynomials over finite fieldsMar 23 2017The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that if $p>(d^2-3d+4)^2$, then there is no complete mapping polynomial $f$ in $\Fp[x]$ of degree $d\ge 2$. For arbitrary finite fields $\Fq$, a similar non-existence result ... More
Reduction of dynatomic curvesMar 12 2017Mar 29 2017The dynatomic modular curves parametrize polynomial maps together with a point of period $n$. It is known that the dynatomic curves $Y_1(n)$ are smooth and irreducible in characteristic 0 for families of polynomial maps of the form $f_c(z) = z^m +c$ where ... More
Positivity of anticanonical divisors from the viewpoint of Fano conic bundlesMar 08 2017We give the first examples of flat fiber type contractions of Fano manifolds onto varieties that are not weak Fano, and we prove that these morphisms are Fano conic bundles. We also review some known results about the interaction between the positivity ... More
The center of the small quantum group II: singular blocksMar 07 2017Dec 01 2017We generalize to the case of singular blocks the result in \cite{BeLa} that describes the center of the principal block of a small quantum group in terms of sheaf cohomology over the Springer resolution. Then using the method developed in \cite{LQ1}, ... More