On the sectional invariants of polarized manifolds

Let (X,L) be a polarized manifold of dimension n. In this paper, for any integer i with 0≤i≤n we introduce the notion of the ith sectional invariant of (X,L). We define the ith sectional Euler number e_i(X,L), the ith sectional Betti number b_i(X,L), and the ith sectional Hodge number of type (j,i−j) of (X,L) and we will study some properties of these.     

A note on the Jacobian Conjecture

In this note, we show that, if the Druzkowski mappings F(X)=X+(AX)^∗3, i.e. F(X)=(x₁+(a₁₁x₁+?+a_1nx_n)³,…,x_n+(a_n1x₁+?+a_nnx_n)³), satisfies TrJ((AX)^∗3)=0, then where δ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension ?9 in the case .     

Precise asymptotics in the deviation probability series of self-normalized sums

Let X,X₁,X₂,… be a sequence of nondegenerate i.i.d. random variables with zero means. Set Sn=X₁+?+Xn and . In the present paper we examine the precise asymptotic behavior for the general deviation probabilities of self-normalized sums, Sn/Wn. For positive functions g(x), ?(x), α(x) and κ(x), we obtain the precise asymptotics for the following deviation probabilities of self-normalized sums:

     

Stanley-Reisner rings and the radicals of lattice ideals

In this article we associate to every lattice ideal I_L,ρ⊂K[x₁,…,x_m] a cone σ and a simplicial complex Δ_σ with vertices the minimal generators of the Stanley-Reisner ideal of σ. We assign a simplicial subcomplex Δ_σ(F) of Δ_σ to every polynomial F. If F₁,…,F_s generate I_L,ρ or they generate rad(I_L,ρ) up to radical, then is a spanning subcomplex of Δ_σ. This result provides a lower bound for the minimal number of generators of I_L,ρ which improves the generalized Krull's principal ideal theorem for lattice ideals. But mainly it provides lower bounds for the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Finally, we show by a family of examples that the given bounds are sharp.     

Galois closure of essentially finite morphisms

Let X be a reduced connected k-scheme pointed at a rational point x∈X(k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphism f:Y→X satisfying the condition H⁰(Y,O_Y)=k; this Galois closure is a torsor dominating f by an X-morphism and universal for this property. Moreover, we show that is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X. We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f:Y→X under a finite group scheme satisfying the condition H⁰(Y,O_Y)=k, Y has a fundamental group scheme π₁(Y,y) fitting in a short exact sequence with π₁(X,x).     

On the independence of Heegner points associated to distinct quadratic imaginary fields

Let E/Q be an elliptic curve with no CM and a fixed modular parametrization and let be Heegner points attached to the rings of integers of distinct quadratic imaginary fields k₁,…,kr. We prove that if the odd parts of the class numbers of k₁,…,kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in .     

Edge-connectivity and edge-disjoint spanning trees

Given a graph G, for an integer c∈{2,…,|V(G)|}, define λ_c(G)=min{|X|:X⊆E(G),ω(G−X)≥c}. For a graph G and for an integer c=1,2,…,|V(G)|−1, define,

     

Classification of systems of Dyson-Schwinger equations in the Hopf algebra of decorated rooted trees

We consider systems of combinatorial Dyson-Schwinger equations (briefly, SDSE) , … , in the Connes-Kreimer Hopf algebra HI of rooted trees decorated by I={1,…,N}, where is the operator of grafting on a root decorated by i, and F₁,…,FN are non-constant formal series. The unique solution X=(X₁,…,XN) of this equation generates a graded subalgebra H(S) of HI. We characterise here all the families of formal series (F₁,…,FN) such that H(S) is a Hopf subalgebra. More precisely, we define three operations on SDSE (change of variables, dilatation and extension) and give two families of SDSE (cyclic and fundamental systems), and prove that any SDSE (S) such that H(S) is Hopf is the concatenation of several fundamental or cyclic systems after the application of a change of variables, a dilatation and iterated extensions.     

The t-median function on graphs

A median of a sequence π=x₁,x₂,…,x_k of elements of a finite metric space (X,d) is an element x for which is minimum. The function M with domain the set of all finite sequences on X and defined by M(π)={x:x is a median of π} is called the median function on X, and is one of the most studied consensus functions. Based on previous characterizations of median sets M(π), a generalization of the median function is introduced and studied on various graphs and ordered sets. In addition, new results are presented for median graphs.     

On the Spaltenstein correspondence

In the article [Spa1], N. Spaltenstein has established a bijection between the irreducible components of _χ, the space of full flags fixed by a nilpotent element χ ? M(n, k), where k is an algebraically closed field, and the standard tableaux associated to the Young diagram of χ. In this present work we determine, when χ is of hook type, for each irreducible component X of _χ, the unique Schubert cell _X of the full flag manifold = (V) (where V is vector space of dimension n over k), such that X ∩ _X is a dense subspace in X. This result will allow us to optimize the computation of _χ and when k = is the complex field, to see that the graph resolution of the partition (2, 1, …, 1) of n is related to the Dynkin diagram of sl(n, ).     

Determination of means by invariance

If M is a mean on and M(f(x₁),f(x₂),…,f(xn))=f(M(x₁,x₂,…,xn)) then we say that M is invariant under f. The problem is to find a class of functions that by invariance determines a mean uniquely. We focus on the geometric mean, which can be transformed to obtain results for other means.     

The Torelli theorem for the moduli spaces of connections on a Riemann surface

Let (X,x₀) be any one-pointed compact connected Riemann surface of genus g, with g≥3. Fix two mutually coprime integers r>1 and d. Let M_X denote the moduli space parametrizing all logarithmic -connections, singular over x₀, on vector bundles over X of degree d. We prove that the isomorphism class of the variety M_X determines the Riemann surface X uniquely up to an isomorphism, although the biholomorphism class of M_X is known to be independent of the complex structure of X. The isomorphism class of the variety M_X is independent of the point x₀∈X. A similar result is proved for the moduli space parametrizing logarithmic -connections, singular over x₀, on vector bundles over X of degree d. The assumption r>1 is necessary for the moduli space of logarithmic -connections to determine the isomorphism class of X uniquely.     

Curves having one place at infinity and linear systems on rational surfaces

Denoting by L_d(m₀,m₁,…,m_r) the linear system of plane curves of degree d passing through r+1 generic points p₀,p₁,…,p_r of the projective plane with multiplicity m_i (or larger) at each p_i, we prove the Harbourne-Hirschowitz Conjecture for linear systems L_d(m₀,m₁,…,m_r) determined by a wide family of systems of multiplicities and arbitrary degree d. Moreover, we provide an algorithm for computing a bound for the regularity of an arbitrary system , and we give its exact value when is in the above family. To do that, we prove an H¹-vanishing theorem for line bundles on surfaces associated with some pencils “at infinity”.     

Noncommutative convexity arises from linear matrix inequalities

This paper concerns polynomials in g noncommutative variables x=(x₁,…,xg), inverses of such polynomials, and more generally noncommutative “rational expressions” with real coefficients which are formally symmetric and “analytic near 0.” The focus is on rational expressions r=r(x) which are “matrix convex” near 0; i.e., those rational expressions r for which there is an ?>0 such that if X=(X₁,…,Xg) is a g-tuple of n×n symmetric matrices satisfying

     

The Frobenius map, rank 2 vector bundles and Kummer's quartic surface in characteristic 2 and 3

Let X be a smooth projective curve of genus g?2 defined over an algebraically closed field k of characteristic p>0. Let M_X(r) be the moduli space of semi-stable vector bundles with fixed trivial determinant. The relative Frobenius map induces by pull-back a rational map . We determine the equations of V in the following two cases (1) (g,r,p)=(2,2,2) and X nonordinary with Hasse-Witt invariant equal to 1 (see math.AG/0005044 for the case X ordinary), and (2) (g,r,p)=(2,2,3). We also show the existence of base points of V, i.e., semi-stable bundles E such that F∗E is not semi-stable, for any triple (g,r,p).     

Projectively simple rings

An infinite-dimensional N-graded k-algebra A is called projectively simple if dim_kA/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,σ). Here X is a smooth projective variety, σ is an automorphism of X with no proper σ-invariant subvariety (we call such automorphisms wild), and L is a σ-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 . 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 is Cohen-Macaulay and Gorenstein.     

On some geometric parameters in Banach spaces

Let X be a normed linear space and be the unit sphere of X. Let , , and J(X)=sup{‖x+y‖∧‖x−y‖}, x and y∈S(X) be the modulus of convexity, the modulus of smoothness, and the modulus of squareness of X, respectively. Let . In this paper we proved some sufficient conditions on δ(?), ρX(?), J(X), E(X), and , where the supremum is taken over all the weakly null sequence xn in X and all the elements x of X for the uniform normal structure.     

An algorithm for unimodular completion over Laurent polynomial rings

We present a new and simple algorithm for completion of unimodular vectors with entries in a multivariate Laurent polynomial ring over an infinite field K. More precisely, given n?3 and a unimodular vector V=^t(v₁,…,v_n)∈Rⁿ (that is, such that 〈v₁,…,v_n〉=R), the algorithm computes a matrix M in M_n(R) whose determinant is a monomial such that MV=^t(1,0,…,0), and thus M^-1 is a completion of V to an invertible matrix.