gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

moduli spaces of hyperelliptic curves with a and d singularities we introduce moduli spaces of quasi admissible hyperelliptic covers with at worst a and d singularities the stability conditions for these moduli spaces depend on two rational parameters describing allowable singularities for the extreme values of the parameters we obtain the stacks of stable limits of a n and d n singularities and the quotients of the miniversal deformation spaces of these singularities by natural mathbb g m actions we interpret the intermediate spaces as log canonical models of the stacks of stable limits of a n and d n singularities

towards a classification of modular compactifications of mathcal m _ g n the moduli space of smooth curves admits a beautiful compactification mathcal m g n subset overline mathcal m g n by the moduli space of stable curves in this paper we undertake a systematic classification of alternate modular compactifications of mathcal m g n let mathcal u g n be the non separated moduli stack of all n pointed reduced connected complete one dimensional schemes of arithmetic genus g when g 0 mathcal u 0 n is irreducible and we classify all open proper substacks of mathcal u 0 n when g 1 mathcal u g n may not be irreducible but there is a unique irreducible component mathcal v g n subset mathcal u g n containing mathcal m g n we classify open proper substacks of mathcal v g n satisfying a certain stability condition

on the distribution of points of bounded height on equivariant compactifications of vector groups we prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space

log canonical models of elliptic surfaces abstract we classify the log canonical models of elliptic surface pairs f x c s f a where f x c is an elliptic fibration s is a section and f a is a weighted sum of marked fibers in particular we show how the log canonical models depend on the choice of the weights we describe a wall and chamber decomposition of the space of weights based on how the log canonical model changes in addition we give a generalized formula for the canonical bundle of an elliptic surface with section and marked fibers this is the first step in constructing compactifications of moduli spaces of elliptic surfaces using the minimal model program

lower bounds on ricci curvature and quantitative behavior of singular sets let yn denote the gromov hausdorff limit m n i stackrel d mathrm gh longrightarrow y n of v noncollapsed riemannian manifolds with mathrm ric m n i geq n 1 the singular set mathcal s subset y has a stratification mathcal s 0 subset mathcal s 1 subset cdots subset mathcal s where y in mathcal s k if no tangent cone at y splits off a factor k 1 isometrically here we define for all 0 0 r 1 the k th effective singular stratum mathcal s k eta r satisfying bigcup eta bigcap r mathcal s k eta r mathcal s k sharpening the known hausdorff dimension bound dim mathcal s k leq k we prove that for all y the volume of the r tubular neighborhood of mathcal s k eta r satisfies mathrm vol t r mathcal s k eta r cap b frac 1 2 y leq c n mathrm v eta r n k eta the proof involves a quantitative differentiation argument this result has applications to einstein manifolds let mathcal b r denote the set of points at which the c2 harmonic radius is r if also the m n i are kahler einstein with l2 curvature bound rm l 2 leq c then mathrm vol mathcal b r cap b frac 1 2 y leq c n mathrm v c r 4 for all y in the kahler einstein case without assuming any integral curvature bound on the m n i we obtain a slightly weaker volume bound on mathcal b r which yields an a priori lp curvature bound for all p 2 the methodology developed in this paper is new and is applicable in many other contexts these include harmonic maps minimal hypersurfaces mean curvature flow and critical sets of solutions to elliptic equations

weighted projective embeddings stability of orbifolds and constant scalar curvature kahler metrics we embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of kodaira embedding dividing by the non reductive automorphisms of weighted projective space then formally gives a moduli space of orbifolds we show how to express this as a reductive quotient and so a git problem thus defining a notion of stability for orbifolds we then prove an orbifold version of donaldson s theorem the existence of an orbifold kahler metric of constant scalar curvature implies k semistability by extending the notion of slope stability to orbifolds we therefore get an explicit obstruction to the existence of constant scalar curvature orbifold kahler metrics we describe the manifold applications of this orbifold result and show how many previously known results troyanov ghigi kollar rollin singer the adscft sasaki einstein obstructions of gauntlett martelli sparks yau fit into this framework

heat kernel and curvature bounds in ricci flows with bounded scalar curvature abstract in this paper we analyze ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time dependent constant on such ricci flows we establish a new time derivative bound for solutions to the heat equation based on this bound we solve several open problems 1 distance distortion estimates 2 the existence of a cutoff function 3 gaussian bounds for heat kernels and 4 a backward pseudolocality theorem which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time using the backward pseudolocality theorem we next establish a uniform l 2 curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity we also obtain a stronger e regularity theorem for ricci flows this result is particularly useful in the study of kahler ricci flows on fano manifolds where it can be used to derive certain convergence results

towards the ample cone of mg n in this paper we study the ample cone of the moduli space m g n of stable n pointed curves of genus g our motivating conjecture is that a divisor on m g n is ample iff it has positive intersection with all 1 dimensional strata the components of the locus of curves with at least 3g n 2 nodes this translates into a simple conjectural description of the cone by linear inequalities and as all the 1 strata are rational includes the conjecture that the mori cone is polyhedral and generated by rational curves our main result is that the conjecture holds iff it holds for g 0 more precisely there is a natural finite map r m0 2g n m g n whose image is the locus rg n of curves with all components rational any 1 strata either lies in rg n or is numerically equivalent to a family e of elliptic tails and we show that a divisor d is nef iff d e 0 and r d is nef we also give results on contractions i e morphisms with connected fibers to projective varieties of m g n for g 1 showing that any fibration factors through a tautological one given by forgetting points and that the exceptional locus of any birational contraction is contained in the boundary

the geometry of stable quotients in genus one stable quotients provide an alternative to stable maps for compactifying spaces of maps when n ge 2 the space overline q g mathbb p n 1 d overline q g g 1 n d compactifies the space of degree d maps of smooth genus g curves to mathbb p n 1 while overline q g g 1 1 d simeq overline m 1 d cdot epsilon s d is a quotient of a hassett weighted pointed space in this paper we study the coarse moduli schemes associated to the smooth proper deligne mumford stacks overline q 1 mathbb p n 1 d for all n ge 1 we show these schemes are projective unirational and have picard number 2 then we give generators for the picard group compute the canonical divisor the cones of ample divisors and in the case n 1 the cones of effective divisors we conclude that overline q 1 mathbb p n 1 d is fano if and only if n d 1 d 2 20 moreover we show that overline q 1 mathbb p n 1 d is a mori fiber space for all n d hence always minimal in the sense of the minimal model program in the case n 1 we write in addition a closed formula for the poincare polynomial

git stability of weighted pointed curves in the late 1970s mumford established chow stability of smooth unpointed genus g curves embedded by complete linear systems of degree d 2g 1 and at about the same time gieseker established asymptotic hilbert stability that is stability of m th hilbert points for some large values of m under the same hypotheses both of them then use an indirect argument to show that nodal deligne mumford stable curves are git stable the case of marked points lay untouched until 2006 when elizabeth baldwin proved that pointed deligne mumford stable curves are asymptotically hilbert stable actually she proved this for stable maps which includes stable curves as a special case her argument is a delicate induction on g and the number of marked points n elliptic tails are glued to the marked points one by one ultimately relating stability of an n pointed genus g curve to gieseker s result for genus g n unpointed curves there are three ways one might wish to improve upon baldwin s results first one might wish to construct moduli spaces of weighted pointed curves or maps it appears that baldwin s proof can accommodate some but not all sets of weights second one might wish to study hilbert stability for small values of m since baldwin s proof uses gieseker s proof as the base case it is not easy to see how it could be modified to yield an approach for small m finally the minimal model program for moduli spaces of curves has generated interest in git for 2 3 or 4 canonical linear systems due to its use of elliptic tails baldwin s proof cannot be used to study these as elliptic tails are known to be git unstable in these cases in this paper i give a direct proof that smooth curves with distinct weighted marked points are asymptotically hilbert stable with respect to a wide range of parameter spaces and linearizations some of these yield the coarse moduli space of deligne mumford stable pointed curves m g n and hassett s moduli spaces of weighted pointed curves m g a while other linearizations may give other quotients which are birational to these and which may admit interpretations as moduli spaces the full construction of the moduli spaces is not contained in this paper only the proof that smooth curves with distinct weighted marked points are stable which is the key new result needed for the construction for this i follow gieseker s approach to reduce to the git problem to a combinatorial problem though the solution is very different

multiple lu factorizations of a singular matrix a singular matrix a may have more than one lu factorizations in this work the set of all lu factorizations of a is explicitly described when the lower triangular matrix l is nonsingular to this purpose a canonical form of a under left multiplication by unit lower triangular matrices is introduced this canonical form allows us to characterize the matrices that have an lu factorization and to parametrize all possible lu factorizations formulae in terms of quotient of minors of a are presented for the entries of this canonical form

ricci flow entropy and optimal transportation let a smooth family of riemannian metrics g tau satisfy the backwards ricer flow equation on a compact oriented n dimensional manifold m suppose two families of normalized n forms omega tau 0 and omega over bar tau 0 satisfy the forwaids in tau heat equation on m generated by the connection laplacian delta g t if these n forms represent two evolving distributions of particles over m the minimum root mean square distance w 2 omega t omega over tilde t t to transport the particles of omega t onto those of omega over tilde t is shown to be non increasing as a function of t without sign conditions on the curvature of mg t moreover this contractivity property is shown to characterize supersolutions to the ricci flow

universal relations on stable map spaces in genus zero we introduce a factorization for the map between moduli spaces of stable maps which forgets one marked point this leads to a study of universal relations in the cohomology of stable map spaces in genus zero

higher genus wall crossing in landau ginzburg theory abstract for a fermat quasi homogeneous polynomial we study the associated weighted fan jarvis ruan witten theory with narrow insertions we prove a wall crossing formula in all genera via localization on a master space which is constructed by introducing an additional tangent vector to the moduli problem this is a landau ginzburg theory analogue of the higher genus quasi map wall crossing formula proved by ciocan fontanine and kim it generalizes the genus 0 result by ross ruan and the genus 1 result by guo ross

generalized fraction free lu factorization for singular systems with kernel extraction abstract linear systems are usually solved with gaussian elimination especially when multiple right hand sides are involved an efficient procedure is to provide a factorization of the left hand side when exact computations are required in an integral domain complete fraction free factorization and forward backward substitutions are useful this article deals with the case where the left hand side may be singular in such a case kernels are required to test a solvability condition and to derive the general form of the solutions the complete fraction free algorithms are therefore extended to deal with singular systems and to provide the kernels with exact computations on the same integral domain where the initial data take their entries

a new look at the moduli space of stable hyperelliptic curves we carry out the log minimal model program for the moduli space bar h g of stable hyperelliptic curves and show that certain log canonical models of bar h g are isomorphic to the proper transform of bar h g in the corresponding log canonical models of bar m g for g 3 we retrieve the compact moduli space bar b 8 of binary forms as a log canonical model and obtain a decomposition of the natural map bar h 3 to bar b 8 into successive divisorial contractions of the boundary divisors as a byproduct we also obtain an isomorphism of bar b 8 with the git quotient of the chow variety of bicanonically embedded hyperelliptic curves of genus three

intersection theory of moduli space of stable n pointed curves of genus zero we give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the chow ring

rank revealing lu factorizations abstract we consider permutations of any given squared matrix and the generalized lu r factorization of the permuted matrix that reveals the rank deficiency of the matrix chan has considered the case with nearly rank deficiency equal to one this paper extends his results to the case with nearly rank deficiency greater than one two applications in constrained optimization are given we are primarily interested in the existence of such factorizations in addition to the theories we also present an efficient two pass rank revealing lu r algorithm

regularity and compactness of harmonic einstein equations let m x g be pointed riemannian manifold with mathrm vol b 1 x ge mathrm v 0 and satisfy harmonic einstein equation mathrm ric g nabla u otimes nabla u lambda g with lambda le n 1 where u m g rightarrow n h is a harmonic map to a fixed compact riemannian manifold n h then for any p 1 we prove the l p curvature estimate open image in new window as a consequence if n h has nonpositive sectional curvature we have mathrm ric le c n mathrm v n that means harmonic einstein equation automatically implies bounded ricci curvature provided nonpositive sectional curvature of n h let x x infty d u infty be the limit of a sequence of harmonic einstein manifolds m i x i g i u i we show that the singular set of x is closed and the convergence is smooth on the regular part we also prove an orbifold type compactness theorem of harmonic einstein equations with bounded int m mathrm rm n 2 without assuming nonpositive sectional curvature of n h which generalizes xu s compactness result in yiyan adv math 231 2 680 708 2012

moduli spaces of weighted pointed stable curves abstract a weighted pointed curve consists of a nodal curve and a sequence of marked smooth points each assigned a number between zero and one a subset of the marked points may coincide if the sum of the corresponding weights is no greater than one we construct moduli spaces for these objects using methods of the log minimal model program and describe the induced birational morphisms between moduli spaces as the weights are varied in the genus zero case we explain the connection to geometric invariant theory quotients of points in the projective line and to compactifications of moduli spaces studied by kapranov keel and losev manin

information transfer between incompatible finite element meshes application to coupled thermo viscoelasticity this article is concerned with information transfer between non matching finite element meshes such a feature is not new in the literature but we focus herein on a geometric approach to transfer solution fields in order to be as application independent as possible moreover the dedicated case concerns the transfer of finite element fields defined at integration points of the meshes and allows iterative exchange of fields in both directions to do so we propose an extension to the mortar technique that fulfills these goals and that does not suffer from a high computational cost the application is the simulation of a strongly coupled thermo viscoelastic problem with phase transition solved with a partitioning technique

the geometry of the moduli space of curves of genus 23 the problem of describing the birational geometry of the moduli space mg of complex curves of genus g has a long history severi already knew in 1915 thatmg is unirational for g 10 cf sev see also ac1 for a modern proof in the same paper severi conjectured thatmg is unirational for all genera g then for a long period this problem seemed intractable mumford writes in mu p 51 whether more mg s g 11 are unirational or not is a very interesting problem but one which looks very hard too especially if g is quite large the breakthrough came in the eighties when eisenbud harris and mumford proved that mg is of general type as soon as g 24 and that the kodaira dimension of m23 is 1 see hm eh3 we note that mg is rational for g 6 see dol for problems concerning the rationality of various moduli spaces severi s proof of the unirationality of mg for small g was based on representing a general curve of genus g as a plane curve of degree d with nodes this is possible when d 2g 3 2 when the number of nodes is small i e d 1 d 2 6 the dominant map from the variety of plane curves of degree d and genus g to mg yields a rational parametrization of the moduli space the two conditions involving d and can be satisfied only when g 10 so severi s argument cannot be extended for other genera however using much more subtle ideas chang ran and sernesi proved the unirationality of mg for g 11 12 13 see cr1 se1 while for g 15 16 they proved that the kodaira dimension is see cr2 4 the remaining cases g 14 and 17 g 23 are still quite mysterious harris and morrison conjectured in hmo that mg is uniruled precisely when g 23 all these facts indicate that m23 is a very interesting transition case our main result is the following

integral points on symmetric varieties and satake compatifications let v be an affine symmetric variety defined over q we compute the asymptotic distribution of the angular components of the integral points in v this distribution is described by a family of invariant measures concentrated on the satake boundary of v in the course of the proof we describe the structure of the satake compactifications for general affine symmetric varieties and compute the asymptotic of the volumes of norm balls

variation of cone metrics on riemann surfaces we discuss the variational properties of the unique conical metric of constant curvature 1 associated to a compact riemann surface together with a weighted divisor 2005 elsevier inc all rights reserved

chow groups of the moduli spaces of weighted pointed stable curves of genus zero abstract the moduli space m a of weighted pointed stable curves of genus zero is stratified according to the degeneration types of such curves we show that the homology groups of m a are generated by the strata of m a and give all additive relations between them we also observe that the chow groups a i m a and the homology groups h 2 i m a are isomorphic this generalizes kontsevich manin s and losev manin s theorems to arbitrary weight data a

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

the κ ring of the moduli of curves of compact type the subalgebra of the tautological ring of the moduli of curves of compact type generated by the classes is studied in all genera relations constructed via the virtual geometry of the moduli of stable quotients are used to obtain minimal sets of generators bases and betti numbers of the rings are computed a universality property relating the higher genus rings to the genus 0 rings is proven using the virtual geometry of the moduli space of stable maps the g formula for hodge integrals arises as the simplest consequence