on nonlinear maps preserving the reduced minimum modulus on differences of matrices abstract we prove that if m n c m n c is a bijective map which preserves the reduced minimum modulus on differences of matrices then is automatically bicontinuous and therefore by a result of mashreghi and stepanyan of a standard form

quivers of sections on toric orbifolds abstract starting from a collection of line bundles on a projective toric orbifold x we introduce a stacky analogue of the classical linear series our first main result extends work of king by building moduli stacks of refined representations of labelled quivers we associate one such stack to any collection of line bundles on x to obtain our notion of a stacky linear series as in the classical case x maps to the ambient stack by evaluating sections of line bundles in the collection as a further application we describe a finite sequence of git wall crossings between a n g and g hilb a n for a finite abelian subgroup g sl n k where n 3

nonlinear maps preserving the reduced minimum modulus of operators abstract let x and y be infinite dimensional complex banach spaces and let b x resp b y denote the algebra of all bounded linear operators on x resp on y we describe bijective bicontinuous maps from b x to b y satisfying s t s t for all s t b x where t is the reduced minimum modulus of an operator t an analogue result for the finite dimensional case is obtained

polytope volume computation a combinatorial form of gram s relation for convex polytopes can be adapted for use in computing polytope volume we present an algorithm for volume computation based on this observation this algorithm is useful in finding the volume of a polytope given as the solution set of a system of linear inequalities p x e r ax b as an illustration we compute a formula for the volume of a projective image of the n cube from this formula we deduce that when a and b have rational entries so that the volume of p is also a rational number the number of binary digits in the denominator of the volume cannot be bounded by a polynomial in the total number of digits in the numerators and denominators of entries of a and b this settles a question posed by dyer and frieze

orbifold quasimap theory we extend to orbifolds the quasimap theory of ciocan fontanine and kim adv math 225 6 3022 3051 2010 j geom phys 75 17 47 2014 as well as the genus zero wall crossing results from algebr geom 1 4 400 448 2014 proceedings of the conference on the occasion of mukai s 60th birthday 2015 as a consequence we obtain generalizations of orbifold mirror theorems in particular of the mirror theorem for toric orbifolds recently proved independently by coates et al a mirror theorem for toric stacks

spectrum nonincreasing maps on matrices abstract maps which do not increase the spectrum on complex matrices in a sense that sp a b sp a b are classified

the logarithmic cotangent complex we define the cotangent complex of a morphism of fine log schemes prove that it is functorial and construct under certain restrictions a transitivity triangle we also discuss its relationship with deformation theory

non archimedean geometry of artin fans abstract the purpose of this article is to study the role of artin fans in tropical and non archimedean geometry artin fans are logarithmic algebraic stacks that can be described completely in terms of combinatorial objects so called kato stacks a stack theoretic generalization of k kato s notion of a fan every logarithmic algebraic stack admits a tautological strict morphism x x a x to an associated artin fan the main result of this article is that on the level of underlying topological spaces the natural functorial tropicalization map of x is nothing but the non archimedean analytic map associated to x by applying thuillier s generic fiber functor using this framework we give a reinterpretation of the main result of abramovich caporaso payne identifying the moduli space of tropical curves with the non archimedean skeleton of the corresponding algebraic moduli space

nonlinear maps preserving the minimum and surjectivity moduli abstract let x and y be infinite dimensional complex banach spaces and let b x resp b y denote the algebra of all bounded linear operators on x resp on y we describe maps from b x onto b y satisfying c s t c s t for all s t b x where c stands either for the minimum modulus or the surjectivity modulus or the maximum modulus we also obtain analog results for the finite dimensional case

ehrhart polynomials and stringy betti numbers we study the connection between stringy betti numbers of gorenstein toric varieties and the generating functions of the ehrhart polynomials of certain polyhedral regions we use this point of view to give counterexamples to hibi s conjecture on the unimodality of vectors of reflexive polytopes

the orbifold chow ring of toric deligne mumford stacks the orbifold chow ring of a deligne mumford stack defined by abramovich graber and vistoli 2 is the algebraic version of the orbifold cohomology ring in troduced by w chen and ruan 7 8 by design this ring incorporates numerical invariants such as the orbifold euler characteristic and the orbifold hodge num bers of the underlying variety the product structure is induced by the degree zero part of the quantum product in particular it involves gromov witten invariants inspired by string theory and results in batyrev 3 and yasuda 28 one expects that in nice situations the orbifold chow ring coincides with the chow ring of a resolution of singularities fantechi and g ttsche 14 and uribe 25 verify this conjecture when the orbifold is symn 5 where 5 is a smooth projective surface with ks 0 and the resolution is hilbn the initial motivation for this project was to compare the orbifold chow ring of a simplicial toric variety with the chow ring of a cr pant resolution to achieve this goal we first develop the theory of toric deligne mumford stacks modeled on simplicial toric varieties a toric deligne mumford stack corresponds to a combinatorial object called a stacky fan as a first approximation this object is a simplicial fan with a distinguished lattice point on each ray in the fan more precisely a stacky fan s is a triple consisting of a finitely generated abelian group n a simplicial fan e in q z n with n rays and a map zn n where the image of the standard basis in zn generates the rays in e a rational simplicial fan e produces a canonical stacky fan s n e where n is the distinguished lattice and is the map defined by the minimal lattice points on the rays hence there is a natural toric deligne mumford stack associated to every simplicial toric variety a stacky fan encodes a group action on a quasi affine variety and the toric deligne mumford stack is the quotient if e corresponds to a smooth toric variety x e and s is the canonical stacky fan associated to e then we simply have x z we show that many of the basic concepts such as open and closed toric substacks line bundles and maps between toric deligne mumford stacks correspond to combinatorial notions we expect that many more results about toric varieties lift to the realm of stacks and hope that toric deligne mumford stacks will serve as a useful testing ground for general theories

mellin barnes integrals as fourier mukai transforms we study the generalized hypergeometric system introduced by gelfand kapranov and zelevinsky and its relationship with the toric deligne mumford dm stacks recently studied by borisov chen and smith we construct series solutions with values in a combinatorial version of the chen ruan orbifold cohomology and in the k theory of the associated dm stacks in the spirit of the homological mirror symmetry conjecture of kontsevich we show that the k theory action of the fourier mukai functors associated to basic toric birational maps of dm stacks are mirrored by analytic continuation transformations of mellin barnes type

toric stacks ii intrinsic characterization of toric stacks the purpose of this paper and its prequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature as well as classical toric varieties while the focus of the prequel is on how to work with toric stacks the focus of this paper is how to show a stack is toric for toric varieties a classical result says that a finite type scheme with an action of a dense open torus arises from a fan if and only if it is normal and separated in the same spirit the main result of this paper is that any artin stack with an action of a dense open torus arises from a stacky fan under reasonable hypotheses

the mori program and non fano toric homological mirror symmetry in the case of toric varieties we continue the pursuit of kontsevich s fundamental insight homological mirror symmetry by unifying it with the mori program we give a refined conjectural version of homological mirror symmetry relating semi orthogonal decompositions of the b model on toric varieties to semi orthogonal decompositions on the a model on the mirror landau ginzburg models as evidence we prove a new case of homological mirror symmetry for a toric surface whose anticanonical bundle is not nef namely a certain blow up of p at three infinitesimally near points

applications of homological mirror symmetry to hypergeometric systems duality conjectures abstract homological mirror symmetry for crepant resolutions of gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of pdes we explain these conjectures and verify them in some cases

gross fibrations syz mirror symmetry and open gromov witten invariants for toric calabi yau orbifolds given a toric calabi yau orbifold x whose underlying toric variety is semi projective we construct and study a non toric lagrangian torus bration on x which we call the gross bration we apply the strominger yau zaslow recipe to the gross bration of a toric modication of x to construct its instanton corrected mirror where the instanton corrections come from genus 0 open orbifold gromov witten invariants which are virtual counts of holomorphic orbi disks inx bounded by bers of the gross bration we explicitly evaluate all these invariants by rst proving an open closed equality and then employing the toric mirror theorem for suitable toric parital compactications of x our calculations are then applied to 1 prove a conjecture of gross siebert on a relation between genus 0 open orbifold gromov witten invariants and mirror maps ofx this is called the open mirror theorem which leads to an enumerative meaning of mirror maps and 2 demonstrate how open orbifold gromov witten invariants for toric calabi yau orb ifolds change under toric crepant resolutions this is an open analogue of ruan s crepant resolution conjecture

lagrangian floer superpotentials and crepant resolutions for toric orbifolds we investigate the relationship between the lagrangian floer superpotentials for a toric orbifold and its toric crepant resolutions more specifically we study an open string version of the crepant resolution conjecture crc which states that the lagrangian floer superpotential of a gorenstein toric orbifold mathcal x and that of its toric crepant resolution y coincide after analytic continuation of quantum parameters and a change of variables relating this conjecture with the closed crc we find that the change of variable formula which appears in closed crc can be explained by relations between open orbifold gromov witten invariants we also discover a geometric explanation in terms of virtual counting of stable orbi discs for the specialization of quantum parameters to roots of unity which appears in ruan s original crc gromov witten theory of spin curves and orbifolds contemp math amer math soc providence ri pp 117 126 2006 we prove the open crc for the weighted projective spaces mathcal x mathbb p 1 ldots 1 n using an equality between open and closed orbifold gromov witten invariants along the way we also prove an open mirror theorem for these toric orbifolds

log canonical models for the moduli space of curves the first divisorial contraction in this paper we initiate our investigation of log canonical models for m g as we decrease from 1 to 0 we prove that for the first critical value 9 11 the log canonical model is isomorphic to the moduli space of pseudostable curves which have nodes and cusps as singularities we also show that 7 10 is the next critical value i e the log canonical model stays the same in the interval 7 10 9 11 in the appendix we develop a theory of log canonical models of stacks that explains how these can be expressed in terms of the coarse moduli space

orbits in pr n and equivariant quantum cohomology abstract we compute the g l r 1 equivariant chow class of the g l r 1 orbit closure of any point x 1 x n p r n in terms of the rank polytope of the matroid represented by x 1 x n p r using these classes and generalizations involving point configurations in higher dimensional projective spaces we define for each d n matrix m an n ary operation m on the small equivariant quantum cohomology ring of p r which is the n ary quantum product when m is an invertible matrix we prove that m m is a valuative matroid polytope association like the quantum product these operations satisfy recursive properties encoding solutions to enumerative problems involving point configurations of given moduli in a relative setting as an application we compute the number of line sections with given moduli of a general degree 2 r 1 hypersurface in p r generalizing the known case of quintic plane curves

global spectra polytopes and stacky invariants given a convex polytope we define its geometric spectrum a stacky version of batyrev s stringy e functions and we prove a stacky version of a formula of libgober and wood about the e polynomial of a smooth projective variety as an application we get a closed formula for the variance of the geometric spectrum and a noether s formula for two dimensional fano polytopes polytopes whose vertices are primitive lattice points a fano polytope is not necessarily smooth we also show that this geometric spectrum is equal to the algebraic spectrum the spectrum at infinity of a tame laurent polynomial whose newton polytope is the polytope alluded to this gives an explanation and some positive answers to hertling s conjecture about the variance of the spectrum of tame regular functions

the quantum orbifold cohomology of weighted projective spaces we calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces we generalize givental s heuristic argument which relates small quantum cohomology to s1 equivariant floer cohomology of loop space to weighted projective spaces and use this to conjecture an explicit formula for the small j function a generating function for certain genus zero gromov witten invariants we prove this conjecture using a method due to bertram this provides the first non trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known in addition we obtain formulas for the small j functions of weighted projective complete intersections satisfying a combinatorial condition this condition naturally singles out the class of orbifolds with terminal singularities

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

canonical artin stacks over log smooth schemes we develop a theory of toric artin stacks extending the theories of toric deligne mumford stacks developed by borisov chen smith fantechi mann nironi and iwanari we also generalize the chevalley shephard todd theorem to the case of diagonalizable group schemes these are both applications of our main theorem which shows that a toroidal embedding x is canonically the good moduli space in the sense of alper of a smooth log smooth artin stack whose stacky structure is supported on the singular locus of x

stable pair tropical and log canonical compactifications of moduli spaces of del pezzo surfaces we give a functorial normal crossing compactification of the moduli space of smooth cubic surfaces entirely analogous to the grothendieck knudsen compactification m 0 n subset overline m 0 n

holomorphic orbi discs and lagrangian floer cohomology of symplectic toric orbifolds we develop floer theory of lagrangian torus fibers in compact symplectic toric orbifolds we first classify holomorphic orbi discs with boundary on lagrangian torus fibers we show that there exists a class of basic discs such that we have one to one correspondences between a smooth basic discs and facets of the moment polytope and b between basic orbi discs and twisted sectors of the toric orbifold we show that there is a smooth lagrangian floer theory of these torus fibers which has a bulk deformation by fundamental classes of twisted sectors of the toric orbifold we show by several examples that such bulk deformation can be used to illustrate the very rigid hamiltonian geometry of orbifolds we define its potential and bulk deformed potential and develop the notion of leading order potential we study leading term equations analogous to the case of toric manifolds by fukaya oh ohta and ono

geometry of chow quotients of grassmannians we consider kapranov s chow quotient compactification of the moduli space of ordered n tuples of hyperplanes in p r 1 in linear general position for r 2 t his is canonically identified with the grothendieck knudsen compactification of m0 n which has among others the following nice properties 1 modular meaning stable pointed rational curves 2 canonical description of limits of one parameter degenerations 3 natural mori theoretic meaning log canonical compactification we generalize 1 and 2 to all r n but we show that 3 which we view as the deepest fails except possibly in the cases 2 n 3 6 3 7 3 8 where we conjecture that it holds

on the better behaved version of the gkz hypergeometric system we consider a version of the generalized hypergeometric system introduced by gelfand kapranov and zelevinsky gkz suited for the case when the underlying lattice is replaced by a finitely generated abelian group in contrast to the usual gkz hypergeometric system the rank of the better behaved gkz hypergeometric system is always the expected one we give largely self contained proofs of many properties of this system the discussion is intimately related to the study of the variations of hodge structures of hypersurfaces in algebraic tori

frobenius morphisms and derived categories on two dimensional toric deligne mumford stacks abstract for a toric deligne mumford dm stack x we can consider a certain generalization of the frobenius endomorphism for such an endomorphism f x x on a 2 dimensional toric dm stack x we show that the push forward f o x of the structure sheaf generates the bounded derived category of coherent sheaves on x we also choose a full strong exceptional collection from the set of direct summands of f o x in several examples of two dimensional toric dm orbifolds x

z linear gale duality and poly weighted spaces pws abstract the present paper is devoted to discussing gale duality from the z linear algebraic point of view this allows us to isolate the class of q factorial complete toric varieties whose class group is torsion free here called poly weighted spaces pws it provides an interesting generalization of weighted projective spaces wps

hamiltonian group actions on symplectic deligne mumford stacks and toric orbifolds abstract we develop differential and symplectic geometry of differentiable deligne mumford stacks orbifolds including hamiltonian group actions and symplectic reduction as an application we construct new examples of symplectic toric dm stacks

an integral structure in quantum cohomology and mirror symmetry for toric orbifolds abstract we introduce an integral structure in orbifold quantum cohomology associated to the k group and the class in the case of compact toric orbifolds we show that this integral structure matches with the natural integral structure for the landau ginzburg model under mirror symmetry by assuming the existence of an integral structure we give a natural explanation for the specialization to a root of unity in y ruan s crepant resolution conjecture yongbin ruan the cohomology ring of crepant resolutions of orbifolds in contemp math vol 403 amer math soc providence ri 2006 pp 117 126

pavages des simplexes schemas de graphes recolles et compactification des pglrn 1 pglr some equivariant compactifications of the quotients pglrn 1 pglr are constructed each one is decomposed into locally closed strata which are smooth are indexed by the entire convex pavings of the simplex of dimension n and admit a modular interpretation deduced from that of the grassmann varieties together they form a simplicial scheme which compactifies the classifying simplicial scheme of pglr consisting of all the quotients pglrn 1 pglr n 0

robust solutions to a general class of approximation problems the solution of robust counterparts of optimization problems with uncertain data is currently attracting much interest in particular this has been considered in the context of approximation using total least squares here we consider the analogue of this in the errors in variables context where attention is focused on the particular errors which arise in the individual independent variable values in data fitting problems in addition to consideration of problems which are linear in the free parameters some suggestions are made for the treatment of nonlinear problems the emphasis throughout is on providing methods which are computationally tractable and the results parallel and extend earlier results on uncertain linear approximation problems

quantum cohomology and toric minimal model programs abstract we give a quantum version of the danilov jurkiewicz presentation of the cohomology of a compact toric orbifold with projective coarse moduli space more precisely we construct a canonical isomorphism from a formal version of the batyrev ring from 4 to the quantum orbifold cohomology at a canonical bulk deformation this isomorphism generalizes results of givental 23 iritani 34 and fukaya oh ohta ono 21 for toric manifolds and coates lee corti tseng 11 for weighted projective spaces the proof uses a quantum version of kirwan surjectivity theorem 2 6 below and an equality of dimensions theorem 4 19 below deduced using a toric minimal model program tmmp we show that there is a natural decomposition of the quantum cohomology where summands correspond to singularities in the tmmp each of which gives rise to a collection of hamiltonian non displaceable lagrangian tori

the integral orbifold chow ring of toric deligne mumford stacks in this paper we study the integral chow ring of toric deligne mumford stacks we prove that the integral chow ring of a semi projective toric deligne mumford stack is isomorphic to the stanley reisner ring of the associated stacky fan the integral orbifold chow ring is also computed our results are illustrated with several examples

orbifold cohomology for global quotients let x be an orbifold that is a global quotient of a manifold y by a finite group g we construct a noncommutative ring h sp ast y g with a g action such that h sp y g sp g is the orbifold cohomology ring of x defined by w chen and y ruan cr when y s sp n with s a surface with trivial canonical class and g mathfrak s sb n we prove that a small modification of the orbifold cohomology of x is naturally isomorphic to the cohomology ring of the hilbert scheme s sp n computed by m lehn and c sorger ls2

combinatorial geometries convex polyhedra and schubert cells this paper is a continuation of gm which was published in the same journal we will explore a remarkable connection between the geometry of the schubert cells in the grassmann manifold the theory of convex polyhedra and the theory of combinatorial geometries in the sense of crapo and rota cr the results in this paper were obtained simultaneously and independently by gelfand and serganova in moscow and by goresky and macpherson at the i h e s in paris as part of larger programs with different purposes see below the geometry of this simple example is so beautiful that we decided to publish it independently of the applications we believe that combinatorial methods will play an increasing role in the future of geometry and topology we consider the grassmann manifold gi of all n k dimensional subspaces of c by fixing the standard basis in c we obtain an action of the torus h i on g pk which is induced from stretching the coordinate axes in c see also sect 1 we will describe not only the trajectories but also the strata of a new and interesting decomposition of the 301 oool 8708 87 7 50

a new cohomology theory of orbifold based on the orbifold string theory model in physics we construct a new cohomology ring for any almost complex orbifold the key theorem is the associativity of this new ring some examples are computed

mori dream stacks we propose a generalisation of mori dream spaces to stacks we show that this notion is preserved under root constructions and taking abelian gerbes unlike the case of mori dream spaces such a stack is not always given as a quotient of the spectrum of its cox ring by the picard group we give a criterion when this is true in terms of mori dream spaces and root constructions finally we compare this notion with that of smooth toric deligne mumford stacks

toric stacks i the theory of stacky fans the purpose of this paper and its sequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature as well as classical toric varieties in this paper we define a toric stack as the stack quotient of a toric variety by a subgroup of its torus we also define a generically stacky version any toric stack arises from a combinatorial gadget called a stacky fan we develop a dictionary between the combinatorics of stacky fans and the geometry of toric stacks stressing stacky phenomena such as canonical stacks and good moduli space morphisms we also show that smooth toric stacks carry a moduli interpretation extending the usual moduli interpretations of p n and a 1 g m indeed smooth toric stacks precisely solve moduli problems specified by generalized effective cartier divisors with given linear relations and given intersection relations smooth toric stacks therefore form a natural closure to the class of moduli problems introduced for smooth toric varieties and smooth toric dm stacks in papers by cox and perroni respectively we include a plethora of examples to illustrate the general theory we hope that this theory of toric stacks can serve as a companion to an introduction to stacks in much the same way that toric varieties can serve as a companion to an introduction to schemes

weighted ehrhart theory and orbifold cohomology we introduce the notion of a weighted d vector of a lattice polytope although the definition is motivated by motivic integration we study weighted d vectors from a combinatorial perspective we present a version of ehrhart reciprocity and prove a change of variables formula we deduce a new geometric interpretation of the coefficients of the ehrhart d vector more specifically they are sums of dimensions of orbifold cohomology groups of a toric stack

compactifications of subvarieties of tori we study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non archimedean amoebas these compactifications have many nice properties for example any k boundary divisors intersect in codimension k we consider some examples including m 0 n subset overline m 0 n and more generally log canonical models of complements of hyperplane arrangements and compact quotients of grassmannians by a maximal torus

monodromy of monomially admissible fukaya seidel categories mirror to toric varieties abstract mirror symmetry for a toric variety involves laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety we show that there is a monodromy action on the monomially admissible fukaya seidel categories of these laurent polynomials as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated in the process we introduce the monomially admissible fukaya seidel category as a new interpretation of the fukaya seidel category of a laurent polynomial on c n which has other potential applications and give evidence of homological mirror symmetry for non compact toric varieties

the orbifold topological vertex abstract we develop a topological vertex formalism for computing the donaldson thomas invariants of calabi yau orbifolds the basic combinatorial object is the orbifold vertex v g a generating function for the number of 3d partitions asymptotic to 2d partitions and colored by representations of a finite abelian group g acting on c 3 in the case where g z n acting on c 3 with transverse a n 1 quotient singularities we give an explicit formula for v g in terms of schur functions we discuss applications of our formalism to the donaldson thomas crepant resolution conjecture and to the orbifold donaldson thomas gromov witten correspondence we also explicitly compute the donaldson thomas partition function for some simple orbifold geometries the local football p a b 1 and the local b z 2 gerbe

framed sheaves on root stacks and supersymmetric gauge theories on ale spaces we develop a new approach to the study of supersymmetric gauge theories on ale spaces using the theory of framed sheaves on root toric stacks which illuminates relations with gauge theories on r 4 and with two dimensional conformal field theory we construct a stacky compactification of the minimal resolution xk of the ak 1 toric singularity c 2 zk which is a projective toric orbifold xk such that xknxk is a zk gerbe we construct moduli spaces of tor sion free sheaves on xk which are framed along the compactification gerbe we prove that this moduli space is a smooth quasi projective variety compute its dimension and classify its fixed points under the natural induced toric action we use this construction to compute the partition functions and correlators of chiral bps operators forn 2 quiver gauge theories on xk with nontrivial holonomies at infinity the partition functions are computed with and without cou plings to bifundamental matter hypermultiplets and expressed in terms of toric blowup formulas which relate them to the corresponding nekrasov partition functions on the affine toric subsets of xk we compare our new partition functions with previous computations explore their con nections to the representation theory of affine lie algebras and find new constraints on fractional instanton charges in the coupling to fundamental matter we show that the partition functions in the low energy limit are characterised by the seiberg witten curves and in some cases also by suitable blowup equations involving riemann theta functions on the seiberg witten curve with characteristics related to the nontrivial holonomies

tropical geometry and correspondence theorems via toric stacks in this paper we generalize correspondence theorems of mikhalkin and nishinou siebert providing a correspondence between algebraic and parameterized tropical curves we also give a description of a canonical tropicalization procedure for algebraic curves motivated by berkovich s construction of skeletons of analytic curves under certain assumptions we construct a one to one correspondence between algebraic curves satisfying toric constraints and certain combinatorially defined objects called stacky tropical reductions that can be enumerated in terms of tropical curves satisfying linear constraints similarly we construct a one to one correspondence between elliptic curves with fixed j invariant satisfying toric constraints and stacky tropical reductions that can be enumerated in terms of tropical elliptic curves with fixed tropical j invariant satisfying linear constraints our theorems generalize previously published correspondence theorems in tropical geometry and our proofs are algebra geometric in particular the theorems hold in large positive characteristic

smooth toric deligne mumford stacks we give a geometric definition of smooth toric deligne mumford stacks using the action of a torus we show that our definition is equivalent to the one of borisov chen and smith in terms of stacky fans in particular we give a geometric interpretation of the combinatorial data contained in a stacky fan we also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks

valuative invariants for polymatroids abstract many important invariants for matroids and polymatroids such as the tutte polynomial the billera jia reiner quasi symmetric function and the invariant g introduced by the first author are valuative in this paper we construct the z modules of all z valued valuative functions for labeled matroids and polymatroids on a fixed ground set and their unlabeled counterparts the z modules of valuative invariants we give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes and explicit formulas for their ranks our results confirm a conjecture of the first author that g is universal for valuative invariants

characteristic classes of singular toric varieties in this paper we compute the motivic chern classes and homology hirzebruch characteristic classes of possibly singular toric varieties which in the context of complete toric varieties fit nicely with a generalized hirzebruch riemann roch theorem as important special cases we obtain new or recover well known formulae for the baum fulton macpherson todd or macpherson s chern classes of toric varieties as well as for the thom milnor l classes of simplicial projective toric varieties we present two different perspectives for the computation of these characteristic classes of toric varieties first we take advantage of the torus orbit decomposition and the motivic properties of the motivic chern and respectively homology hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the dual todd classes of closures of orbits this method even applies to torus invariant subspaces of a given toric variety the obtained formula is then applied to weighted lattice point counting in lattice polytopes and their subcomplexes yielding generalized pick type formulae second in the case of simplicial toric varieties we compute our characteristic classes by using the lefschetz riemann roch theorem of edidin graham in the context of the geometric quotient description of such varieties in this setting we define mock hirzebruch classes of simplicial toric varieties which specialize to the mock chern mock todd and mock l classes of such varieties and investigate the difference between the actual homology hirzebruch class and the mock hirzebruch class we show that this difference is localized on the singular locus and we obtain a formula for it in which the contribution of each singular cone is identified explicitly finally the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by cappell and shaneson in the early 1990s 2015 wiley periodicals inc

mathbb c 2 mathbb z _ n fractional branes and monodromy we construct geometric representatives for the mathbb c 2 mathbb z n fractional branes in terms of branes wrapping certain exceptional cycles of the resolution in the process we use large radius and conifold type monodromies and also check some of the orbifold quantum symmetries we find the explicit seiberg duality which connects our fractional branes to the ones given by the mckay correspondence we also comment on the harvey moore bps algebras

on clifford double mirrors of toric complete intersections abstract we present a construction of noncommutative double mirrors to complete intersections in toric varieties this construction unifies existing sporadic examples and explains the underlying combinatorial and physical reasons for their existence

the crepant transformation conjecture for toric complete intersections abstract let x and y be k equivalent toric deligne mumford stacks related by a single toric wall crossing we prove the crepant transformation conjecture in this case fully equivariantly and in genus zero that is we show that the equivariant quantum connections for x and y become gauge equivalent after analytic continuation in quantum parameters furthermore we identify the gauge transformation involved which can be thought of as a linear symplectomorphism between the givental spaces for x and y with a fourier mukai transformation between the k groups of x and y via an equivariant version of the gamma integral structure on quantum cohomology we prove similar results for toric complete intersections we impose only very weak geometric hypotheses on x and y they can be non compact for example and need not be weak fano or have gorenstein coarse moduli space our main tools are the mirror theorems for toric deligne mumford stacks and toric complete intersections and the mellin barnes method for analytic continuation of hypergeometric functions

on the conjecture of king for smooth toric deligne mumford stacks abstract we construct full strong exceptional collections of line bundles on smooth toric fano deligne mumford stacks of picard number at most two and of any picard number in dimension two it is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef fano deligne mumford stacks

a derham model for chen ruan cohomology ring of abelian orbifolds we present a derham model for chen ruan cohomology ring of abelian orbifolds we introduce the notion of twist factors so that formally the stringy cohomology ring can be defined without going through pseudo holomorphic orbifold curves thus our model can be viewed as the classical description of chen ruan cohomology for abelian orbifolds the model simplifies computation of chen ruan cohomology ring using our model we give a version of wall crossing formula

maps on matrices that preserve the spectrum abstract let t be a continuous map of the space of complex n n matrices into itself satisfying t 0 0 such that the spectrum of t x t y is always a subset of the spectrum of x y there exists then an invertible n n matrix u such that either t a uau 1 for all a or t a ua t u 1 for all a we arrive at the same conclusion by supposing that the spectrum of x y is always a subset of the spectrum of t x tt y without the continuity assumption on t

computing genus zero twisted gromov witten invariants twisted gromov witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold x which depend in addition on a vector bundle over x and an invertible multiplicative characteristic class special cases are closely related to local gromov witten invariants of the bundle and to genus zero one point invariants of complete intersections in x we develop tools for computing genus zero twisted gromov witten invariants of orbifolds and apply them to several examples we prove a quantum lefschetz theorem that expresses genus zero one point gromov witten invariants of a complete intersection in terms of those of the ambient orbifold x we determine the genus zero gromov witten potential of the type a surface singularity c2 zn we also compute some genus zero invariants of c3 z3 verifying predictions of aganagic bouchard and klemm 3 in a self contained appendix we determine the relationship between the quantum cohomology of the an surface singularity and that of its crepant resolution thereby proving the crepant resolution conjectures of ruan and of bryan and graber 12 in this case

weighted blowups and mirror symmetry for toric surfaces abstract this paper explores homological mirror symmetry for weighted blowups of toric varietes it will be shown that both the a model and b model categories have natural semi orthogonal decompositions an explicit equivalence of the right orthogonal categories will be shown for the case of toric surfaces

a matroid invariant via the k theory of the grassmannian abstract let g d n denote the grassmannian of d planes in c n and let t be the torus c n diag c which acts on g d n let x be a point of g d n and let t x be the closure of the t orbit through x then the class of the structure sheaf of t x in the k theory of g d n depends only on which plucker coordinates of x are nonzero combinatorial data known as the matroid of x in this paper we will define a certain map of additive groups from k g d n to z t letting g x t denote the image of 1 n dim t x o t x g x behaves nicely under the standard constructions of matroid theory such as direct sum two sum duality and series and parallel extensions we use this invariant to prove bounds on the complexity of kapranov s lie complexes m kapranov chow quotients of grassmannians i adv soviet math 16 2 1993 29 110 hacking keel and tevelev s very stable pairs p hacking s keel e tevelev compactification of the moduli space of hyperplane arrangements j algebraic geom 15 2006 657 680 and the author s tropical linear spaces when they are realizable in characteristic zero d speyer tropical linear spaces siam j discrete math 22 4 2008 1527 1558 namely in characteristic zero a lie complex or the underlying d 1 dimensional scheme of a very stable pair can have at most n i 1 d i n d i i 1 strata of dimensions n i and d i respectively this prove the author s f vector conjecture from d speyer tropical linear spaces siam j discrete math 22 4 2008 1527 1558 in the case of a tropical linear space realizable in characteristic 0

parameter estimation with multiple sources and levels of uncertainties least squares designs are sensitive to errors in the data which can be due to several factors including the approximation of complex models by simpler ones the presence of unavoidable experimental errors when collecting data or even due to unknown or unmodeled effects we formulate a new design criterion that treats multiple sources of uncertainties in the data with possibly varied degrees of intensity we show that the solution has a regularized form with one regularization parameter for each source of uncertainty the parameters turn out to be model dependent and can be determined optimally as the nonnegative roots of certain coupled equations applications in array signal processing and image processing are considered

framed sheaves on projective stacks abstract given a normal projective irreducible stack x over an algebraically closed field of characteristic zero we consider framed sheaves on x i e pairs e e where e is a coherent sheaf on x and e is a morphism from e to a fixed coherent sheaf f after introducing a suitable notion of semi stability we construct a projective scheme which is a moduli space for semistable framed sheaves with fixed hilbert polynomial and an open subset of it which is a fine moduli space for stable framed sheaves if x is a projective irreducible orbifold of dimension two and f a locally free sheaf on a smooth divisor d x satisfying certain conditions we consider d f framed sheaves i e framed sheaves e e with e a torsion free sheaf which is locally free in a neighbourhood of d and e d an isomorphism these pairs are stable for a suitable choice of a parameter entering the semi stability condition and of the polarization of x this implies the existence of a fine moduli space parameterizing isomorphism classes of d f framed sheaves on x with fixed hilbert polynomial which is a quasi projective scheme in an appendix we develop the example of stacky hirzebruch surfaces this is the first paper of a project aimed to provide an algebro geometric approach to the study of gauge theories on a wide class of 4 dimensional riemannian manifolds by means of framed sheaves on stacky compactifications of them in particular in a subsequent paper 20 these results are used to study gauge theories on ale spaces of type a k