the integral monodromy of hyperelliptic and trielliptic curves we compute the mathbb z ell and mathbb z ell monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves in particular we provide a proof that the mathbb z ell monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group rm s p2g mathbb z ell we prove that the mathbb z ell monodromy of the moduli space of trielliptic curves with signature r s is the special unitary group rm su r s mathbb z ell otimes mathbb z zeta 3

projectively simple rings abstract an infinite dimensional n graded k algebra a is called projectively simple if dim k a i for every nonzero two sided ideal i a we show that if a projectively simple ring a is strongly noetherian is generated in degree 1 and has a point module then a is equal in large degree to a twisted homogeneous coordinate ring b b x l here x is a smooth projective variety is an automorphism of x with no proper invariant subvariety we call such automorphisms wild and l is a ample line bundle we conjecture that if x admits a wild automorphism then every irreducible component of x is an abelian variety we prove several results in support of this conjecture in particular we show that the conjecture is true if dim x 2 in the case where x is an abelian variety we describe all wild automorphisms of x finally we show that if a is projectively simple and admits a balanced dualizing complex then proj a is cohen macaulay and gorenstein

residuation of linear series and the effective cone of cd we obtain new information about divisors on the dth symmetric power cd of a general curve c of genus g 4 this includes a complete description of the effective cone of cg 1 and a partial computation of the volume function on one of its non nef subcones as well as new bounds for the effective and movable cones of cd in the range g 1 d g 2 we also obtain for each g 5 a divisor on cg 1 with non equidimensional stable base locus for a general hyperelliptic curve c of genus g we obtain a complete description of the effective cone of cd for 2 d g and an integral divisor on cg 1 which has non integral volume whenever g is not a power of 2 1 introduction let c be a smooth complex projective algebraic curve and let d 1 be an integer the dth symmetric power cd c d sd is a smooth d dimensional complex projective variety which is a fine moduli space parametriz ing effective divisors of degree d on c in this paper we study the cone of effective divisors on cd and its refinements there are two natural divisor classes on cd for each p c the image of the embedding

primitive algebras with arbitrary gelfand kirillov dimension abstract we construct for every real 2 a primitive affine algebra with gelfand kirillov dimension unlike earlier constructions there are no assumptions on the base field in particular this is the first construction over or l given a recursive sequence v n of elements in a free monoid we investigate the quotient of the free associative algebra by the ideal generated by all nonsubwords in v n we bound the dimension of the resulting algebra in terms of the growth of v n in particular if n is less than doubly exponential then the dimension is 2 this also answers affirmatively a conjecture of salwa 1997 comm algebra 25 3965 3972

divisors of secant planes to curves abstract inside the symmetric product of a very general curve we consider the codimension one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension we compute the classes of such divisors and thus obtain improved bounds for the slope of the cone of effective divisor classes on symmetric products of a very general curve by letting the moduli of the curve vary we study more generally the classes of the related divisors inside the moduli space of stable pointed curves

on the tate and mumford tate conjectures in codimension 1 for varieties with h2 0 1 we prove the tate conjecture for divisor classes and the mumford tate conjecture for the cohomology in degree 2 for varieties with h 1 over a finitely generated field of characteristic 0 under a mild assumption on their moduli as an application of this general result we prove the tate and mumford tate conjectures for several classes of algebraic surfaces with pg 1 ams 2010 mathematics subject classification 14c 14d 14j20

on symmetric products of curves let c be a smooth complex projective curve of genus g and let c be its second symmetric product this paper concerns the study of some attempts at extending to c the notion of gonality in particular we prove that the degree of irrationality of c is at least g 1 when c is generic and that the minimum gonality of curves through the generic point of c equals the gonality of c in order to produce the main results we deal with correspondences on the k fold symmetric product of c with some interesting linear subspaces of p enjoying a condition of cayley bacharach type and with monodromy of rational maps as an application we also give new bounds on the ample cone of c when c is a generic curve of genus 6 g 8

asymptotically periodic solutions of second order difference equations abstract in this paper we obtain sufficient conditions for the existence of asymptotically periodic solutions of the second order difference equations of the form r n x n a n f x n b n moreover we give sufficient conditions under which every bounded solution of the above equation is asymptotically periodic our approach allows us to control the degree of approximation

rigid and non smoothable schubert classes a schubert class in the grassmannian is rigid if the only proper subvarieties representing that class are schubert varieties the hyperplane class 1 is not rigid because a codimension one schubert cycle can be deformed to a smooth hyperplane section in this paper we show that this phenomenon accounts for the failure of rigidity in schubert classes more precisely we prove that a schubert class in g k n is not rigid if and only if the partition 1 k defining the class has a part i such that n k i 1 i and i i 1 1 under these assumptions on the parts i and i 1 determine a partial flag isomorphic to one defining a hyperplane class in another grassmannian g k n using a deformation of the hyperplane in g k n we can deform the schubert cycle otherwise the schubert class is rigid we also show that if the partition contains the partition defining a rigid and singular schubert cycle in some grassmannian as a sub partition then cannot be represented by a smooth subvariety of g k n more precisely if does not have the form 1 j1 n k i i 1 1 for j1 i j2 and i j2 for i j2 then the schubert class cannot be represented by a smooth subvariety of g k n

morphisms and faces of pseudo effective cones let x y be a morphism of projective varieties and suppose that is a pseudo effective numerical cycle class satisfying 0 a conjecture of debarre jiang and voisin predicts that is a limit of classes of effective cycles contracted by we establish new cases of the conjecture for higher codimension cycles in particular we prove a strong version when x is a fourfold and has relative dimension one

rigidity of smooth schubert varieties in hermitian symmetric spaces in this paper we study the space z k g p r x w of effective k cycles x in g p with the homology class equal to an integral multiple of the homology class of schubert variety x w of type w when x w is a proper linear subspace p k k n of a linear space p n in g p p v we know that z k p n r p k is already complicated we will show that for a smooth schubert variety x w in a hermitian symmetric space any irreducible subvariety x with the homology class x r x w r z is again a schubert variety of type w unless x w is a non maximal linear space in particular any local deformation of such a smooth schubert variety in hermitian symmetric space g p is obtained by the action of the lie group g

divisors on symmetric products of curves for a curve with general moduli the neron severi group of its symmetric products is generated by the classes of two divisors x and in this paper we give bounds for the cones of effective and ample divisors in the x plane

asymptotic behavior of solutions of second order nonlinear difference equations 0 2 j 1 b n u n g n y y h 1 1 or equivalently e y a0 by g n y ey where e is the operator defined by ey n y n 1 2 let z z2 be a fundamental system of solutions of the linear difference equation lz e2z f aez bz 0 3 satisfying z o 1 z l 0 zz o 0 22 l 1 4 we will prove that there exists a ball b 0 p c r2 such that if y o y 1 e b 0 p then the solution y of equation 1 is defined on all n and for n co it satisfies

on the nef cone of symmetric products of a generic curve the nef cone of the symmetric product c of a curve c with general moduli is easily determined when k is at least equal to the gonality gon c of the curve in this paper we describe the nef cone for k gon c 1 when the genus of c is even in this case the boundary of the nef cone has rational slope and is determined using curves on c associated to the g s of c 1 introduction let c be a smooth irreducible curve of genus g 1 for any k 2 denote by c its mi symmetric product that parametrizes effective degree k divisors z x x2 on c let ni c k resp nl c k be the set of r linear combinations of 1 cycles resp divisors on k modulo numerical equivalence nx k is called the n ron severi group of c and denoted ns c k n k and nl k are finite dimensional vector spaces and the bilinear form given by the intersection pairing is nondegenerate if c has general moduli n c and nl c have rank two an interesting problem is to try to determine inside n c the convex cone of classes of effective 1 cycles denoted by ne c and called the cone of curves or mori cone dually one can study inside nl c the nef cone nef k i e the closure of the convex cone of classes of ample divisors by kleiman s criterion of ampleness

on monomial algebras of finite global dimension let g be an associative monomial k algebra if g is assumed to be finitely presented then either g contains a free subalgebra on two monomials or else g has polynomial growth if instead g is assumed to have finite global dimension then either g contains a free subalgebra or else g has a finite presentation and polynomial growth also a graded hopf algebra with generators in degree one and relations in degree two contains a free hopf subalgebra if the number of relations is small enough

e adic and x adic representations associated to abelian varieties defined over number fields 0 introduction let a be an abelian variety over a number field k of dimension n 1 and let g gal kik for each prime number let vi a be the qe adic tate module of a one general problem is to study the image of the e adic representation vr a of g let ge be the e adic lie algebra of the image of this f adic representation fix an embedding cr k c c and denote by a c the abelian variety a x c in 12 mumford and tate conjectured that ge m 0q qe where mn is the lie algebra of the mumford tate group mt a c cf 9 section 3 as is now well known in general g is contained in m 0q qe cf 9 16 various results toward this general problem have been obtained for some classes of abelian varieties cf 22 section 4 more recently some important general results on e adic representations attached to abelian varieties have been established especially one has the following results i the rank of g is independent of c serre 24 zarhin 34 ii geis algebraic and contains the homotheties bogomolov 1 iii g is reductive and end ve a endk a 0 qe faltings 10 these make the determination of g possible for some other classes of abelian varieties for example when d dim a is odd and endt a z j p serre has proved that ge m 0q qe sp 2d qe 3 qe id where id is the 2d x 2d identity matrix cf 25 the purpose of this article is to extend serre s method to some other cases as a result ge can be determined for the following types of simple abelian varieties

pseudo effective and nef cones on spherical varieties we show that nef cycle classes on smooth complete spherical varieties are effective and the products of nef cycle classes are also nef let x be a smooth projective spherical variety such that its effective cycle classes of codimension k are nef where 1 le k le text dim x 1 we study the properties of x in particular if x is a toric variety then x is isomorphic to the product of some projective spaces if x is toroidal then x is isomorphic to a rational homogeneous space if x is horospherical text dim x ge 3 and k 2 then effective divisors on x are nef if x is horospherical and effective divisors on x are nef then there is a morphism from x to a rational homogeneous space such that each fiber is isomorphic to the product of some horospherical varieties of picard number one

the cones of effective cycles on projective bundles over curves the study of the cones of curves or divisors on complete varieties is a classical subject in algebraic geometry cf 4 9 10 and it still is an active research topic cf 1 11 or 2 however little is known if we pass to higher co dimension in this paper we study this problem in the case of projective bundles over curves and describe the cones of effective cycles in terms of the numerical data appearing in a harder narasimhan filtration this generalizes to higher codimension results of miyaoka and others 3 15 for the case of divisors an application to projective bundles over a smooth base of arbitrary dimension is also given given a smooth complex projective variety x of dimension n consider the vector spaces

on the kurosh problem for algebras of polynomial growth over a general field lenagan and smoktunowicz 2007 ls see also lenagan smoktunowicz and young in press lsy gave an example of a nil algebra of finite gelfand kirillov dimension their construction requires a countable base field however we show that for any field k and any monotonically increasing function f n which grows super polynomially but subexponentially there exists an infinite dimensional finitely generated nil k algebra whose growth is asymptotically bounded by f n this construction gives the first examples of nil algebras of subexponential growth over uncountable fields

ample subvarieties and q ample divisors abstract we introduce a notion of ampleness for subschemes of any codimension using the theory of q ample line bundles we also investigate certain geometric properties satisfied by ample subvarieties e g the lefschetz hyperplane theorems and numerical positivity using these properties we also construct a counterexample to the converse of the andreotti grauert vanishing theorem

extremal higher codimension cycles on moduli spaces of curves we show that certain geometrically defined higher codimension cycles are ex tremal in the effective cone of the moduli space mg n of stable genus g curves with n ordered marked points in particular we prove that codimension two boundary strata are extremal and exhibit extremal boundary strata of higher codimension we also show that the locus of hyperelliptic curves with a marked weierstrass point in m3 1 and the locus of hyperel liptic curves in m4 are extremal cycles in addition we exhibit infinitely many extremal codimension two cycles in m1 n for n 5 and in m2 n for n 2

representations of solvable lie algebras v on the gelfand kirillov dimension of simple modules abstract examples are given of simple noetherian integral domains which have simple modules of widely differing gk dimensions there is also an example of a simple integral domain whose gk dimension is one

structure of prime finitely presented monomial algebras abstract the structure of a finitely presented monomial algebra k x k i over a field k is described here x is a finitely generated free monoid and i is a prime ideal of x that is finitely generated as an application a new structural proof of the recent result of bell and pekcagliyan j bell p pekcagliyan primitivity of finitely presented monomial algebras preprint arxiv 0712 0815v1 on the primitivity of such algebras is presented which yields a positive solution to the trichotomy problem raised by bell and smoktunowicz j bell a smoktunowicz the prime spectrum of algebras of quadratic growth j algebra 319 2008 414 431 in the finitely presented case our approach is based on a new result on the form of prime rees factors of semigroups satisfying the ascending chain condition on one sided annihilators and on its refinement in the case of finitely presented factors of the form x i

an application of darbo s fixed point theorem in the investigation of periodicity of solutions of difference equations using the technique of darbo s fixed point theorem we investigate a nonlinear second order difference equation of the form d r n dx n a nf x n 1 where x n 0 r a r n 0 r and f r r is a continuous function additionally the sturm liouville difference equation is considered as a special case of the above equation sufficient conditions for the existence of an asymptotically periodic solution of this equation are obtained an example illustrates the result next the conditions under which this equation has no asymptotically periodic solutions are presented