A note on pseudo-metrics on the plane

Every c-finite measure Μ on the set G of the lines on the plane such that $$(0){\text{ }}\mu {\text{(\{ g}} \in G:{\text{ }}P \in {\text{g\} ) = 0}}$$ for every point P?R ² generates a pseudo-metric F on the plane when one puts F P ₁, P ₂= \(\tfrac{1}{2}\) μ({g∈G:g separates the points P ₁ and P ₂}) The pseudo-metrics which are generated in this way possess the property of linear additivity, that is F(P ₁,P ₃)=F(P ₁,P ₂)+F(P ₂,P ₃) for P ₁,P ₂,P ₃ on a line, P ₂ between P ₁ and P ₃, and are continuous with respect to the Euclidean topology in R ² × R ². In this paper we prove the converse: every linear additive and continuous pseudo-metric F is generated as above by some c-finite measure Μ on G for which (0) holds. The method of proof shows that values of linearly additive and continuous pseudo-metric F inside every bounded convex polygon C are determined completely by the values of F on (δC)². The representation of pseudo-metrics by measures is useful in derivation of inequalities for the former.     

A note on families of hyperelliptic curves

We give a stack-theoretic proof for some results on families of hyperelliptic curves. Received: 5 February 2008     

A note on coverings of plane graphs

For any plane graph G the number of edges in a minimum edge covering of the faces of G is at most the vertex independence number of G and the numbre of vertices in a minimum vertex covering of the faces of G is at most the edge independence number of G. © 1995 John Wiley & Sons, Inc.     

A note on characterizing maximally looped complex curves

In this paper characterization of maximally looped complex curves is discussed and a relation between the number of double points and the number of loops is presented. Also we calculate the rotation number of maximally looped complex curves and compute the number σ of double tangents for those without inflection points.     

A note on Griffiths infinitesimal invariant for curves

Given a generic curve of genus $g\ge 4$ and a smooth point $L\in W_{g-1}^{1}(C)$ , whose linear system is base-point free, we consider the Abel–Jacobi normal function associated with $L^{\otimes 2}\otimes \omega _{C}^{-1}$ , when $(C,L)$ varies in moduli. We prove that its infinitesimal invariant reconstructs the couple $(C,L)$ . When $g=4$ , we obtain the generic Torelli theorem proved by Griffiths.     

A note on polylogarithms on curves and abelian schemes

In this note we study the polylogarithm extension on curves and abelian schemes in the étale realization. The main result shows that the polylog on the jacobian of a curve is given as the cup-product of the polylog on the curve with the fundamental class of the curve.     

A note on simultaneous Diophantine approximation on planar curves

Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type. The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University Research Fellow. For Iona and Ayesha on No. 3.     

A note on tropical triangles in the plane

We define transversal tropical triangles （affine and projective） and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically independent and they tropically span a classical hexagon whose sides have slopes ∞, 0, 1. Using this classical hexagon, we determine a parameter space for transversal tropical triangles. The coordinates of the vertices of a transversal tropical triangle determine a tropically regular matrix. Triangulations of the tropical plane are obtained.     

A note on constant geodesic curvature curves on surfaces

In this paper we are concerned with the structure of curves on surfaces whose geodesic curvature is a large constant. We first discuss the relation between closed curves with large constant geodesic curvature and the critical points of Gauss curvature. Then, we consider the case where a curve with large constant geodesic curvature is immersed in a domain which does not contain any critical point of the Gauss curvature.     

A note on biharmonic curves in Sasakian space forms

We classify the proper-biharmonic non-Legendre curves in a Sasakian space form for which the angle between the tangent vector field and the characteristic vector field is constant and then obtain explicit examples of such curves in \mathbbR²ⁿ⁺¹(-3){\mathbb{R}^{2n+1}(-3)}.     

A note on the postulation of maximal rank curves

Summary Lets be the lowest degree of a surface containing a maximal rank curveY. We want to compare the dimension of the space of the surfaces of degrees containingY with the dimension of the space of the surfaces of degrees+1. We apply the bound that we find for getting a bound for the third Chern class of a rank-two reflexive sheaf onP _k ³ with seminatural cohomology, thus answering to a conjecture proposed byR. Hartshorne.

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Riassunto Sias il più basso grado di una superficie contenente una curva di rango massimoY. Si vuole confrontare la dimensione dello spazio delle superfici di grados contenentiY con la dimensione dello spazio delle superfici di grados+1. Applichiamo la limitazione trovata per ottenere una limitazione per la terza classe di Chern di un fascio riflessivo di rango 2 suP _k ³ con coomologia seminaturale, rispondendo così ad una congettura posta daR. Hartshorne.

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Résumé Soits le plus petit degré d’une surface contenante une curveY de rang maximum. Nous comparons la dimension de l’espace des surfaces de degrés contenantesY avec la dimension de l’espace des surfaces de degrés+1. Nous appliquons la borne trouvée pour obtenir une borne pour la troisième classe de Chern d’un faisceau réflexif de rang deux surP _k ³ à cohomologie seminaturelle. Ceci répond à une conjecture posée parR. Hartshorne.

     

Ultradifferentiable functions on smooth plane curves

The regularity, in the sense of ultradifferentiability, of real functions of two variables is determined in terms of the regularity of their restrictions to a given family of smooth plane curves. The special case of line segments reduces to the main result in [Proc. Amer. Math. Soc. 127 (1999) 2099-2104]. As a consequence, the Bochnak-Siciak theorem on real analyticity is obtained. A formal analog of one of the results provides a generalization of the two-variable case of the Abhyankar and Moh [J. Reine Angew. Math. 241 (1970) 27-33] theorem.