on the stability of extensions of tangent sheaves on kahler einstein fano calabi yau pairs let s be a smooth projective variety and delta a simple normal crossing mathbb q divisor with coefficients in 0 1 for any ample mathbb q line bundle l over s we denote by mathscr e l the extension sheaf of the orbifold tangent sheaf t s log delta by the structure sheaf mathcal o s with the extension class c 1 l we prove the following two results these results generalize tian s result where k s is ample and delta emptyset we give two applications of these results the first is to study a question by borbon spotti about the relationship between local euler numbers and normalized volumes of log canonical surface singularities we prove that the two invariants differ only by a factor 4 when the log canonical pair is an orbifold cone over a marked riemann surface in particular we complete the computation of langer s local euler numbers for any line arrangements in mathbb c 2 the second application is to derive miyaoka yau type inequalities on k semistable log smooth fano pairs and calabi yau pairs which generalize some chern number inequalities proved by song wang

symplectic automorphisms of prime order on k3 surfaces abstract we study algebraic k3 surfaces defined over the complex number field with a symplectic automorphism of prime order in particular we consider the action of the automorphism on the second cohomology with integer coefficients by a result of nikulin this action is independent on the choice of the k3 surface with the help of elliptic fibrations we determine the invariant sublattice and its perpendicular complement and show that the latter coincides with the coxeter todd lattice in the case of automorphism of order three

the index of an eisenstein ideal and multiplicity one mazur s fundamental work on the eisenstein ideal of prime level has a variety of arithmetic applications in this article we generalize some of his work to square free level more specifically we attempt to compute the index of an eisenstein ideal and the dimension of the mathfrak m torsion of the modular jacobian variety where mathfrak m is an eisenstein maximal ideal in many cases the dimension of the mathfrak m torsion is 2 in other words a multiplicity one theorem holds

a divisorial valuation with irrational volume of an m primary ideal a r in fact if qm a for a fixed ideal a then it is evident that e a vol v the volume of a valuation has implicitly been studied already in 4 but it was first explicitly defined in 5 the terminology is intended to emphasize the relation with global invariants of linear series on projective varieties a natural question is to what extent the properties of vol v mirror those of the samuel multiplicity results of 5 9 assert that

on multiplicities of graded sequences of ideals abstract we generalize a result from l ein et al math ag 0202303 proving that for an arbitrary graded sequence of zero dimensional ideals the multiplicity of the sequence is equal to its volume this is done using a deformation to monomial ideals as a consequence of our result we obtain a formula which computes the multiplicity of an ideal i in terms of the multiplicities of the initial monomial ideals of the powers i m we use this to give a new proof of the inequality between multiplicity and the log canonical threshold from de fernex et al j alg geom to appear

construction of nikulin configurations on some kummer surfaces and applications a nikulin configuration is the data of 16 disjoint smooth rational curves on a k3 surface according to a well known result of nikulin if a k3 surface contains a nikulin configuration mathcal c then x is a kummer surface x mathrm km b where b is an abelian surface determined by mathcal c let b be a generic abelian surface having a polarization m with m 2 k k 1 for k 0 an integer and let x mathrm km b be the associated kummer surface to the natural nikulin configuration mathcal c on x mathrm km b we associate another nikulin configuration mathcal c we denote by b the abelian surface associated to mathcal c so that we have also x mathrm km b for k ge 2 we prove that b and b are not isomorphic we then construct an infinite order automorphism of the kummer surface x that occurs naturally from our situation associated to the two nikulin configurations mathcal c mathcal c there exists a natural bi double cover s rightarrow x which is a surface of general type we study this surface which is a lagrangian surface in the sense of bogomolov tschinkel and for k 2 is a schoen surface

analytic approximations of plurisubharmonic singularities we study several classes of isolated singularities of plurisubharmonic functions that can be approximated by analytic singularities with control over their residual monge ampere masses they are characterized in terms of green functions for demailly s approximations relative types and valuations furthermore the classes are shown to appear when studying graded families of ideals of analytic functions and the corresponding asymptotic multiplier ideals

cuspidal class number of the tower of modular curves x1 npn we consider a cuspidal class number which is the order of a subgroup of the full cuspidal divisor class group of x1 npn with p nmid n and n 1 by studying the second generalized bernoulli numbers we obtain results similar to ones ferrero and washington in ann math 2 109 2 377 395 1979 washington in invent math 49 87 97 1978 about the relative class numbers of cyclotomic mathbb z p extension of an abelian number field

epsilon entropy and critical distortion of random fields the link between epsilon entropy for binary random fields and the lee yang theorem in statistical mechanics makes it possible to prove the existence of positive critical distortion for most binary random fields with pair interactions and some with many body interactions lower bounds for critical distortion of random fields on two dimensional 2 d lattices have been obtained in particular the new lower bounds for some 2 d ising models improve upon previously known bounds

asymptotic multiplicities of graded families of ideals and linear series abstract we find simple necessary and sufficient conditions on a local ring r of dimension d for the limit lim n l r r i n n d to exist whenever i n is a graded family of m r primary ideals and give a number of applications we also give simple necessary and sufficient conditions on projective schemes over a field k for asymptotic limits of the growth of all graded linear series of a fixed kodaira iitaka dimension to exist

sur les groupes de transformations birationnelles des surfaces ni agissant par transformations bi rationnelles sur les surfaces complexes compactes kahl eriennes nous mon trons a que le groupe des transformations birationnelles d une surface satisfait l alternative de tits b que les actions birationnelles de groupes de kazhdan sur les surfaces sont toutes birationnellement conjugu eesa des actions homographiques sur le plan projectif et c que si f et g sont deux transformations birationnelles de surfaces qui commutent alors ou bien f pr eserve un pinceau de courbes ou bien l un des it er es g m de g m 0 concide avec un it er e f n de f n 2 z let s be any compact complex kahler surface and bir s the group of birational transformations of s we study the structure of nitely genera ted subgroups of bir s and prove three main results a bir s satises the tits alternative b if a group with kazhdan property t acts on a compact kahler surface by birational transformations then the action is conjugate to an action by linear projective transformations on the projec tive plane and c if f is an element of bir s which does not preserve any pencil of curves and if g commutes with f then a positive iterate of g coincides with an iterate of f the second statement provides a positive answer to zimmer s conjecture for birational actions on surfaces

ordinary representations and modular forms this paper establishes criteria for a representation gal q q gl2 qp to be modular in the sense that there is a newform f such that f fontaine and mazur have conjectured that is modular provided it is unramified outside of a finite set of primes irreducible has odd determinant and the restriction to a decomposition group dp at p satisfies certain conditions the following conjecture is a special case of fm95 ref 1 conjecture 3c

some mathematical properties of classical many body systems with repulsive interactions it is shown that for systems with repulsive non negative interactions the yang lee zeros of the grand canonical partition function lie on the real z axis as a result a series of successively improving lower and upper bounds for the thermodynamic ratio of density to activity is obtained in addition bounds on the transition activity and density are given in terms of the cluster coefficients

a decomposition theorem for binary markov random fields consider a binary markov random field whose neighbor structure is specified by a countable graph with nodes of uniformly bounded degree under a minimal assumption we prove a decomposition theorem to the effect that such a markov random field can be represented as the nodewise modulo 2 sum of two independent binary random fields one of which is white binary noise of positive weight said decomposition provides the information theorist with an exact expression for the per site rate distortion function of the random field over an interval of distortions not exceeding this weight we mention possible implications for communication theory probability theory and statistical physics

deformations of canonical singularities m reid defined the concept of canonical singularities as those which appear on canonical models of algebraic varieties whose canonical rings are finitely generated from the view point of the classification theory of algebraic varieties it is expected that small deformations of a variety which has only canonical singularities have only canonical singularities the main result of this paper states that this is the case

kahler einstein metrics and volume minimization abstract we prove that if a polarized q fano variety admits a kahler einstein metric then the normalized volume functional over the associated affine cone is globally minimized at the canonical divisorial valuation obtained by blowing up the vertex this is also generalized to the logarithmic and the orbifold settings a refinement of this result together with the result in 41 gives an equivalent characterization of k semistability for any smooth fano manifold we also prove that the valuation associated to the reeb vector field of a smooth sasaki einstein metric minimizes the normalized volume over the corresponding kahler cone these results strengthen and generalize the minimization result of martelli sparks yau

positivity and complexity of ideal sheaves the problem of bounding the complexity of a polynomial ideal in terms of the degrees of its generators has attracted a great deal of interest in recent years results in this direction go back at least as far as the classical work 17 of hermann on the ideal membership problem and the effective nullstellensatz of brownawell 4 and kollar 21 marks a major recent advance with the development of computational algebraic geometry the question has taken on increasing importance and it came into particularly clear focus through the influential paper 3 of bayer andmumford more recently the theorem of 8 and 20 concerning regularity of powers raises the question of bounding the complexity of powers of an ideal and suggests that asymptotically the picture should become very clean the aim of the present paper is to examine some of the results and questions of 3 30 and 7 from a geometric perspective in the spirit of 12 our thesis is that much of this material is clarified and parts rendered transparent when viewed through the lenses of vanishing theorems and intersection theory specifically motivated by thework 25 26 of paolettiwe introduce an invariant s j thatmeasures in effect howmuchonehas to twist an idealj in order tomake

vanishing sequences and okounkov bodies we define and study the vanishing sequence along a real valuation of sections of a line bundle on a normal projective variety building on previous work of the first author with huayi chen we prove an equidistribution result for vanishing sequences of large powers of a big line bundle and study the limit measure in particular the latter is described in terms of restricted volumes for divisorial valuations we also show on an example that the associated concave function on the okounkov body can be discontinuous at boundary points

deformations of certain reducible galois representations ii we examine the deformation theory of two dimensional mod p reducible indecomposable galois representations removing or relaxing many of the hypotheses of part i we are able to prescribe local conditions on our deformations by allowing ramification at a set of primes congruent to 1 mod p not 1 mod p 2 and satisfying some other splitting conditions the new hypotheses admit irreducible ordinary characteristic zero lifts of many mod p representations which by skinner wiles allows us to conclude the modularity of such representations

thresholds valuations and k stability abstract let x be a normal complex projective variety with at worst klt singularities and l a big line bundle on x we use valuations to study the log canonical threshold of l as well as another invariant the stability threshold the latter generalizes a notion by fujita and odaka and can be used to characterize when a q fano variety is k semistable or uniformly k stable it can also be used to generalize volume bounds due to fujita and liu the two thresholds can be written as infima of certain functionals on the space of valuations on x when l is ample we prove that these infima are attained in the toric case toric valuations achieve these infima and we obtain simple expressions for the two thresholds in terms of the moment polytope of l

invariance of plurigenera in this paper we give a proof of the following long conjectured result on the invariance of the plurigenera main theorem let p x d be a smooth projective family of compact complex manifolds parametrized by the open unit 1 disk d assume that the bers xt py1 t t 2 d are of general type then for every positive integer m the plurigenus dimc c xt mkxt is independent of t 2 d where kxt is the canonical line bundle of xt

uniform bounds and symbolic powers on smooth varieties the purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety in particular weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety let

the alternating group of degree 6 in the geometry of the leech lattice and k3 surfaces the alternating group of degree 6 is located at the junction of three series of simple non commutative groups simple sporadic groups alternating groups and simple groups of lie type it plays a very special role in the theory of finite groups we shall study its new roles both in a finite geometry of a certain pentagon in the leech lattice and also in the complex algebraic geometry of k3 surfaces

degenerations of amoebae and berkovich spaces we prove a continuity result for the fibers of the berkovich analytification of a complex algebraic variety with respect to the maximum of the archimedean norm and the trivial norm as a consequence we obtain generalizations of a result of mikhalkin and rullgard about degenerations of amoebae onto tropical varieties

valuations and multiplier ideals this article is the third of a series of work on a new approach to the study of singularities of various objects in a local two dimensional setting our focus in the present paper is on multiplier ideals and singularity exponents in the discussion below we fix an equicharacteristic zero two dimensional regular local ring r m with algebraically closed residue field an important example is the ring r do o holomorphic germs at the origin in c2 in fj1 we introduced the space v consisting of all r u 00 valued valuations on r centered at m and normalized by m 1 this space is naturally a tree it is a union of uncountably many real segments patched together in such a way that v remains homotopic to a point it is also an r tree in the classical sense for a natural metric we hence call v the valuative tree it encodes in a natural way all possible blowups of r centered at m and therefore gives a way of measuring quite precisely singularities of different kinds of objects the points in v that are not ends form the subtree vqm of quasimonomial valuations these valuations which can alternatively be characterized as abhyankar valuations of rank 1 or as

on simultaneous signal estimation and parameter identification using a generalized likelihood approach a common approach to blind deconvolution of bernoulli gaussian processes consists of performing both signal restoration and hyperparameter identification through maximization of a single generalized likelihood criterion it is shown on a simple example that the resulting hyperparameter estimates may not converge toward any meaningful value therefore other more reliable approaches should be adopted whenever possible

a finite group acting on the moduli space of k3 surfaces we consider the natural action of a finite group on the moduli space of polarized k3 surfaces which induces a duality defined by mukai for surfaces of this type we show that the group permutes polarized fourier mukai partners of polarized k3 surfaces and we study the divisors in the fixed loci of the elements of this finite group

uniform approximation of abhyankar valuation ideals in smooth function fields fix a rank one valuation centered at a smooth point x on an algebraic variety over a field of characteristic zero assume that is abhyankar that is that its rational rank plus its transcendence degree equal the dimension of the variety let a m denote the ideal of elements in the local ring of x whose valuations are at least m our main theorem is that there exists k 0 such that a mn is contained in a m k n for all m and n this can be viewed as a greatly strengthened form of izumi s theorem for abhyankar valuations centered on smooth complex varieties the proof uses the theory of asymptotic multiplier ideals

nikulin involutions and the chl string we study certain even eight curve configurations on a specific class of jacobian elliptic k3 surfaces with lattice polarizations of rank ten these configurations are associated with k3 double covers some of which are elliptic but not jacobian elliptic several non generic cases corresponding to k3 surfaces of higher picard rank are also discussed finally the results and the construction in question are interpreted in the context of the string dualities linked with the eight dimensional chl string

bauer furuta invariants under mathbb z _2 actions s bauer and m furuta defined a stable cohomotopy refinement of the seiberg witten invariants in this paper we prove a vanishing theorem of bauer furuta invariants for 4 manifolds with smooth mathbb z 2 actions as an application we give a constraint on smooth mathbb z 2 actions on homotopy k3 k3 and construct a nonsmoothable locally linear mathbb z 2 action on k3 k3 we also construct a nonsmoothable locally linear mathbb z 2 action on k3

prym varieties and moduli of polarized nikulin surfaces abstract we present a structure theorem for the moduli space r 7 of prym curves of genus 7 as a projective bundle over the moduli space of 7 nodal rational curves the existence of this parametrization implies the unirationality of r 7 and that of the moduli space of nikulin surfaces of genus 7 as well as the rationality of the moduli space of nikulin surfaces of genus 7 with a distinguished line using the results in genus 7 we then establish that r 8 is uniruled

nontrivial elements of sha explained through k3 surfaces we present a new method to show that a principal homogeneous space of the jacobian of a curve of genus two is nontrivial the idea is to exhibit a brauer manin obstruction to the existence of rational points on a quotient of this principal homogeneous space in an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose jacobians have nontrivial tate shafarevich group

strong modularity of reducible galois representations in this paper we call strongly modular those reducible semi simple odd mod l galois representations for which the conclusion of the strongest form of serre s original modularity conjecture holds under the assumption that the serre weight k satisfies l k 1 we give a precise characterization of strongly modular representations hence generalizing a classical theorem of ribet pertaining to the case of conductor 1 when the representation rho is not strongly modular we give a necessary and sufficient condition on the primes p not dividing nl for which it arises in level np where n denotes the conductor of rho this generalizes a result of mazur on the case n k 1 2

equivalences of twisted k3 surfaces we prove that two derived equivalent twisted k3 surfaces have isomorphic periods the converse is shown for k3 surfaces with large picard number it is also shown that all possible twisted derived equivalences between arbitrary twisted k3 surfaces form a subgroup of the group of all orthogonal transformations of the cohomology of a k3 surface

minimizing normalized volumes of valuations for any mathbb q gorenstein klt singularity x o we introduce a normalized volume function widehat mathrm vol that is defined on the space of real valuations centered at o and consider the problem of minimizing widehat mathrm vol we prove that the normalized volume has a uniform positive lower bound by proving an izumi type estimate for any mathbb q gorenstein klt singularity furthermore by proving a properness estimate we show that the set of real valuations with uniformly bounded normalized volumes is compact and hence reduce the existence of minimizers for widehat mathrm vol to a conjectural lower semicontinuity property for a smooth point we use an inequality of de fernex ein musta to show that the divisorial valuation associated to the exceptional divisor of the standard blow up is a minimizer of widehat mathrm vol finally the relation to fujita s work on divisorial stability is pointed out

a new method to estimate the critical distortion of random fields it was shown by c m newman and g a baker 1991 that the critical distortion d sub c of the ising model on z sup k is related to the radius of convergence r of the classical mayer series by d sub c r 1 r an approach from statistical mechanics to estimate r as well as d sub c by using the matrix representation of mayer series is found

moduli of theta characteristics via nikulin surfaces we study moduli spaces of k3 surfaces endowed with a nikulin involution and their image in the moduli space r g of prym curves of genus g we observe a striking analogy with mukai s well known work on ordinary k3 surfaces many of mukai s results have a very precise prym nikulin analogue for instance a general prym curve from r g is a section of a nikulin surface if and only if g 7 and g 6 furthermore r 7 has the structure of a fibre space over the corresponding moduli space of polarized nikulin surfaces we then use these results to study the geometry of the moduli space s g of even spin curves with special emphasis on the transition case of s 8 which is a 21 dimensional calabi yau variety

a toeplitz representation for repulsive systems a new method is presented for the calculation of thermodynamic properties of ensembles of particles interacting via repulsive forces a tridiagonal matrix representation is constructed in terms of the cluster integrals its asymptotic form is utilised to obtain very good numerical estimates of the thermodynamic properties moreover it is shown that the fluid branch may terminate at a critical activity zc which is identical to the activity at the fluid solid transition a relation is found between zc and the radius of convergence of the activity series via the asymptotic matrix elements

multiplier ideal sheaves and kahler einstein metrics of positive scalar curvature we present a method for proving the existence of kiihler einstein metrics of positive scalar curvature on certain compact complex manifolds and use the method to produce a large class of examples of compact kiihler einstein manifolds of positive scalar curvature suppose that m is a compact complex manifold of positive first chern class as is well known the existence of a kiihler einstein metric on m is equivalent to the existence of a solution to a certain complex monge ampere equation on m to solve this complex mongeampere equation by the method of continuity one needs only to establish the appropriate zeroth order a priori estimate suppose now that m does not admit a kiihler einstein metric so that the zeroth order a priori estimate fails to hold from this lack of an estimate we extract various global algebro geometric properties of m by introducing a coherent sheaf of ideals j on m called the multiplier ideal sheaf which carefully measures the extent to which the estimate fails the sheaf y is analogous to the subelliptic multiplier ideal sheaf that j j kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d neumann problem now j is a global algebro geometric object on m and it so happens that j satisfies a number of highly nontrivial global algebro geometric conditions including a cohomology vanishing theorem in particular the complex analytic subspace v c m cut out by j is nonempty connected and has arithmetic genus zero if v is zero dimensional then it is a single reduced point while if v is one dimensional then its support is a tree of smooth rational curves the logarithmic geometric genus of m v always vanishes these considerations place nontrivial global algebro geometric restric

convex bodies appearing as okounkov bodies of divisors abstract based on the work of okounkov okounkov 1996 15 2003 16 lazarsfeld and musta 2009 13 and kaveh and khovanskii preprint 10 have independently associated a convex body called the okounkov body to a big divisor on a smooth projective variety with respect to a complete flag in this paper we consider the following question what can be said about the set of convex bodies that appear as okounkov bodies we show first that the set of convex bodies appearing as okounkov bodies of big line bundles on smooth projective varieties with respect to admissible flags is countable we then give a complete characterisation of the set of convex bodies that arise as okounkov bodies of r divisors on smooth projective surfaces such okounkov bodies are always polygons satisfying certain combinatorial criteria finally we construct two examples of non polyhedral okounkov bodies in the first one the variety we deal with is fano and the line bundle is ample in the second one we find a mori dream space variety such that under small perturbations of the flag the okounkov body remains non polyhedral

on deformation rings of residually reducible galois representations and r t theorems we introduce a new method of proof for r t theorems in the residually reducible case we study the crystalline universal deformation ring r and its ideal of reducibility i of a mod p galois representation 0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non isomorphic constituents 1 and 2 under some assumptions on selmer groups associated with 1 and 2 we show that r i is cyclic and often finite using ideas and results of but somewhat different assumptions from bellaiche and chenevier we prove that i is principal for essentially self dual representations and deduce statements about the structure of r using a new commutative algebra criterion we show that given enough information on the hecke side one gets an r t theorem we then apply the technique to modularity problems for 2 dimensional representations over an imaginary quadratic field and a 4 dimensional representation over q

modularity of residual galois extensions and the eisenstein ideal for a totally real field f a finite extension f of fp and a galois character gf f unramified away from a finite set of places p p consider the bloch kato selmer group h h1 f 1 in an earlier paper of the authors it was proved that the number d of isomorphism classes of non semisimple reducible residual representations giving rise to lines in h which are modular by some f also unramified outside satisfies d n dimfh this was proved under the assumption that the order of a congruence module is greater than or equal to that of a divisible selmer group we show here that if in addition the relevant local eisenstein ideal j is non principal then d n when f q we prove the desired bounds on the congruence module and the selmer group we also formulate a congruence condition implying the non principality of j that can be checked in practice allowing us to furnish an example where d n

mmp for moduli of sheaves on k3s via wall crossing nef and movable cones lagrangian fibrations we use wall crossing with respect to bridgeland stability conditions to systematically study the birational geometry of a moduli space m of stable sheaves on a k3 surface x a we describe the nef cone the movable cone and the effective cone of m in terms of the mukai lattice of x b we establish a long standing conjecture that predicts the existence of a birational lagrangian fibration on m whenever m admits an integral divisor class d of square zero with respect to the beauville bogomolov form these results are proved using a natural map from the space of bridgeland stability conditions mathop mathrm stab nolimits x to the cone mathop mathrm mov nolimits x of movable divisors on m this map relates wall crossing in mathop mathrm stab nolimits x to birational transformations of m in particular every minimal model of m appears as a moduli space of bridgeland stable objects on x

the volume of an isolated singularity we introduce a notion of volume of a normal isolated singularity that gener alizes wahl s characteristic number of surface singularities to arbitrary dimensions we prove a basic monotonicity property of this volume under finite morphisms we draw sev eral consequences regarding the existence of non invertible finite endomorphisms fixing an isolated singularity using a cone construction we deduce that the anticanonical di visor of any smooth projective variety carrying a non invertible polarized endomorphism is pseudoeffective our techniques build on shokurov s b divisors we define the notion of nef weil b divisors and of nef envelopes of b divisors we relate the latter notion to the pull back of weil divisors introduced by de fernex and hacon using the subadditivity theorem for multiplier ideals with respect to pairs recently obtained by takagi we carry over to the isolated singularity case the intersection theory of nef weil b divisors formerly developed by boucksom favre and jonsson in the smooth case

rational surfaces with a large group of automorphisms we classify rational surfaces for which the image of the automorphisms group in the group of linear transformations of the picard group is the largest possible this answers a question raised by arthur coble in 1928 and can be rephrased in terms of periodic orbits of a birational action of an infinite coxeter group on ordered point sets in the projective plane modulo projective equivalence we also outline the classification of non rational surfaces with large automorphism groups

on critical distortion for markov sources corresp the authors underbound the critical distortion below which the rate distortion function of a markov chain equals its shannon lower bound the bound is tight in all cases in which the exact answer is currently known

local okounkov bodies and limits in prime characteristic this article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic these sequences which we call p families of ideals are ubiquitous in prime characteristic commutative algebra e g they occur naturally in the theories of tight closure hilbert kunz multiplicity and f signature we associate to each p family of ideals an object in euclidean space that is analogous to the newton okounkov body of a graded family of ideals which we call a p body generalizing the methods used to establish volume formulas for the hilbert kunz multiplicity and f signature of semigroup rings we relate the volume of a p body to a certain asymptotic invariant determined by the corresponding p family of ideals we apply these methods to obtain new existence results for limits in positive characteristic an analogue of the brunn minkowski theorem for hilbert kunz multiplicity and a uniformity result concerning the positivity of a p family

the classification of purely non symplectic automorphisms of high order on k3 surfaces abstract an automorphism of order n of a k3 surface is called purely non symplectic if it multiplies the holomorphic symplectic form by a primitive n th root of unity we give the classification of purely non symplectic automorphisms with n 12 where denotes the euler totient function

on buchi s k3 surface we study the buchi k3 surface proving that it belongs to the one dimensional family of kummer surfaces associated to genus two curves with automorphism group d 4 we compute its picard lattice and show that the rational points of the surface are zariski dense moreover we provide analogous results for the kummer surface associated to any genus two curve whose automorphism group contains a non hyperelliptic involution

decomposition of binary random fields and zeros of partition functions on montre qu il existe une relation etroite entre le probleme de decomposition pour des champs eleatoires binaires etudies dans un article recent de hajek et berger 1987 et le probleme de mecanique statistique de determination de regions independantes du zero pour des fonctions de partition