brauer groups and quotient stacks a natural question is to determine which algebraic stacks are quotient stacks in this paper we give some partial answers and relate it to the question of whether for a scheme x the natural map from the brauer group equivalence classes of azumaya algebras to the cohomological brauer group the torsion subgroup of h 2 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i m is surjective

fibration de hitchin et endoscopie we propose a geometric interpretation of the theory of elliptic endoscopy due to langlands and kottwitz in terms of the hitchin fibration as applications we prove a global analog of a purity conjecture due to goresky kottwitz and macpherson for unitary groups this global purity statement has been used in a joint work with g laumon to prove the fundamental lemma over a local fields of equal characteristics

le lemme fondamental pondere ii enonces cohomologiques dans cet article on etudie la cohomologie de la fibration de hitchin tronquee introduite dans un article precedent on etend les principaux theoremes de ngo sur la cohomologie de la partie elliptique de la fibration de hitchin comme consequence on obtient une demonstration du lemme fondamental pondere d arthur in this paper we study the cohomology of the truncated hitchin fibration which was introduced in a previous paper we extend ngo s main theorems on the cohomology of the elliptic part of the hitchin fibration as a consequence we get a proof of arthur s weighted fundamental lemma

the generic green lazarsfeld secant conjecture using lattice theory on special k3 surfaces calculations on moduli stacks of pointed curves and voisin s proof of green s conjecture on syzygies of canonical curves we prove the prym green conjecture on the naturality of the resolution of a general prym canonical curve of odd genus as well as many cases of the green lazarsfeld secant conjecture on syzygies of non special line bundles on general curves

on the projective normality of double coverings over a rational surface abstract we study the projective normality of a minimal surface x which is a ramified double covering over a rational surface s with dim k s 1 in particular horikawa surfaces the minimal surfaces of general type with k x 2 2 p g x 4 are of this type up to resolution of singularities let be the covering map from x to s we show that the z 2 invariant adjoint divisors k x r a are normally generated where the integer r 3 and a is an ample divisor on s

on lagrangian schemes for porous medium type generalized diffusion equations a discrete energetic variational approach abstract in this paper we present a systematic framework to derive a variational lagrangian scheme for porous medium type generalized diffusion equations by employing a discrete energetic variational approach such discrete energetic variational approaches are analogous to energetic variational approaches 39 25 in a semidiscrete level which provide a basis of deriving the semi discrete equations and can be applied to a large class of partial differential equations with energy dissipation laws and kinematic relations the numerical schemes derived by this framework can inherit the variational structure from the continuous energy dissipation law as an illustration we develop two variational lagrangian schemes for the multidimensional porous medium equations pme based on two different energy dissipation laws we focus on the numerical scheme based on the energy dissipation law with 1 2 u 2 d x as the dissipation functional several numerical experiments demonstrate the accuracy of this scheme as well as its ability in capturing the free boundary and estimating the waiting time for the pme in both 1d and 2d

secant varieties and birational geometry we show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations these were first constructed as flips in the case of curves by m thaddeus via geometric invariant theory and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper we expose the finer structure of a second flip again for varieties of arbitrary dimension we also prove a result on the cubic generation of the secant variety and give some conjectures on the behavior of equations defining the higher secant varieties

a variational finite volume scheme for wasserstein gradient flows we propose a variational finite volume scheme to approximate the solutions to wasserstein gradient flows the time discretization is based on an implicit linearization of the wasserstein distance expressed thanks to benamou brenier formula whereas space discretization relies on upstream mobility two point flux approximation finite volumes our scheme is based on a first discretize then optimize approach in order to preserve the variational structure of the continuous model at the discrete level our scheme can be applied to a wide range of energies guarantees non negativity of the discrete solutions as well as decay of the energy we show that our scheme admits a unique solution whatever the convex energy involved in the continuous problem and we prove its convergence in the case of the linear fokker planck equation with positive initial density numerical illustrations show that it is first order accurate in both time and space and robust with respect to both the energy and the initial profile

uniruledness of strata of holomorphic differentials in small genus abstract we address the question concerning the birational geometry of the strata of holomorphic and quadratic differentials we show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on k3 and del pezzo surfaces respectively restricting to genus 3 g 6 we construct projective bundles over a rational varieties that dominate the holomorphic strata with length at most g 1 hence showing in addition that these strata are unirational

the geometry of the moduli space of odd spin curves the spin moduli spacesg is the parameter space of theta characteristics spin structures on stable curves of genus g it has two connected components s g ands g depending on the parity of the spin structure we establish a complete birational classication by kodaira dimension of the odd components g of the spin moduli space we show thats g is uniruled for g 12 and even unirational for g 8 in this range introducing the concept of cluster for the mukai variety whose one dimensional linear sections are general canonical curves of genus g we construct new birational models ofs g these we then use to explicitly describe the birational structure of s g for instance s 8 is birational to a locally trivial p 7 bundle over the moduli space of elliptic curves with seven pairs of marked points for g 12 we prove thats g is a variety of general type in genus 12 this requires the construction of a counterexample to the slope conjecture on eective divisors on the moduli space of stable curves of genus 12

le lemma fondamental pour les groupes unitaires let g be an unramified reductive group over a nonarchimedian local field f the so called langlands fundamental lemma is a family of conjectural identities between orbital integrals for g f and orbital integrals for endoscopic groups of g in this paper we prove the langlands fundamental lemma in the particular case where f is a finite extension of fp t g is a unitary group and p rank g waldspurger has shown that this particular case implies the langlands fundamental lemma for unitary groups of rank p when f is any finite extension of qp we follow in part a strategy initiated by goresky kottwitz and macpherson our main new tool is a deformation of orbital integrals which is constructed with the help of the hitchin fibration for unitary groups over projective curves

multigraded linear series and recollement given a scheme y equipped with a collection of globally generated vector bundles e 1 ldots e n we study the universal morphism from y to a fine moduli space mathcal m e of cyclic modules over the endomorphism algebra of e mathcal o y oplus e 1 oplus cdots oplus e n this generalises the classical morphism to the linear series of a basepoint free line bundle on a scheme we describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category when the morphism is surjective this gives a fine moduli space interpretation of the image and as an application we show that for a small finite subgroup g subset mathrm gl 2 mathbb k every sub minimal partial resolution of mathbb a 2 mathbb k g is isomorphic to a fine moduli space mathcal m e c where e c is a summand of the bundle e defining the reconstruction algebra we also consider applications to gorenstein affine threefolds where reid s recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution

homology of affine springer fibers in the unramified case assuming a certain purity conjecture we derive a formula for the complex cohomology groups of the affine springer fiber corresponding to any unramified regular semisimple element we use this calculation to present a complex analog of the fundamental lemma for function fields we show that the kappa orbital integral that arises in the fundamental lemma is equal to the lefschetz trace of the frobenius acting on the etale cohomology of a related variety

curves and symmetric spaces ii let sym 3 c p k sym 3 k sym 3 k k p 13 a 1 a a det a be the veronese embedding of the space of symmetric matrices of degree 3 where a is the cofactor matrix of a the closure spg 3 6 of this image is a 6 dimensional homogeneous variety of the symplectic group sp 3 a canonical curve c 16 c p 8 of genus 9 over a perfect field k is isomorphic to a complete linear section of this projective variety spg 3 6 c p 13 unless c k k k being the algebraic closure is a covering of degree at most 5 of the projective line we prove this by means of linear systems of higher rank

the mercat conjecture for stable rank 2 vector bundles on generic curves we prove the mercat conjecture for rank 2 vector bundles on generic curves of every genus for odd genus we identify the effective divisor in mg where the mercat conjecture fails and compute its slope the clifford index of a smooth curve c defined as the non negative quantity cliff c min deg l 2r l l pic c h c l 2 deg l g 1 stratifies the moduli space mg of curves of genus g with the smallest stratum being that consisting of hyperelliptic curves which have clifford index zero while the definition is obviously inspired by the classical clifford theorem the notion came to real prominence in the works of green lazarsfeld g gl voisin v2 and many others in the context of syzygies of canonical curves for a general curve c of genus g one has cliff c bg 1 2 c the locus of curves c mg with cliff c bg 1 2 c is a subvariety of codimension 1 respectively 2 when g is odd respectively even much studied by harris and mumford hm it has been a long standing problem to find an adequate definition of the clifford index for higher rank vector bundles on curves a higher rank clifford index should not only capture the behavior of the generic curve from the point of view of special higher rank vector bundles but also provide a geometrically meaningful stratification ofmg an interesting definition has been put forward by lange and newstead ln1 for a semistable vector bundle e of rank 2 and slope e on a curve c of genus g one defines its clifford index as cliff e e h c e 2 0 the rank 2 clifford index of c is then defined as the quantity cliff2 c min cliff e e uc 2 d d 2g 2 h c e 4 observe that if l is a line bundle on c with deg l 2h0 c l 2 cliff c that is l computes the clifford index of c then cliff l l cliff c in particular the inequality cliff2 c cliff c holds for every curve c of genus g the main achievement of this paper is the proof of the following result theorem 1 for a general curve c of genus g the following equality holds cliff2 c cliff c g 1 2 equivalently if e is a semistable rank 2 vector bundle on c contributing to cliff2 c then h c e e 2 cliff c mercat me conjectured the equality cliff2 c cliff c for every smooth curve c mg counterexamples to this expectation have been found using noether lefschetz special k3 surfaces in fo1 fo2 and ln2 however these examples are special in moduli and theorem 1 proves mercat s conjecture for generic curves of every genus

motivic integration over deligne mumford stacks abstract the aim of this article is to develop the theory of motivic integration over deligne mumford stacks and to apply it to the birational geometry of deligne mumford stacks

a fully discrete variational scheme for solving nonlinear fokker planck equations in multiple space dimensions we introduce a novel spatio temporal discretization for nonlinear fokker planck equations on the multidimensional unit cube this discretization is based on two structural properties of these equations the first is the representation as a gradient flow of an entropy functional in the l 2 wasserstein metric the second is the lagrangian nature meaning that solutions can be written as the push forward transformation of the initial density under suitable flow maps the resulting numerical scheme is entropy diminishing and mass conserving further we are able to prove consistency in the sense that if the discrete solutions are regular independently of the mesh size then they converge to a classical solution finally we present results from numerical experiments in space dimension d 2

the unirationality of the moduli space of k 3 surfaces of genus 22 using the connection discovered by hassett between the noether lefschetz moduli space mathcal c 42 of special cubic fourfolds of discriminant 42 and the moduli space mathcal f 22 of polarized k3 surfaces of genus 22 we show that the universal k3 surface over mathcal f 22 is unirational

mirror symmetry langlands duality and the hitchin system among the major mathematical approaches to mirror symmetry are those of batyrev borisov and strominger yau zaslow syz the first is explicit and amenable to computation but is not clearly related to the physical motivation the second is the opposite furthermore it is far from obvious that mirror partners in one sense will also be mirror partners in the other this paper concerns a class of examples that can be shown to satisfy the requirements of syz but whose hodge numbers are also equal this provides significant evidence in support of syz moreover the examples are of great interest in their own right they are spaces of flat slr connections on a smooth curve the mirror is the corresponding space for the langlands dual group pglr these examples therefore throw a bridge from mirror symmetry to the duality theory of lie groups and more broadly to the geometric langlands program

smooth varieties of almost minimal degree abstract in this article we study non linearly normal smooth projective varieties x p r of deg x codim x p r 2 we first give geometric characterizations for x theorem 1 1 indeed x is the image of an isomorphic projection of smooth varieties x p r 1 of minimal degree also if x is not the veronese surface then there exists a smooth rational normal scroll y p r which contains x as a divisor linearly equivalent to h 2 f where h is the hyperplane section of y and f is a fiber of the projection morphism y p 1 by using these characterizations 1 we determine all the possible types of y from the type of x theorem 1 2 and 2 we investigate the relation between the betti diagram of x and the type of y theorem 1 3 in particular we clarify the relation between the number of generators of the homogeneous ideal of x and the type of y as an application we construct non linearly normal examples where the converse to theorem 1 1 in d eisenbud m green k hulek s popescu restriction linear syzygies algebra and geometry compos math 141 2005 1460 1478 fails to hold remark 2

hom stacks and restriction of scalars fix an algebraic space s and let x and y be separated artin stacks of finite presentation over s with finite diagonals over s we define a stack homs x y classifying morphisms between x and y assume that x is proper and flat over s and assume fppf locally on s that there exists a finite finitely presented flat cover z x with z an algebraic space then we show that homs x y is an artin stack with quasi compact and separated diagonal