Normal generation and Clifford index on algebraic curves

For a smooth curve C it is known that a very ample line bundle on C is normally generated if Cliff() < Cliff(C) and there exist extremal line bundles (:non-normally generated very ample line bundle with Cliff() = Cliff(C)) with . However it has been unknown whether there exists an extremal line bundle with . In this paper, we prove that for any positive integers (g, c) with g = 2c + 5 and (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle with . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle with . More generally, if C has a line bundle computing the Clifford index c of C with , then C has such an extremal line bundle . For all authors, this work was supported by Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Reasearch Promotion Fund)(KRF-2005-070-C00005).     

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 spinorial analogue of Aubin’s inequality

Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric conformal to g, we denote by the first positive eigenvalue of the Dirac operator on . We show that

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 .     

Kaplansky classes and derived categories

We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semi-separated scheme X. The approach generalizes and simplifies the method used by the author in (Trans Am Math Soc 356(8) 3369–3390, 2004) and (Trans Am Math Soc 358(7), 2855–2874, 2006) to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed, much of the paper is dedicated to showing that in any Grothendieck category , any nice enough class of objects induces a model structure on the category Ch() of chain complexes. The main technical requirement on is the existence of a regular cardinal κ such that every object satisfies the following property: Each κ-generated subobject of F is contained in another κ-generated subobject S for which . Such a class is called a Kaplansky class. Kaplansky classes first appeared in Enochs and López-Ramos (Rend Sem Mat Univ Padova 107, 67–79, 2002) in the context of modules over a ring R. We study in detail the connection between Kaplansky classes and model categories. We also find simple conditions to put on which will guarantee that our model structure is monoidal. We will see that in several categories the class of flat objects form such Kaplansky classes, and hence induce monoidal model structures on the associated chain complex categories. We will also see that in any Grothendieck category , the class of all objects is a Kaplansky class which induces the usual (non-monoidal) injective model structure on Ch().     

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.     

Quasi-locally finite polynomial endomorphisms

If F is a polynomial endomorphism of , let denote the field of rational functions such that . We will say that F is quasi-locally finite if there exists a nonzero such that p(F) = 0. This terminology comes out from the fact that this definition is less restrictive than the one of locally finite endomorphisms made in Furter, Maubach (J Pure Appl Algebra 211(2):445–458, 2007). Indeed, F is called locally finite if there exists a nonzero such that p(F) = 0. In the present paper, we show that F is quasi-locally finite if and only if for each the sequence is a linear recurrent sequence. Therefore, this notion is in some sense natural. We also give a few basic results on such endomorphisms. For example: they satisfy the Jacobian conjecture.     

Positivity of the spherically averaged atomic one-electron density

We investigate the positivity of the spherically averaged atomic one-electron density . For a which stems from a physical ground state we prove that for r ≥ 0. This article may be reproduced in its entirety for non-commercial purposes.     

Quantization for a fourth order equation with critical exponential growth

For concentrating solutions weakly in H ²(Ω) to the equation on a domain with Navier boundary conditions the concentration energy is shown to be strictly quantized in multiples of the number .     

Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

Ortho and Causal Closure Operations in Ordered Vector Spaces

On a non-trivial partially ordered real vector space V the orthogonality relation is defined by incomparability and is a complete lattice of double orthoclosed sets. In an earlier paper we defined an integrally open ordered vector space V and proved orthomodularity of . We shall say that is an orthogonal set when for all with , we have . We consider two different closure operations and (ortho and causal closure) and prove: V is integrally open iff for every orthogonal set . Hence follows: if V is integrally open, then . Received July 6, 2007; accepted in final form July 31, 2007.     

Calderón–Zygmund estimates for higher order systems with p(x) growth

For weak solutions of higher order systems of the type , for all , with variable growth exponent p : Ω → (1,∞) we prove that if with , then . We should note that we prove this implication both in the non-degenerate (μ > 0) and in the degenerate case (μ = 0).     

Asymptotic analysis of solutions to parabolic systems

We study asymptotics as of solutions to a linear, parabolic system of equations with time-dependent coefficients in , where is a bounded domain. On we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function . This includes in particular situations when the coefficients may take different values on different parts of and the boundaries between them can move with t but stabilize as . The main result is an asymptotic representation of solutions for large t. A consequence is that for , the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.     

Algebraic-geometry codes, one-point codes, and evaluation codes

One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed–Muller codes. Given an -algebra A, an order function on A and given a surjective -morphism of algebras , the ith evaluation code with respect to is defined as the code . In this work it is shown that under a certain hypothesis on the -algebra A, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on A is that its field of fractions is a function field over and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G _i with associated divisors is the sequence of evaluation codes associated to some -algebra A, some order function and some surjective morphism with if and only if it is a sequence of one-point codes.      

Solutions with multiple spike patterns for an elliptic system

We consider a system of the form , in an open domain of , with Dirichlet conditions at the boundary (if any). We suppose that f and g are power-type non-linearities, having superlinear and subcritical growth at infinity. We prove the existence of positive solutions and which concentrate, as , at a prescribed finite number of local minimum points of V(x), possibly degenerate.     

Gelfand–Zeitlin actions and rational maps

We observe that the analogue of the Gelfand–Zeitlin action on , which exists on any symplectic manifold M with an Hamiltonian action of , has a natural interpretation as a residual action, after we identify M with a symplectic quotient of . We also show that the Gelfand–Zeitlin actions on and on the regular part of can be identified with natural Hamiltonian actions on spaces of rational maps into full flag manifolds, while the Gelfand–Zeitlin action on the whole corresponds to a natural action on a space of rational maps into the manifold of half-full flags in . The research of the first author is supported by the Alexander von Humboldt Foundation.     

A Slow Transient Diffusion in a Drifted Stable Potential

We consider a diffusion process X in a random potential of the form , where is a positive drift and is a strictly stable process of index with positive jumps. Then the diffusion is transient and converges in law towards an exponential distribution. This behaviour contrasts with the case where is a drifted Brownian motion and provides an example of a transient diffusion in a random potential which is as “slow” as in the recurrent setting.      

Analytic properties in the spectrum of certain Banach algebras

We show a sufficient condition for a domain in to be a H ^∞-domain of holomorphy. Furthermore if a domain has the Gleason property at a point and the projection of the n − 1th order generalized Shilov boundary does not coincide with Ω then is schlicht. We also give two examples of pseudoconvex domains in which the spectrum is non-schlicht and satisfy several other interesting properties.      

Non-free isometric immersions of Riemannian manifolds

Let (V, g) be a Riemannian manifold and let be the isometric immersion operator which, to a map , associates the induced metric on V, where denotes the Euclidean scalar product in . By Nash–Gromov implicit function theorem is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions . We show that the operator (where denotes the space of C ^∞- smooth quadratic forms on ) is infinitesimally invertible over a non-empty open subset of and therefore is an open map in the respective fine topologies.