Algebraic cycles on products of elliptic curves over <Emphasis Type="Italic">p</Emphasis>-adic fields

We show that for any odd prime p there is a smooth projective threefold W defined over a p-adic field such that the Chow group CH²(W)/ and the Griffiths group Griff²(W)/ are infinite for suitable primes . We further give examples of smooth projective fourfolds over these p-adic fields for which the -torsion subgroup CH³ is infinite.     

Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three

A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of such that holds for any , , where . A CAC with maximum code size for given n and k is called optimal. Furthermore, an optimal CAC C is said to be tight equi-difference if holds and any codeword has the form . The concept of a CAC is motivated from applications in multiple-access communication systems. In this paper, we give a necessary and sufficient condition to construct tight equi-difference CACs of weight k = 3 and characterize the code length n’s admitting the condition through a number theoretical approach.      

Symplectic structure and reduction on the space of Riemannian metrics

The space of Riemannian metrics ${\mathfrak{Met}}MThe space of Riemannian metrics on an oriented compact manifold M of dimension n = 4k − 2 is endowed with a canonical presymplectic structure and a moment map [cf. Ferreiro Pérez and Mu?oz Masqué, Preprint (arXiv: math.DG/0507075)]. The fiber is characterized as the space of solutions to a differential equation. In dimension 2, the symplectic reduction of is analyzed and the construction presented here is compared with that introduced in Donaldson (Fields Medallists’ Lectures, 1997) and Fujiki (Sugaku Expositions 5(2):173–191, 1992). Finally, conformally flat metrics and, for n = 6, K?hler metrics of constant holomorphic sectional curvature are shown to be contained in .      

The symplectomorphism group of a blow up

We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp and Diff of its one point blow up . There are three main arguments. The first shows that for any oriented M the natural map from to is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group that persist into the space of self-homotopy equivalences of . Since it uses purely homological arguments, it applies to c-symplectic manifolds (M, a), that is, to manifolds of dimension 2n that support a class such that . The third argument uses the symplectic structure on M and detects nontrivial elements in the (higher) homology of BSymp, M using characteristic classes defined by parametric Gromov–Witten invariants. Some results about many point blow ups are also obtained. For example we show that if M is the four-torus with k-fold blow up (where k > 0) then is not generated by the groups as ranges over the set of all symplectic forms on . Partially supported by NSF grants DMS 0305939 and 0604769.     

Norm-preserving extensions of functionals and denting points of convex sets

Let W and Z be Banach spaces, and let and be closed subspaces. Let be a subspace of , the Banach space of bounded linear operators from W* to Z**, containing . We describe, for and , all norm-preserving extensions of to the space in terms of convergence of convex combinations. We also characterize denting points of bounded convex subsets of Banach spaces in similar terms. Various applications are presented. Supported by Estonian Science Foundation Grant 5704.     

A speciality theorem for curves in P<Superscript>5</Superscript>

Let be an integral projective curve. One defines the speciality index e(C) of C as the maximal integer t such that , where ω _C denotes the dualizing sheaf of . Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if is an integral degree d curve not contained in any surface of degree  < s, in any threefold of degree  < t, and in any fourfold of degree  < u, and if , then Moreover equality holds if and only if C is a complete intersection of hypersurfaces of degrees u, , and . We give also some partial results in the general case , .      

Fundamental domains in Lorentzian geometry

We consider a discrete subgroup Γ of the simply connected Lie group of finite level, i.e. the subgroup intersects the centre of in a subgroup of finite index, this index is called the level of the group. The Killing form induces a Lorentzian metric of constant curvature on the Lie group . The discrete subgroup Γ acts on by left translations. We describe the Lorentz space form by constructing a fundamental domain F for Γ. We want F to be a polyhedron with totally geodesic faces. We construct such F for all Γ satisfying the following condition: The image of Γ in PSU(1,1) has a fixed point u in the unit disk of order larger than the index of Γ. The construction depends on the group Γ and on the orbit Γ(u) of the fixed point u.      

Well-posedness of constrained minimization problems via saddle-points

In this paper, it is proved a very general well-posedness result for a class of constrained minimization problems of which the following is a particular case: Let X be a Hausdorff topological space and let be two non-constant functions such that, for each , the function has sequentially compact sub-level sets and admits a unique global minimum in X. Then, for each , the restriction of J to has a unique global minimum, say , toward which every minimizing sequence converges. Moreover, the functions and are continuous in .     

The integral monodromy of hyperelliptic and trielliptic curves

We compute the and monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group . We prove that the monodromy of the moduli space of trielliptic curves with signature (r,s) is the special unitary group . Rachel Pries was partially supported by NSF grant DMS-04-00461.     

On the number of Galois points for a plane curve in positive characteristic,II

For a smooth plane curve , we call a point a Galois point if the point projection at P is a Galois covering. We study Galois points in positive characteristic. We give a complete classification of the Galois group given by a Galois point and estimate the number of Galois points for C in most cases.