non connected toric hilbert schemes we construct small 50 and 26 points respectively point sets in dimension 5 whose graphs of triangulations are not connected these examples improve our construction in j amer math soc 13 3 2000 611 637 not only in size but also in that the associated toric hilbert schemes are not connected either a question left open in that article additionally the point sets can easily be put into convex position providing examples of 5 dimensional polytopes with non connected graph of triangulations

from real affine geometry to complex geometry we construct from a real ane manifold with singularities a tropical manifold a degeneration of calabi yau manifolds this solves a fundamental problem in mirror symmetry furthermore a striking feature of our approach is that it yields an explicit and canonical order by order description of the degeneration via families of tropical trees this gives complete control of the b model side of mirror symmetry in terms of tropical geometry for example we expect that our deformation parameter is a canonical coordinate and expect period calculations to be expressible in terms of tropical curves we anticipate this will lead to a proof of mirror symmetry via tropical methods

computing invariants of semi log canonical surfaces we describe some general methods to compute fundamental groups co homology and irregularity of semi log canonical surfaces as an application we show that there are exactly two irregular gorenstein stable surfaces with k 2 1 which have chi x 0 and mathrm pic 0 x mathbb c but different homotopy type

on the tropical torelli map abstract we construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties working in the category of what we call stacky fans we define the tropical torelli map between these two moduli spaces and we study the fibers tropical torelli theorem and the image of this map tropical schottky problem finally we determine the image of the planar tropical curves via the tropical torelli map and we use it to give a positive answer to a question raised by namikawa on the compactified classical torelli map

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

comparing perfect and 2nd voronoi decompositions the matroidal locus we compare two rational polyhedral admissible decompositions of the cone of positive definite quadratic forms the perfect cone decomposition and the 2nd voronoi decomposition we determine which cones belong to both the decompositions thus providing a positive answer to a conjecture of alexeev and brunyate invent math doi 10 1007 s00222 011 0347 2 2011 as an application we compare the two associated toroidal compactifications of the moduli space of principal polarized abelian varieties the perfect cone compactification and the 2nd voronoi compactification

voronoi and delaunay cells of root lattices classification of their faces and facets by coxeter dynkin diagrams the voronoi domains their duals delaunay domains and all their faces of any dimension are classified and described in terms of the weyl group action on a representative of each type of face the representative of a face type is specified by a decoration of the corresponding coxeter dynkin diagram the rules of domain description are uniform for root lattices of simple lie groups of all types an explicit description of the representatives of all faces is carried out for the domains of root lattices of the four classical series and for the five exceptional simple lie groups the coxeter dynkin diagrams required here are the diagrams extended by the highest short root each diagram is partitioned into two subdiagrams one describing completely a d face of the voronoi domain its complement completely describing the dual of the d face the applicability of the authors classification method to generalized kaleidoscopes is explained

end to end distance distributions and asymptotic behaviour of self avoiding walks in two and three dimensions we use monte carlo methods to study the reduced moments and full end to end distance distributions of self avoiding walks in two and three dimensions we find that the reduced moments scale with length via delta pg a b n delta pq with corrections to the scaling exponents that vary with the order of the moment we also find that the complete end to end distance distributions are well described by a redner des cloizeaux rdc model qn x cx theta n exp kx tn x being the rescaled length we develop a method that allows reliable estimation of the exponents theta n and tn from the extrapolated reduced moments and use this method to extrapolate to chain lengths beyond those investigated here we find that in three dimensions the optimal tn for n 1000 is smaller than the theoretically expected value t 2 445 this implies that care must be taken in using the rdc ansatz to interpret the behaviour of self avoiding walks even in the asymptotic limit

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

toric hilbert schemes we introduce and study the toric hilbert scheme that parametrizes all ideals with the same multigraded hilbert function as a given toric ideal

the local structure of compactified jacobians this paper studies the local geometry of compactified jacobians the main result is a presentation of the completed local ring of the compactified jacobian of a nodal curve as an explicit ring of invariants described in terms of the dual graph of the curve the authors have investigated the geometric and combinatorial properties of these rings in previous work and consequences for compactified jacobians are presented in this paper similar results are given for the local structure of the universal compactified jacobian over the moduli space of stable curves

the class of the locus of intermediate jacobians of cubic threefolds we study the locus of intermediate jacobians of cubic threefolds within the moduli space mathcal a 5 of complex principally polarized abelian fivefolds and its generalization to arbitrary genus the locus of abelian varieties with a singular odd two torsion point on the theta divisor assuming that this locus has expected codimension g which we show to be true for g 5 and conjecturally for any g we compute the class of this locus and of its closure in the perfect cone toroidal compactification mathcal a g mathrm perf in the chow homology and the tautological ring

a riemann singularity theorem for integral curves we prove results generalizing the classical riemann singularity theorem to the case of integral singular curves the main result is a computation of the multiplicity of the theta divisor of an integral nodal curve at an arbitrary point we also suggest a general formula for the multiplicity of the theta divisor of a singular integral curve at a point and present some evidence that this formula should hold our results give a partial answer to a question posed by lucia caporaso in a recent paper

a hyper kahler compactification of the intermediate jacobian fibration associated with a cubic 4 fold let x be a general cubic fourfold it was observed by donagi and markman that the relative intermediate jacobian fibration j u with u p5 x associated to the smooth hyperplane sections of x carries a natural holomorphic symplectic form making the fibration lagrangian in this paper we obtain a smooth projective compactification j of j with the property that the holomorphic symplectic form on j extends to a holomorphic symplectic form on j in particular j is a 10 dimensional compact hyper kahler manifold which we show to be deformation equivalent to the exceptional example of o grady this proves a conjecture by markushevich

the zero section of the universal semiabelian variety and the double ramification cycle we study the chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties ppav we describe the subring generated by divisor classes and we compute the class of the partial compactification of the universal zero section which turns out to lie in this subring our formula extends the results for the zero section of the universal uncompactified family the partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of ag our formula provides a first step in a program to understand the chow groups of a g especially of the perfect cone compactification by induction on genus by restricting to the image of mg under the torelli map our results extend the results of hain on the double ramification cycle answering eliashberg s question

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

compactification of the moduli of polarized abelian varieties and mirror symmetry we show that martin olsson s compactification of moduli space of polarized abelian varieties in ols08 can be interpreted in terms of ksba stable pairs we find that there is a canonical set of divisors s k2 associated with each cusp near the cusp a polarized semiabelic scheme x g l is the canonical degeneration given by the compactification if and only if x g is an object in a pg d for any s k2 moreover we give an alternative construction of the compactification by using mirror symmetry we construct a toroidal compactification a m g that is isomorphic to olsson s compactification over characteristic zero the data needed for a toroidal compactification is a collection of fans we obtain the collection of fans from the mori fans of the minimal models of the mirror families

study of monte carlo methods for generating self avoiding walks the concept of fractal dimensionality is used to study different statistical methods for generating self avoiding walks saws the reliability of saws traced by the enrichment technique and the dynamic monte carlo technique is verified the number of dynamic cycles which represent a single independent saw ofn0 steps is found to be about 0 1n03 we show that the enrichment process for generating saws may be presented as a critical phenomenon

fractal dimensionality of polymer chains the authors suggest that the concept of fractal dimensionality provides a useful characterisation of the configurational properties of a single polymer from numerical studies of long polymers traced out by self avoiding walks on a planar lattice they find that the fractal dimensionality is well defined and has a constant value for most scales of length of the chain renormalisation group theory provides a theoretical basis for the concept of fractal dimensionality in polymers

a point set whose space of triangulations is disconnected in this paper we explicitly construct a triangulation of a 6 dimensional point configuration of 324 points which admits no geometric bistellar operations or flips for short this triangulation is an isolated element in the graph of triangulations of the point configuration it has been a central open question in polytope combinatorics in the last decade whether point configurations exist for which this graph is not connected see e g 37 question 1 2 and 48 challenge 3 we also construct a 234 dimensional polytope with 552 vertices whose graph of triangulations has an isolated element our construction is likely to have an impact in algebraic geometry too via the connections between lattice polytopes and toric varieties 21 23 31 43 for example in 2 section 2 and 24 section 4 the different authors study algebraic schemes based on the poset of subdivisions of an integer point configuration the connectivity of these schemes and of the graph of triangulations are equivalent see section 4 3 in particular corollary 4 9 the graph of triangulations is also related to the baues poset which appears in oriented matroid theory zonotopal tilings and hyperplane arrangements so our result has implications in these areas let a be a finite point set in the real affine space r a polyhedral subdivision of a is a geometric polyhedral complex with vertices in a which covers the convex hull of a if all the cells are simplices then it is a triangulation more combinatorial definitions are convenient if a is not in convex position i e if some element of a is not a vertex of the convex hull see definitions 4 1 and 1 1 for details and also 6 21 chapter 7 36 47 chapter 9 or the monograph in preparation 14 there are at least the following three ways to give a structure to the collection of all triangulations of a point configuration a a flips geometric bistellar operations or flips are the minimal changes which can be made in a triangulation of a to produce a new one see definition 1 3 a particular case is the familiar diagonal edge flip in two dimensional triangulations of frequent use in computational geometry and geometric combinatorics

the nef cone of toroidal compactifications of cal a _4 the moduli space of principally polarised abelian 4 folds can be compactified in several different ways by toroidal methods here we consider in detail the igusa compactification and the second voronoi compactification we describe in both cases the cone of nef cartier divisors the proof depends on a detailed description of the voronoi compactification which makes it possible to proceed by induction using the known description of the nef cone for compactifications of mathcal a 3 the igusa compactification has a non mathbb q factorial singularity which is resolved by a single blow up this resolution is the voronoi compactification the exceptional divisor e is a toric fano variety of dimension 9 the other boundary divisor d corresponds to degenerations with corank 1 after imposing a level structure in order to avoid certain technical complications we show that the closure of d in the voronoi compactification maps to the voronoi compactification of mathcal a 3 the toric description of the exceptional divisor allows us to describe the map in sufficient detail to estimate the intersection numbers needed this inductive process is only valid for the voronoi compactification the result for the igusa compactification is deduced from the voronoi compactification

gromov witten theory of deligne mumford stacks given a smooth complex deligne mumford stack cal x with a projective coarse moduli space we introduce gromov witten invariants of cal x and prove some of their basic properties including the wdvv equation

new approach to self avoiding walks as a critical phenomenon a new approach is developed to self avoiding walks as a critical phenomenon the approach is based on a simple assumption made on the step step correlation function this leads to a single scaling field related to the degree of polymerisation as a result all critical exponents are related to a single exponent moreover the order parameter and critical fields have a clear physical meaning the present interpretation of self avoiding walks as critical suggests an additional meaning of the exponent y in other critical systems

new method of analysing self avoiding walks in four dimensions a new method based on the concept of fractal dimensionality is used to study the problem of self avoiding walks in a four dimensional lattice the authors find from monte carlo simulations that the confluent logarithmic exponent related to the end to end distance is 1 8 or 0 01 in excellent agreement with the prediction derived from the n to 0 vector model

a virtual fundamental class construction for the moduli space of c n equivariant morphisms abstract let x be a smooth projective variety with the action of c n the article describes the moduli space of c n equivariant morphisms from stable toric varieties into x as the inverse limit of the git quotients of x and their flips when these spaces are enhanced with a naturally associated deligne mumford stack structure this description is used for constructing a class in the chow group of the moduli space of dimension dim x n which is invariant under equivariant deformations of x

the self avoiding walk on the honeycomb lattice the author analyses the chain generating function for the self avoiding walk problem on the honeycomb lattice she uses the exact critical point value of nienhuis and finds exponent values of gamma approximately 1 344 and delta 1 approximately 0 93 for the dominant and first confluent exponents the former value is in agreement with nienhuis exact result second and third confluent exponents delta 2 approximately 1 2 and delta 3 approximately 1 5 are also identified these exponents being consistent with monte carlo results and nienhuis exact irrelevant eigenvalue respectively

toroidal crossings and logarithmic structures abstract we generalize friedman s notion of d semistability which is a necessary condition for spaces with normal crossings to admit smoothings with regular total space our generalization deals with spaces that locally look like the boundary divisor in gorenstein toroidal embeddings in this situation we replace d semistability by the existence of global log structures for a given gerbe of local log structures this leads to cohomological descriptions for the obstructions existence and automorphisms of log structures we also apply toroidal crossings to mirror symmetry by giving a duality construction involving toroidal crossing varieties whose irreducible components are toric varieties this duality reproduces a version of batyrev s construction of mirror pairs for hypersurfaces in toric varieties but it applies to a larger class including degenerate abelian varieties

a generalization of voronoi s reduction theory and its application we consider voronoi s reduction theory of positive definite quadratic forms which is based on delone subdivision we extend it to forms and delone subdivisions having a prescribed symmetry group even more general the theory is developed for forms which are restricted to a linear subspace in the space of quadratic forms we apply the new theory to complete the classification of totally real thin algebraic number fields which was recently initiated by bayer fluckiger and nebe moreover we apply it to construct new best known sphere coverings in dimensions 9 15

complete moduli in the presence of semiabelian group action i prove the existence and describe the structure of moduli space of pairs p consisting of a projective variety p with semiabelian group action and an ample cartier divisor on it satisfying a few simple conditions every connected component of this moduli space is proper a component containing a projective toric variety is described by a configuration of several polytopes the main one of which is the secondary polytope on the other hand the component containing a principally polarized abelian variety provides a moduli compactification of a g the main irreducible component of this compactification is described by an infinite periodic analog of the secondary polytope and coincides with the toroidal compactification of a g for the second voronoi decomposition

the uniformization of the moduli space of principally polarized abelian 6 folds author s alexeev valery donagi ron farkas gavril izadi elham ortega angela abstract abstractstarting from a beautiful idea of kanev we construct a uniformization of the moduli space mathcal a 6 of principally polarized abelian 6 folds in terms of curves and monodromy data we show that the general principally polarized abelian variety of dimension 6 is a prym tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the weyl group of the e 6 lattice along the way we establish numerous facts concerning the geometry of the hurwitz space of such e 6 covers including 1 a proof that the canonical class of the hurwitz space is big 2 a concrete geometric description of the hodge hurwitz eigenbundles with respect to the kanev correspondence and 3 a description of the ramification divisor of the prym tyurin map from the hurwitz space to mathcal a 6 in the terms of syzygies of the abel prym tyurin curve

extensions of the universal theta divisor abstract the jacobian varieties of smooth curves fit together to form a family the universal jacobian over the moduli space of smooth pointed curves and the theta divisors of these curves form a divisor in the universal jacobian in this paper we describe how to extend these families over the moduli space of stable pointed curves using a stability parameter we then prove a wall crossing formula describing how the theta divisor varies with this parameter we use this result to analyze divisors on the moduli space of smooth pointed curves that have recently been studied by grushevsky zakharov hain and muller finally we compute the pullback of the theta divisor studied in alexeev s work on stable semiabelic varieties and in caporaso s work on theta divisors of compactified jacobians

contact cohomology of the projective plane we construct an associative ring which is a deformation of the quantum cohomology ring of the projective plane just as the quantum cohomology encodes the incidence characteris tic numbers of rational plane curves the contact cohomology encodes the tangency characteristic numbers 1 introduction in this paper we construct an associative ring which we call the contact cohomology ring of the projective plane we believe that an analogous construction should work for all homogeneous varieties but in our proof of associativity we rely on certain technical results from our earlier paper on characteristic numbers of rational plane curves ek98 as we formulate it in section 3 of the present paper the quantum product is actually a whole family of products parametrized by elements of the chow ring

fractals in monte carlo simulations of a short polyelectrolyte fractal dimensionalities for charged polymers are calculated over a continuous range of bjerrum length for chains with various numbers of beads n drastically decreasing values of characteristic of a phase transition are found as increases from zero in the range a the fractal dimensionality approaches unity as predicted by de gennes et al de gennes p g pincus p valesco r m and brochard f 1976 j physique 37 1461 close to the onset of order a newly developed smoothing algorithm yields a substantial improvement in the mc results and reveals an interesting fan shaped behaviour of exponents and prefactor components at slightly above zero it also enables discernment of a non monotonic behaviour of versus n at and close to zero differences found in bending between short and long chains may provide an additional explanation for the stability of benzene and the low stability of conjugated rings of n 6 based on fractal concepts monomer densities are derived at various and it is suggested that the drastic density changes at a are evidence of the first order nature of this phase transition

on three dimensional self avoiding walks long self avoiding walks of up to 2400 steps have been generated on the sc and bcc lattices using an improved monte carlo technique analysis of the asymptotic length dependence of the end to end distance and radius of gyration of the walks leads to values in the range 0 591 0 593 for the critical exponent nu the simple power law dependence on length provides an excellent fit to the data over walk lengths between 120 and 2400 with fluctuations around the asymptotic results averaging to a mere 0 13 systematic deviations from asymptotic behaviour in short self avoiding walks have also been examined using monte carlo generated walks ranging in length from 12 to 60 the results do not support the existence of the non analytic correction predicted by the renormalisation group in the light of this unexpected result available series expansions for walks on the fcc lattice have been re examined and previous claims to have observed non analytic behaviour questioned equal if not better convergence of the extrapolated series can be obtained without resorting to non analytic correction terms finally an analysis has been made of the radius of gyration series to which several new terms have been added

stable reductive varieties ii projective case abstract we construct a moduli space of stable projective pairs with a nontrivial action of a connected reductive group these stable reductive pairs are higher dimensional analogs of stable n pointed curves and generalize to the non commutative case a functorial compactification of moduli of abelian varieties and of toric pairs we prove that confirming a prediction of log minimal program stable reductive pairs have semi log canonical singularities

the gauss map of a genus three theta divisor a smooth complex curve is determined by the gauss map of the theta divisor of the jacobian variety of the curve the gauss map is invariant with respect to the 1 map of the jacobian we show that for a generic genus three curve the gauss map is locally z 2 stable one method of proof is to analyze the first order z 2 deformations of the gauss map of a hyperelliptic theta divisor

theta characteristics on algebraic curves the theory of theta characteristics is developed algebraically so that it may be applied to possibly singular and or reducible algebraic curves the configuration of theta characteristics on a curve is described in terms of its singularities with applications to the geometry of plane quartic curves and the problem of appolonius some results on gorenstein local rings are appended 0 introduction the theory of theta characteristics has been studied from different points of view and often inadvertently cf for example the 19th century literature on bitangents to smooth plane quartic curves e g 4 and 10 until recently however virtually all treatments of theta characteristics per se involved the transcendental theory of the theta function associated to a compact riemann surface then in 1971 mumford in 8 gave a completely algebraic treatment of the subject formulating the main results and proving them without reference to the transcendental theory this in turn suggested that theta characteristics could be studied in a more general context for example on possibly singular and or reducible algebraic curves in this paper we undertake such an analysis in 1 following mumford we establish some of the basic facts about theta characteristics algebraically so as to apply to more general curves in fact 1 10 i and 1 13 in their present generality are due to mumford while he assumes throughout 8 that his curve x is smooth his proof seems to apply as given the argument for 1 13 here is somewhat different in order to make the proof more elementary and selfcontained one bonus is the result 1 11 on the subvariety of the moduli space 9thg consisting of curves with large semicanonical linear series this fact seemingly can be proved only via the algebraic set up in 2 we investigate the number and configuration of odd and even theta characteristics on a singular curve the results are expressed in the two main theorems 2 12 and 2 22 giving a more or less complete answer to these questions in terms of the singularities of the curve interestingly a recurrent motif in these arguments is one introduced by mumford in 8 the behavior of the maximal isotropic subspaces for a nondegenerate quadratic form on an even dimensional vector space here the mainspring of theorems 1 13 2 12 and 2 22 is assertion 1 4 below that the sum of the dimensions of the pairwise intersections of any three such subspaces is congruent to n mod 2 received by the editors july 11 1980 and in revised form october 22 1981 1980 mathematics subject classification primary 14h99 14h30 secondary 14h20 14h40 14k25 research partially supported by nsf grant mcs78 04008 aol dl 982 american mathematical society 0002 9947 8 1 0000 0371 10 25