effect of symmetry to the structure of positive solutions in nonlinear elliptic problems ii abstract we consider the problem u hu f u 0 in r u 0 on r u 0 in r where r x r n r 1

multiple solutions for a henon like equation on the annulus abstract for the equation u x 2 u p 1 1 x 3 we prove the existence of two solutions for large and of two additional solutions when p is close to the critical sobolev exponent 2 2 n n 2 a symmetry breaking phenomenon appears showing that the least energy solutions cannot be radial functions

grobner geometry of schubert polynomials given a permutation w sn we consider a determinantal ideal iw whose generators are certain minors in the generic n n matrix filled with independent variables using emultidegrees f as simple algebraic substitutes for torus equivariant cohomology classes on vector spaces our main theorems describe for each ideal iw

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

existence of many positive solutions of semilinear elliptic equations on annulus abstract we study the existence of many nonradial positive solutions in an annulus of rn it is an improvement of results in c v coffman a nonlinear boundary value problem with many positive solutions j differential equations 54 1984 429 437 an annulus is invariant under many group actions and these group actions act naturaly on the functional and leave the functional invariant therefore when we look for critical points of the functional we can restrict the functional to the manifold consisting of all fixed points under the group action we are going to use different group actions to obtain critical points and furthermore we can distinguish them

twisted cubics on singular cubic fourfolds on starr s fibration we show that the hilbert scheme compactification of the total space of starr s fibration on the space of twisted cubics on a cubic hypersurface in mathbb p 5 not containing a plane admits a contraction to a singular projective symplectic variety of dimension eight which has a crepant resolution deformation equivalent to the symplectic eightfold constructed from twisted cubics on a smooth cubic fourfold this yields another proof that the symplectic eightfold and the hilbert scheme of four points on a k3 surface are deformation equivalent as a byproduct we obtain similar results for the variety of lines on a singular cubic fourfold

existence of clustering high dimensional bump solutions of superlinear elliptic problems on expanding annuli abstract we consider the nonlinear elliptic problem u u p in r u 0 in r u 0 in r where p 1 and r x r n r x r 1 with n 3 it is known that as r the number of nonequivalent solutions of the above problem goes to when p 1 n 2 n 2 n 3 here we prove the same phenomenon for any p 1 by finding o n 1 symmetric clustering bump solutions which concentrate near the set x 1 x n r x n 0 for large r 0

subword complexes in coxeter groups abstract let be a coxeter system an ordered list of elements in and an element in determine a subword complex as introduced in knutson and miller ann of math 2 2003 to appear subword complexes are demonstrated here to be homeomorphic to balls or spheres and their hilbert series are shown to reflect combinatorial properties of reduced expressions in coxeter groups two formulae for double grothendieck polynomials one of which appeared in fomin and kirillov proceedings of the sixth conference in formal power series and algebraic combinatorics dimacs 1994 pp 183 190 are recovered in the context of simplicial topology for subword complexes some open questions related to subword complexes are presented

asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli we study the asymptotic behavior of positive solutions of the semilinear elliptic equation u u 0 in a u 0 on a where a x of rn a x a 1 are expanding annuli as a and is positive and superlinear at both 0 and we first show that there are a priori bounds for some positive solutions ua x as a then if we fix any direction after a suitable translation of ua the limiting solutions are non negative solutions on the infinite strip we can obtain more detailed descriptions of these limits if ua is radially symmetric least energy or least energy with a particular symmetry

on the lowest eigenvalue of the laplacian for the intersection of two domains ifa andb are two bounded domains in n and a b are the lowest eigenvalues of with dirichlet boundary conditions then there is some translate b x ofb such that a b x a b a similar inequality holds for lambda p a inf parallel nabla f parallel p p parallel f parallel p p f in w 0 1 p a there are two corollaries of this theorem i a lower bound for sup x volume a b x in terms of a whenb is a ball ii a compactness lemma for certain sequences inw 1 p n

borel weil theorem for configuration varieties and schur modules we present a generalization of the classical schur modules ofgl n exhibiting the same interplay among algebra geometry and combinatorics a generalized young diagramdis an arbitrary finite subset ofn n for eachd we define the schur modulesdofgl n we introduce a projective variety fdand a line bundle ld and describe the schur module in terms of sections of ld for diagrams with the northeast property i1 j1 i2 j2 d min i1 i2 max j1 j2 d which includes the skew diagrams we resolve the singularities of fdand show analogues of bott s and kempf s vanishing theorems finally we apply the atiyah bott fixed point theorem to establish a weyl type character formula of the form charsd x txwt t i j 1 xix 1j dij t wheretruns over certain standard tableaux ofd our results are valid over fields of arbitrary characteristic

the nil hecke ring and singularity of schubert varieties let g be a semi simple simply connected complex algebraic group and t b a maximal torus and a borel subgroup respectively let lie t be the cartan subalgebra of the lie algebra lie g and w n t t the weyl group associated to the pair g t where n t is the normalizer of t in g we can view any element omega overline omega mod t i w as the element denoted by the corresponding german character xo of g b denned as m overline omega b for any w i w there is associated the schubert variety rm x omega overline b omega b b subset g b and the t fixed points of x w under the canonical left action are precisely rm i omega left o upsilon in wand upsilon leqslant omega right

when is a schubert variety gorenstein abstract a normal variety is gorenstein if it is cohen macaulay and its canonical sheaf is a line bundle this property is a weakening of smoothness which implies that a variety behaves like a smooth one for various algebro geometric purposes we give a combinatorial characterization of which schubert varieties are gorenstein using in part a new generalization of pattern avoidance we also give an explicit description as a line bundle of the canonical sheaf of a gorenstein schubert variety

on the separation property of orbits in representation spaces a subset x of a vector space v is said to have the separation property if it separates linear forms in the following sense given a pair of linearly independent linear forms on v there is a point x on x such that x 0 and x 0 a more geometric way to express this is the following every linear subspace h v of codimension 1 is linearly spanned by its intersection with x the separation property was first asked for conjugacy classes in simple lie algebras in connection with some classification problems we give a general answer for orbits in representation spaces of algebraic groups and discuss in detail some special cases we also introduce a strong and a weak separation property which come up very naturally in our setting it turns out that these separation properties have a number of very nice features for example we discovered the surprising fact that in an irreducible representation of a connected semisimple group every linear hyperplane meets every orbit and we show that a generic orbit always has the separation property

richardson varieties and equivariant k theory we generalize standard monomial theory smt to intersections of schubert varieties and opposite schubert varieties such varieties are called richardson varieties the aim of this article is to get closer to a geometric interpretation of the standard monomial theory as constructed in p littelmann j amer math soc 11 1998 551 567 in fact the construction given here is very close to the ideas in p lakshmibai c s seshadri j algebra 100 1986 462 557 our methods show that in order to develop a smt for a certain class of subvarieties in g b which includes g b it suffices to have the following three ingredients a basis for h0 g b l compatibility of such a basis with the varieties in the class certain quadratic relations in the monomials in the basis elements an important tool as in p lakshmibai c s seshadri j algebra 100 1986 462 557 will be the construction of nice filtrations of the vanishing ideal of the boundary of the varieties above this provides a direct connection to the equivariant k theory products of classes of structure sheaves with classes of line bundles where the combinatorially defined notion of standardness gets a geometric interpretation

positivity in the grothendieck group of complex flag varieties abstract we prove a conjecture of a s buch concerning the structure constants of the grothendieck ring of a flag variety with respect to its basis of schubert structure sheaves for this we show that the coefficients in this basis of the structure sheaf of any subvariety with rational singularities have alternating signs equivalently the class of the dualizing sheaf of such a subvariety is a nonnegative combination of classes of dualizing sheaves of schubert varieties

standing waves with a critical frequency for nonlinear schrodinger equations this paper is concerned with the existence and qualitative property of standing wave solutions for the nonlinear schrodinger equation with e being a critical frequency in the sense that we show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of v and whose amplitude goes to 0 as moreover depending upon the local behaviour of the potential function v x near the minimum points the limiting profile of the standing wave solutions will be shown to exhibit quite different characteristic features this is in striking contrast with the non critical frequency case which has been extensively studied in recent years

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

arithmetic cohen macaulayness and arithmetic normality for schubert varieties let g be a semisimple simply connected chevalley group over a field k let t be a maximal torus b a borel subgroup d t and w the weyl group of g relative to t let p be a maximal parabolic subgroup of classical type cf 7 8 i e a maximal parabolic subgroup such that if w is the associated fundamental weight then i w av i 1 2 w c a is 2 for all roots ae being a w invariant scalar product on x t q where x t character group of t for t e w wp let x t denote the schubert variety in g p associated to t namely x t is the zariski closure btp mod p in g p of the bruhat cell endowed with the canonical reduced structure for the canonical projective embedding g p in p h g p l let rt denote the homogeneous co ordinate ring of x t then the main result of this paper is given by

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