the thick thin decomposition and the bilipschitz classification of normal surface singularities we describe a natural decomposition of a normal complex surface singularity x 0 into its thick and thin parts the former is essentially metrically conical while the latter shrinks rapidly in thickness as it approaches the origin the thin part is empty if and only if the singularity is metrically conical the link of the singularity is then seifert fibered in general the thin part will not be empty in which case it always carries essential topology our decomposition has some analogy with the margulis thick thin decomposition for a negatively curved manifold however the geometric behavior is very different for example often most of the topology of a normal surface singularity is concentrated in the thin parts

good filtrations and strong f regularity of the ring of u p invariants abstract let k be an algebraically closed field of positive characteristic g a reductive group over k and v a finite dimensional g module let p be a parabolic subgroup of g and u p its unipotent radical we prove that if s sym v has a good filtration then s u p is strongly f regular

theta caracteristiques des courbes tracees sur une surface lisse nous generalisons la notion de theta caracterisque sur une courbe lisse a une courbe de gorenstein puis construisons un espace de modules t au sens grossier pour les theta caracteristiques des courbes de degre d sur une surface algebrique projective lisse nous montrons que le schema t est projectif et determinons l ensemble de ses points fermes ensuite nous generalisons le theoreme d atiyah et mumford sur l invariance sous deformation de la dimension modulo 2 de l espace des sections globales d une theta caracterique a notre situation ce qui donne en corollaire que t est reunion de deux fermes disjoints puis nous donnons un critere de lissite et calculons l espace tangent en un point stable a t enfin nous considerons le cas ou la surface est le plan projectif nous montrons que t n a que des singularites rationnelles et calculons sa dimension ensuite nous decomposons t en comosantes irreductibles en utilisant la decomposition sur l ouvert des courbes lisses de beauville et catanese puis donnons une description generique de ces composantes irreductibles

cyclic covers of rings with rational singularities we examine some recent work of phillip griffith on etale covers and fibered products from the point of view of tight closure theory while it is known that cyclic covers of gorenstein rings with rational singularities are cohen macaulay we show this is not true in general in the absence of the gorenstein hypothesis specifically we show that the canonical cover of a q gorenstein ring with rational singularities need not be cohen macaulay

a characterization of rational singularities is given bythe reynolds operator a well known and widely used theoremstates that in characteristic 0 canonicalsingularities are cohen macaulay and therefore rational see e and kmm the original proofs are based on a very clever use of grothendieck duality simulta neously for a resolution and its restriction onto the exceptional divisor and on a doubleloop induction kollar gave a simpler proof in k2 11 without using derived cate gories but still relying on a technically hard vanishing theorem recently kollar andmori found a simple proof allowing nonempty boundaries they do not use derivedcategories either but restrict to the projective case see km 5 18 these proofs areingenious but one would like to have a simple natural proof at least in the classical case when the boundary is empty as an application of theorem1 a simple proof is given here in the classical case but without the projective assumption this proof seems even simpler than that ofkollar and mori derived categories and grothendieck duality are used but in sucha simple way that one is tempted to say that this proof is the most natural one notealso that everything used here was already available when the question was raised forthe rst time a statement similar to theorem 1 was given in k2 3 12 some ideas of the

the quantization conjecture revisited a strong version of the quantization conjecture of guillemin and sternberg is proved for a reductive group action on a smooth compact polarized variety x l the cohomologies of l over the git quotient x g equal the invariant part of the cohomologies over x this generalizes the theorem of gs on global sections and strengthens its subsequent extensions jk li to riemannroch numbers remarkable by products are the invariance of cohomology of vector bundles over x g under a small change in the defining polarization or under shift desingularization as well as a new proof of boutot s theorem also studied are equivariant holomorphic forms and the equivariant hodgeto de rham spectral sequences for x and its strata whose collapse is shown one application is a new proof of the borel weil bott theorem of ti for the moduli stack of c bundles over a curve and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures collapse of the hodge to de rham sequences for these stacks is also shown

on the littlewood richardson polynomials we prove the equivalence of several descriptions of generators of rings of semiinvariants of quivers due to domokos and zubkov schofield and van den bergh and our earlier work we also show that the dimensions of semi invariants of weights n depend polynomially on n 2002 elsevier science usa all rights reserved

liaison and castelnuovo mumford regularity in this article we establish bounds for the castelnuovo mumford regularity of projective schemes in terms of the degrees of their defining equations the main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities when applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals

a generalization of tight closure and multiplier ideals we introduce a new variant of tight closure associated to any fixed ideal a which we call tight closure and study various properties thereof in our theory the annihilator ideal r a of all tight closure relations which is a generalization of the test ideal in the usual tight closure theory plays a particularly important role we prove the correspondence of the ideal a and the multiplier ideal associated to a or the adjoint of a in lipman s sense in normal q gorenstein rings reduced from characteristic zero to characteristic p 0 also in fixed prime characteristic we establish some properties of r a similar to those of multiplier ideals e g a briancon skoda type theorem subadditivity etc with considerably simple proofs and study the relationship between the ideal and the f rationality of rees algebras

cox rings and combinatorics given a variety x with a finitely generated total coordinate ring we describe basic geometric properties of x in terms of certain combinatorial structures living in the divisor class group of x for example we describe the singularities we calculate the ample cone and we give simple fano criteria as we show by means of several examples the results allow explicit computations as immediate applications we obtain an effective version of the kleiman chevalley quasiprojectivity criterion and the following observation on surfaces a normal complete surface with finitely generated total coordinate ring is projective if and only if any two of its non factorial singularities admit a common affine neighbourhood

the core of zero dimensional monomial ideals the core of an ideal is the intersection of all its reductions we describe the core of a zero dimensional monomial ideal i as the largest monomial ideal contained in a general reduction of i this provides a new interpretation of the core in the monomial case as well as an efficient algorithm for computing it we relate the core to adjoints and first coefficient ideals and in dimension two and three we give explicit formulas

the ring of sections of a complete symmetric variety we study the ring of sections a x of a complete symmetric variety x that is of the wonderful completion of g h where g is an adjoint semisimple group and h is the fixed subgroup for an involutorial automorphism of g we find generators for pic x we generalize the prv conjecture to complete symmetric varieties and construct a standard monomial theory for a x that is compatible with g orbit closures in x this gives a degeneration result and the rational singularityness for a x 2003 elsevier science usa all rights reserved

on a non vanishing conjecture of kawamata and the core of an ideal we show under suitable hypothesis which are sharp in a certain sense that the core of an m primary ideal in a regular local ring of dimension d is equal to the adjoint or multiplier ideal of its d th power this generalizes the fundamental formula for the core of an integrally closed ideal in a two dimensional regular local ring due to huneke and swanson we also find a generalization of this result to singular non regular settings which we show to be intimately related to the problem of finding non zero sections of ample line bundles on projective varieties in particular we show that a graded analog of our formula for core would imply a remarkable conjecture of kawamata predicting that every adjoint ample line bundle on a smooth variety admits a non zero section

algebras defined by powers of determinantal ideals let x be an m x n matrix of indeterminates over a field k a k x and i z x the ideal generated by the t minors of x the main objects of this article are the rees algebra 2 y these characteristics will be called non exceptional we show how this result generalizes to an arbitrary integral domain of coefficients the intersection of primary ideals which gives i in non exceptional characteristics always is the integral closure of i it follows immediately from the primary decomposition that the powers i are integrally closed in non exceptional characteristics therefore is a normal domain and the primary decomposition of i turns out easy giving some insight into the structure of y an interesting observation the primary decomposition of the ideals i can be computed very quickly if one knows in advance that all these ideals are integrally closed the best results are obtained in characteristic 0 since one has a multiplicity free action of the linearly reductive group gl m k x gl n k on k x under which z is stable cf dep or bv sect 111 applying the theory of u invariants kraft kr one shows that g has rational singularities in particular and hence 9 are cohen macaulay rings we have no doubt that and 3 are cohen macaulay in arbitrary non exceptional characteristic it seems however that in exceptional characteristic they are as far as possible from this property the case t 2

dynamical behavior of an epidemic model with a nonlinear incidence rate abstract in this paper we study the global dynamics of an epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action by carrying out global qualitative and bifurcation analyses it is shown that either the number of infective individuals tends to zero as time evolves or there is a region such that the disease will be persistent if the initial position lies in the region and the disease will disappear if the initial position lies outside this region when such a region exists it is shown that the model undergoes a bogdanov takens bifurcation i e it exhibits a saddle node bifurcation hopf bifurcations and a homoclinic bifurcation existence of none one or two limit cycles is also discussed

half canonical series on algebraic curves preprint enviat per a la seva publicacio en una revista cientifica transactions of the american mathematical society vol 302 no 1 1987 pp 99 115 https www jstor org stable 2000899

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

automorphismes reels d un fibre et operateurs de cauchy riemann soit n c n un fibre vectoriel complexe muni d une structure reelle au dessus d une courbe reelle sigma g c sigma de genre g in mathbb n nous etudions le signe de l action des automorphismes de n c n relevant l identite de sigma g sur les orientations du fibre determinant au dessus de l espace des operateurs de cauchy riemann reels de n c n ce signe s obtient comme le produit de deux termes le premier calcule la signature des permutations induite par les automorphismes sur les structures pin pm de la partie reelle de n c n le second provient de l action des automorphismes du fibre sur les classes de bordisme de structures spin reelles de sigma g c sigma

occupation times in systems of null recurrent markov processes we study asymptotic properties of differences of occupation times for infinite systems of noninteracting markovian particles under a suitable normalisation we prove convergence in law to a nondegerate gaussian field we also obtain large deviations properties these results generalise previous results obtained separately by both authors

large deviations for independent random walks on the line for a system of infinitely many independent symmetric random walks on z let k n x be the number of visits to x z from time 0 to n 1 the probabilities of some rare events involving k n 0 k n 1 are estimated as n and the corresponding large deviation rate functions are derived for both deterministic and invariant initial distributions the dependence on the initial distributions is discussed a simple method is used for guessing at the rate functions this method is effective for independent random walks on the line and is worth exploring in more general settings

long term analysis of a stochastic sirs model with general incidence rates this paper investigates a stochastic sirs epidemic model with an incidence rate that is sufficiently general and that covers many incidence rate models considered to date in the literature we clas

pure subrings of regular rings are pseudo rational we prove a generalization conjectured by aschenbrenner and schoutens 2003 of the hochster roberts boutot kawamata theorem let r s be a pure homomorphism of equicharacteristic zero noetherian local rings if s is regular then r is pseudo rational and if r is moreover q gorenstein then it is pseudo log terminal

moderately ramified actions in positive characteristic in characteristic 2 and dimension 2 wild mathbb z 2 mathbb z actions on k u v ramified precisely at the origin were classified by artin who showed in particular that they induce hypersurface singularities we introduce in this article a new class of wild quotient singularities in any characteristic p 0 and dimension n ge 2 arising from certain non linear actions of mathbb z p mathbb z on the formal power series ring k u 1 dots u n these actions are ramified precisely at the origin and their rings of invariants in dimension 2 are hypersurface singularities with an equation of a form similar to the form found by artin when p 2 in higher dimension the rings of invariants are not local complete intersection in general but remain quasi gorenstein we establish several structure results for such actions and their corresponding rings of invariants

cohomology of the hilbert scheme of points on a surface with values in representations of tautological bundles let x a smooth quasi projective algebraic surface l a line bundle on x let x n the hilbert scheme of n points on x and l n the tautological bundle on x n naturally associated to the line bundle l on x we explicitely compute the image l n of the tautological bundle l n for the bridgeland king reid equivalence d b x n d bn x n in terms of a complex cof sn equivariant sheaves in d bn x n we give moreover a characterization of the image l n l n in terms of of the hyperderived spectral sequence e p q 1 associated to the derived k fold tensor power of the complex c l the study of the sn invariants of this spectral sequence allows to get the derived direct images of the double tensor power and of the general k fold exterior power of the tautological bundle for the hilbert chow morphism providing danila brion type formulas in these two cases this yields easily the computation of the cohomology of x n with values in l n l n andkl n

local cohomology with support in ideals of maximal minors abstract suppose that k is a field of characteristic zero x is an r s matrix of indeterminates where r s and r k x is the polynomial ring over k in the entries of x we study the local cohomology modules h i i r where i is the ideal of r generated by the maximal minors of x we identify the indices i for which these modules vanish compute h i i r at the highest nonvanishing index i r s r 1 and characterize all nonzero ones as submodules of certain indecomposable injective modules these results are consequences of more general theorems regarding linearly reductive groups acting on local cohomology modules of polynomial rings

factorisation of generalised theta functions i we prove a version of factorisation relating the space of sections of theta bundles on the moduli spaces of parabolic rank 2 vector bundles on curves of genusg andg 1

der kanonische modul eines invariantenrings abstract we calculate the canonical module of a ring of invariants r of a reductive group acting on an affine variety from that we derive a criterion for the gorenstein property to hold for r in case r is the ring of invariants of a graded ring we obtain lower and upper bounds for the degree of the generating function of r

large deviations for independent random walks we consider a system of independent random walks on let n x be the number of particles atx at timen and letl n x 0 x n x be the total occupation time ofx by timen in this paper we study the large deviations ofl n 0 l n 1 the behavior we find is much different from that ofl n 0 we investigate the limiting behavior when the initial configurations has asymptotic density 1 and when 0 x are i i d poisson mean 1 finding that the asymptotics are different in these two cases

large deviations for the three dimensional super brownian motion let t dx denote a three dimensional super brownian motion with deterministic initial state 0 dx dx the lebesgue measure let v r 3 r be holder continuous with compact support not identically zero and such that r 3v x dx 0 we show that log p 0 t r 3v x s dx ds bt 3 4 is of order t 1 2 as t for b 0 this should be compared with the known result for the case r 3v x dx 0 in that case the normalization bt 3 4 b 0 must be replaced by bt b r 3v x dx in order that the same statement hold true while this result only captures the logarithmic order the method of proof enables us to obtain complete results for the corresponding moderate deviations and central limit theorems

d modules bernstein sato polynomials and f invariants of direct summands abstract we study the structure of d modules over a ring r which is a direct summand of a polynomial or a power series ring s with coefficients over a field we relate properties of d modules over r to d modules over s we show that the localization r f and the local cohomology module h i i r have finite length as d modules over r furthermore we show the existence of the bernstein sato polynomial for elements in r in positive characteristic we use this relation between d modules over r and s to show that the set of f jumping numbers of an ideal i r is contained in the set of f jumping numbers of its extension in s as a consequence the f jumping numbers of i in r form a discrete set of rational numbers we also relate the bernstein sato polynomial in r with the f thresholds and the f jumping numbers in r

absolute integral closures are big cohen macaulay algebras in characteristic p throughout this paper ring means commutative ring with identity and modules are unital our main interest is in local rings i e noetherian rings r m with a unique maximal ideal m in such a ring xx xn e m is a system of parameters if m has a power in the ideal x xn r and n is the krull dimension of r when r is complete and contains a field this means that r is module finite over a formal power series subring k xx xn a i is called cohen macaulay if some equivalently every system of parameters is a regular sequence in i which means that every xi x is a nonzerodivisor on r xx xt r for 0 i n 1 in the case where r is module finite over the formal power series subring a this means that r is a free 4 module for many theorems of commutative algebra and of algebraic geometry the cohen macaulay condition possibly on the local rings of a variety is just what is needed to make the theory work our main result which is given below in an algebraic form in theorem 1 and in a geometric form in theorem 2 asserts that under mild conditions on a local ring r of positive prime characteristic p one can correct the failure of the cohen macaulay condition in r itself by passing to a very large integral extension of r it is worth emphasizing that this is quite false in characteristic 0 and comes as a surprise we believe in characteristic p one can get an idea of how far these theorems are from the truth over a field of characteristic 0 from the following observation in the situation of theorem 2 when x is projectively normal the maps of cohomology are always injective by an easy trace argument despite the fact that both theorem 1 and theorem 2

rees algebras of f regular type abstract we study the f regularity of rees algebras r i a it in terms of the global f regularity of the blowing up x proj r i of spec a as it reads global f regularity is a global analog of strong f regularity defined via splitting of frobenius maps in prime characteristic and these notions are extended to characteristic zero by reduction modulo p 0 we study in detail the case where a m is a two dimensional local ring and i is an m primary ideal in characteristic zero the condition for r i to have f regular type is described in terms of the dual graph of a resolution x on which i o x is invertible we also prove some miscellaneous results concerning singularities of rees algebras and extended rees algebras of higher dimension

a stochastic sirs epidemic model with nonlinear incidence rate a stochastic sirs model with ratio dependent incidence rate is developed the global dynamics of the deterministic model is shown the stochastic dynamics of the sde model is given the existence of a unique stationary distribution of the sde model is displayed in this paper we investigate the global dynamics of a general sirs epidemic model with a ratio dependent incidence rate and its corresponding stochastic differential equation version for the deterministic model we show that the basic reproduction number r0 determines whether there is an endemic outbreak or not if r0 1 the disease persists for the stochastic model we show that its related reproduction number r0s can determine whether there is a unique disease free stationary distribution or a unique endemic stationary distribution in addition we provide analytic results regarding the stochastic boundedness and permanence extinction one of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak which can provide us some useful control strategies to regulate disease dynamics

character varieties of free groups are gorenstein but not always factorial abstract fix a rank g free group f g and a connected reductive complex algebraic group g let x f g g be the g character variety of f g when the derived subgroup d g g is simply connected we show that x f g g is factorial which implies it is gorenstein and provide examples to show that when dg is not simply connected x f g g need not even be locally factorial despite the general failure of factoriality of these moduli spaces using different methods we show that x f g g is always gorenstein

hankel determinantal rings have rational singularities abstract hankel determinantal rings i e determinantal rings defined by minors of hankel matrices of indeterminates arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves they may also be viewed as linear specializations of generic determinantal rings we prove that over fields of characteristic zero hankel determinantal rings have rational singularities in the case of positive prime characteristic we prove that they are f pure independent of the characteristic we give a complete description of the divisor class groups of these rings and show that each divisor class group element is the class of a maximal cohen macaulay module

multi symbolic rees algebras and strong f regularity let i be a divisorial ideal of a strongly f regular ring a k i watanabe raised the question whether the symbolic rees algebra mathcal r s i oplus n ge 0 i n is cohen macaulay whenever it is noetherian we develop the notion of multi symbolic rees algebras and use this to show that mathcal r s i is indeed cohen macaulay whenever a certain auxiliary ring is finitely generated over a

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

generalized theta functions strange duality and odd orthogonal bundles on curves this paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial stiefel whitney class and the associated space of twisted spin bundles in particular we prove a verlinde type formula and a dimension equality that was conjectured by oxbury wilson modifying hitchin s argument we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection we furthermore address the issue of strange duality for odd orthogonal bundles and we demonstrate that the naive conjecture fails in general a consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the hitchin connection for these moduli spaces finally we answer a question of nakanishi tsuchiya about rank level duality for conformal blocks on the pointed projective line with spin weights

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

representation growth and rational singularities of the moduli space of local systems let g be a semisimple algebraic group defined over mathbb q p and let gamma be a compact open subgroup of g mathbb q p we relate the asymptotic representation theory of gamma and the singularities of the moduli space of g local systems on a smooth projective curve proving new theorems about both 1 we prove that there is a constant c independent of g such that the number of n dimensional representations of gamma grows slower than n c confirming a conjecture of larsen and lubotzky in fact we can take c 3 cdot text dim e 8 1 745 we also prove the same bounds for groups over local fields of large enough characteristic 2 we prove that the coarse moduli space of g local systems on a smooth projective curve of genus at least lceil c 2 rceil 1 374 has rational singularities for the proof we study the analytic properties of push forwards of smooth measures under algebraic maps more precisely we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities

f rationality of rees algebras in this paper using the notion of the tight integral closure we will give a criterion for f rationality of rees algebras of m primary ideals in a cohen macaulay local ring as its application we prove the following results 1 in dimension two if a is f rational and i is integrally closed then the rees algebra r i is f rational on the other hand in higher dimensions we construct many examples of cohen macaulay normal rees algebras which are not f rational 2 if both a and r i are f rational then so is the extended rees algebra r i 3 if r i is f rational and a g i 1 then a is f rational on the other hand using resolution of singularities we will prove that a two dimensional rational singularity always admits f rational rees algebras in particular this theorem gives another way than that devised by watanabe 1997 j pure appl algebra122 323 328 to construct counterexamples to the boutot type theorem for f rational rings