The Rees algebra of a positive normal affine semigroup ring

Let R be a positive normal affine semigroup ring of dimension d and let be the maximal homogeneous ideal of R. We show that the integral closure of is equal to for all n ∈ℕ with n ≥ d − 2. From this we derive that the Rees algebra R[t] is normal in case that d ≤ 3. If emb dim(R) = d + 1, we can give a necessary and sufficient condition for R[t] to be normal.     

The theory of energy for sub-Laplacians with an application to quasi-continuity

In this paper, we provide a suitable theory for the energy where μ is a Radon measure and Γ is the fundamental solution of a sub-Laplacian on a stratified group As a significant application, we prove the quasi-continuity of superharmonic functions related to . The proofs are elementary and mostly rely on the use of appropriate mean-value formulas and mean-integral operators relevant to the Potential Theory for .     

Tannaka duality on quotient stacks

Let = [X/G] be the quotient stack of a scheme X by an affine group scheme G over a field k. Assume that there is a line bundle on whose underlying line bundle on X is very ample. Let VB() be the category of vector bundles on .We show that is canonically isomorphic to the stack of fiber functors on VB(). This is an analogue of the Tannaka duality for affine groups. Partially supported by CNCSIS contract no. 33079/2004     

On the rank of abelian varieties over function fields

Let be a smooth projective curve defined over a number field k, A/k() an abelian variety and (τ, B) the k()/k-trace of A. We estimate how the rank of A(k())/τB(k) varies when we take a finite geometrically abelian cover defined over k. This work was partially supported by CNPq research grant 304424/2003-0, Pronex 41.96.0830.00 and CNPq Edital Universal 470099/2003-8. I would like to thank Douglas Ulmer for comments on how to treat the case of arbitrary ramification, but the conductor prime to the ramification locus, in the case of elliptic fibrations. I would also like to thank Marc Hindry for comments on the inequality comparing the conductors of A and A'. Finally, I also thank the referee for his comments and criticisms.     

Restriction of the cotangent bundle to elliptic curves and Hilbert-Kunz functions

We describe the possible restrictions of the cotangent bundle to an elliptic curve . We apply this in positive characteristic to the computation of the Hilbert-Kunz function of a homogeneous R₊-primary ideal in the graded section ring .     

Conjugacy classes of affine automorphisms of and linear automorphisms of ℙ n in the Cremona groups

We describe the conjugacy classes of affine automorphisms in the group Aut(n,) (respectively Bir()) of automorphisms (respectively of birational maps) of . From this we deduce also the classification of conjugacy classes of automorphisms of ℙⁿ in the Cremona group Bir().     

Invertible modules for commutative -algebras with residue fields

The aim of this note is to understand under which conditions invertible modules over a commutative -algebra in the sense of Elmendorf, Kriz, Mandell & May give rise to elements in the algebraic Picard group of invertible graded modules over the coefficient ring by taking homotopy groups. If a connective commutative -algebra R has coherent localizations for every maximal ideal , then for every invertible R-module U, U_*=π_*U is an invertible graded R_*-module. In some non-connective cases we can carry the result over under the additional assumption that the commutative -algebra has ‘residue fields’ for all maximal ideals if the global dimension of R_* is small or if R is 2-periodic with underlying Noetherian complete local regular ring R₀. We apply these results to finite abelian Galois extensions of Lubin-Tate spectra.     

Estimates for stable hypersurfaces of prescribed F-mean curvature

We consider immersed hypersurfaces :Mⁿ→ℝⁿ⁺¹ with prescribed anisotropic mean curvature . Such hypersurfaces can be characterized as critical points of parametric functionals of the type with an elliptic Lagrangian F depending on normal directions and a smooth vectorfield Q satisfying . We establish curvature estimates for stable hypersurfaces of dimension n≤5, provided F is C³-close to the area integrand.     

Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence

Let G be a connected linear semisimple Lie group with Lie algebra , and let be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that Ω is a nilpotent G-orbit in and is the nilpotent -orbit in associated to Ω by the Kostant-Sekiguchi correspondence. We show that the corank of the Hamiltonian K-space Ω is twice the complexity of the variety .     

Iwasawa invariants of galois deformations

Fix a residual ordinary representation :G_F→GL_n(k) of the absolute Galois group of a number field F. Generalizing work of Greenberg–Vatsal and Emerton–Pollack–Weston, we show that the Iwasawa invariants of Selmer groups of deformations of depends only on and the ramification of the deformation.     

Associated primes and cofiniteness of local cohomology modules

Let be an ideal of Noetherian ring R and let s be a non-negative integer. Let M be an R-module such that is finite R-module. If s is the first integer such that the local cohomology module is non -cofinite, then we show that is finite. In particular, the set of associated primes of is finite. Let be a local Noetherian ring and let M be a finite R-module. We study the last integer n such that the local cohomology module is not -cofinite and show that n just depends on the support of M.The research of the first author was supported in part by a grant from IPM (No. 83130114).The second author was supported by a grant from University of Tehran (No. 6103023/1/01).     

Fonctions plurisousharmoniques et courants positifs de type (1,1) sur une variété presque complexe

If (X, J) is an almost complex manifold, then a function u is said to be plurisubharmonic on X if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is conjectured that plurisubharmonicity is equivalent to the positivity of the (1,1)-current , (the (1,1)-current need not be closed here!). The conjecture is trivial if u is of class The result is elementary in the complex integrable case because the operator can be written as an operator with constant coefficients in complex coordinates. Hence the positivity of the current is preserved by regularising with usual convolution kernels. This is not possible in the almost complex non integrable case and the proof of the result requires a much more intrinsic study. In this chapter we prove the necessity of the positivity of the (1,1)-current . We prove also the sufficiency of the positivity in the particular case of an upper semi-continuous function f which is continuous in the complement of the singular locus f⁻¹(−∞).

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Résume Une fonction semi-continue supérieurement u sur une variété presque complexe (X, J) est dite plurisousharmonique si la restriction à toute courbe pseudo-holomorphe locale est sous-harmonique. Comme dans le cas analytique complexe, nous conjecturons que la notion de plurisousharmonicité pour une fonction u est équivalente à la positivité du (1,1)-courant , (lequel n'est pas forcément fermé dans le cas non intégrable). La conjecture est triviale dans le cas d'une fonction u de classe Le résultat en question est élémentaire dans le cas complexe intégrable car l'opérateur s'écrit comme un opérateur à coefficients constants dans des coordonnées complexes. On peut donc facilement conserver la positivité du courant en régularisant avec des noyaux usuels. Dans le cas presque complexe non intégrable ceci ce n'est pas possible et la preuve du résultat exige un étude beaucoup plus intrinsèque. Nous montrons la nécessité de la positivité du (1,1)-courant en utilisant la théorie locale des courbes J-holomorphes. Nous montrons aussi la suffisance de la positivité dans le cas particulier d'une fonction f semi-continue supérieurement et continue en dehors du lieu singulier f⁻¹(−∞).

     

Bielliptic modular curves X 1(M,N)

In this work, we determine all modular curves X₁(M,N) which are bielliptic, and then we also discuss the problem of finding the curves X₁(M,N) which admit infinitely many quartic points over .     

On the logarithmic plurigenera of complements of plane curves

Let B be a (not necessarily irreducible) plane curve in ². In the present article, we prove that if and only if Moreover, we determine the curve B when and Mathematics Subject Classification (2000): 14R05, 14H50, 14J26     

Infinitely many solutions of a semilinear problem for the Heisenberg Laplacian on the Heisenberg group

We study the semilinear equationwhere is the Heisenberg Laplacian and is the Heisenberg group. The function f C²(×, ) is supposed to satisfy some (subcritical) growth conditions and to be left invariant under the action of the subgroup of consisting of points with integer coordinates.. We show the existence of infinitely many solutions in the space S₁²(), which is the Heisenberg analogue of the Sobolev space W^1,2(^N).Mathematics Subject Classification (2000): 22E30, 22E27     

Mean-value theorems for multiplicative functions on Beurling’s generalized integers (I)

Halász’s general mean-value theorem for multiplicative functions on ℕ is classical in probabilistic number theory. We extend this theorem to functions f, defined on a set of generalized integers associated with a set of generalized primes in Beurling’s sense, which satisfies Halász’s conditions, in particular,Assume that the distribution function N(x) of satisfieswith γ>γ₀, where ρ₁<ρ₂<···<ρ_m are constants with ρ_m≥1 and A₁,···,A_m are real constants with A_m>0. Also, assume that the Chebyshev function ψ(x) of satisfieswith M>M₀. Then the asymptoticimplieswhere τ is a positive constant with τ≥1 and L(u) is a slowly oscillating function with |L(u)|=1.     

On the exact Hausdorff dimension of the set of Liouville numbers. II

Let denote the set of Liouville numbers. For a dimension function h, we write for the h-dimensional Hausdorff measure of . In previous work, the exact ``cut-point' at which the Hausdorff measure of drops from infinity to zero has been located for various classes of dimension functions h satisfying certain rather restrictive growth conditions. In the paper, we locate the exact ``cut-point' at which the Hausdorff measure of drops from infinity to zero for all dimension functions h. Namely, if h is a dimension function for which the function increases faster than any power function near 0, then , and if h is a dimension function for which the function increases slower than some power function near 0, then . This provides a complete characterization of all Hausdorff measures of without assuming anything about the dimension function h, and answers a question asked by R. D. Mauldin. We also show that if then does not have σ-finite measure. This answers another question asked by R. D. Mauldin. This work was done while Dave L. Renfro was at the Department of Mathematics at Central Michigan University.     

Eilenberg-Moore spectral sequence calculation of function space cohomology

The purpose of this article is to introduce an Eilenberg-Moore spectral sequence converging to the cohomology algebra of a function space with an adjunction space as its source. Computability of the spectral sequence is shown by determining explicitly the mod p cohomology algebra of the function space of maps from a non-orientable surface S to the classifying space of a simply-connected Lie group G whose homology is p-torsion free. Let M be a closed orientable 3-dimensional manifold. Applying the spectral sequence obtained from a Heegaard splitting of the manifold M, we also prove that is a direct summand of .Mathematics Subject Classification (2000):55T20, 57T35This research was partially supported by a Grant-in-Aid for Scientific Research (C)14540095 from Japan Society for the Promotion of Science.     

Polynomial closure in essential domains

Let D be a domain, E a subset of its quotient field K, and Int. The polynomial closure of E is the set We compare the polynomial closure with the divisorial closure in a general setting and then in an essential domain. Especially, we show that these two closures of ideals are the same if D is a Krull-type domain.This work was done while the first author was visiting KIAS. She would like to thank KIAS for the hospitality and support.