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”.     

On random perturbation of some intersective sets

Given a subset S of Z and a sequence I = (I_n)_n=1^∞ of intervals of increasing length contained in Z, let

     

Numerically effective line bundles associated to a stable bundle over a curve

Let G be a complex semisimple group and χ a character of a parabolic subgroup P⊂G such that the associated line bundle on G/P is ample. For a general stable G-bundle E_G over a compact Riemann surface of genus at least two, the line bundle over E_G/P defined by χ has the property that the restriction of  to any closed subvariety of E_G/P of smaller dimension is ample, although is not ample.     

Soluble groups with their centralizer factor groups of bounded rank

For a group class X, a group G is said to be a CX-group if the factor group G/C_G(g^G)∈X for all g∈G, where C_G(g^G) is the centralizer in G of the normal closure of g in G. For the class F_f of groups of finite order less than or equal to f, a classical result of B.H. Neumann [Groups with finite classes of conjugate elements, Proc. London Math. Soc. 1 (1951) 178-187] states that if G∈CF_f, the commutator group G^′ belongs to F_f^′ for some f^′ depending only on f. We prove that a similar result holds for the class , the class of soluble groups of derived length at most d which have Prüfer rank at most r. Namely, if , then for some r^′ depending only on r. Moreover, if , then for some r^′ and f^′ depending only on r,d and f.     

Higher syzygies of hyperelliptic curves

Let X be a hyperelliptic curve of arithmetic genus g and let f:X→P¹ be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d≤2g. Note that the minimal free resolution of X of degree ≥2g+1 is already completely known. Let A=f^∗O_P¹(1), and let L be a very ample line bundle on X of degree d≤2g. For , we call the pair (m,d−2m)the factorization type ofL. Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by |L| are precisely determined by the factorization type of L.     

On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ?₁,?₂,… of positive numbers there exist disjoint open collections V₁,V₂,… of open subsets of X, with diameters of members of Vi less than ?i and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X² is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].     

Coding and tiling of Julia sets for subhyperbolic rational maps

Let be a subhyperbolic rational map of degree d. We construct a set of “proper” coding maps Cod^°(f)={π_r:Σ→J}_r of the Julia set J by geometric coding trees, where the parameter r ranges over mappings from a certain tree to the Riemann sphere. Using the universal covering space for the corresponding orbifold, we lift the inverse of f to an iterated function system I=(g_i)_i=1,2,…,d. For the purpose of studying the structure of Cod^°(f), we generalize Kenyon and Lagarias-Wang's results : If the attractor K of I has positive measure, then K tiles φ^-1(J), and the multiplicity of π_r is well-defined. Moreover, we see that the equivalence relation induced by π_r is described by a finite directed graph, and give a necessary and sufficient condition for two coding maps π_r and π_r^′ to be equal.     

Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank

We define the multiple zeta function of the free Abelian group Z^d as

ζ_Z^d(s₁,…,s_d)=∑_|Z^d:H|<∞α₁(H)^−s₁?α_d(H)^−s_d,     

On soluble skew linear groups over finite-dimensional division algebras

Let D be a division ring of finite degree d and let n be a positive integer. If G is any soluble subgroup of , we prove that G has derived length at most

9+log₂d+(11/3)log₂n     

Cohomology of regular differential forms for affine curves

Let C be a complex affine reduced curve, and denote by H¹(C) its first truncated cohomology group, i.e. the quotient of all regular differential 1-forms by exact 1-forms. First we introduce a nonnegative invariant μ^′(C,x) that measures the complexity of the singularity of C at the point x, and we establish the following formula:

     

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.     

The stability of the equilibrium of the damped oscillator with damping changing sign

In this paper, we prove a sufficient and necessary condition for the stability of the equilibrium x=x^′=0 of the damped oscillator with damping changing sign

     

Brill-Gordan loci, transvectants and an analogue of the Foulkes conjecture

The hypersurfaces of degree d in the projective space Pn correspond to points of PN, where . Now assume d=2e is even, and let X(n,d)⊆PN denote the subvariety of two e-fold hyperplanes. We exhibit an upper bound on the Castelnuovo regularity of the ideal of X(n,d), and show that this variety is r-normal for r?2. The latter result is representation-theoretic, and says that a certain GLn+1-equivariant morphism

Sr(S2e(Cn+1))→S₂(Sre(Cn+1))     

Chen-Ruan cohomology of some moduli spaces of parabolic vector bundles

Let (X,D) be an ?-pointed compact Riemann surface of genus at least two. For each point x∈D, fix parabolic weights such that . Fix a holomorphic line bundle ξ over X of degree one. Let PMξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights . The group of order two line bundles over X acts on PMξ by the rule E_∗⊗L?E_∗⊗L. We compute the Chen-Ruan cohomology ring of the corresponding orbifold.     

Borel-plus-powers monomial ideals

Let S=K[x₁,…,x_n] be a standard graded polynomial ring over a field K. In this paper, we show that the lex-plus-powers ideal has the largest graded Betti numbers among all Borel-plus-powers monomial ideals with the same Hilbert function. In addition in the case of characteristic 0, by using this result, we prove the lex-plus-powers conjecture for graded ideals containing , where p is a prime number.     

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.     

Weakly Birkhoff recurrent switching signals, almost sure and partial stability of linear switched dynamical systems

Let {A₁,…,AK}⊂Cd×d be arbitrary K matrices, where K and d both ?2. For any 0<Δ<∞, we denote by the set of all switching sequences u=(λ.,t.):N→{1,…,K}×R₊ satisfying tj−tj−1?Δ and

     

Generic initial ideals and graded Artinian-level algebras not having the Weak-Lefschetz Property

We find a sufficient condition that is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function cannot be level if h_d≤2d+3, and that there exists a level O-sequence of codimension 3 of type for h_d≥2d+k for k≥4. Furthermore, we show that is not level if , and also prove that any codimension 3 Artinian graded algebra A=R/I cannot be level if . In this case, the Hilbert function of A does not have to satisfy the condition h_d−1>h_d=h_d+1.Moreover, we show that every codimension n graded Artinian level algebra having the Weak-Lefschetz Property has a strictly unimodal Hilbert function having a growth condition on (h_d−1−h_d)≤(n−1)(h_d−h_d+1) for every d>θ where

h₀<h₁<<h_α==h_θ>>h_s−1>h_s.