Isotrivially fibred isles in the moduli space of surfaces of general type

We complement Catanese's results on isotrivially fibred surfaces by completely describing the components containing an isotrivial surface with monodromy group . We also give an example for deformation equivalent isotrivial surfaces with different monodromy group.

     

Enumerative tropical algebraic geometry in

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629-634.

The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the real space and holomorphic curves with certain piecewise-linear graphs there.

     

The Minimal Length of a Closed Geodesic Net on a Riemannian Manifold with a Nontrivial Second Homology Group

Let Mⁿ be a closed Riemannian manifold with a nontrivial second homology group. In this paper we prove that there exists a geodesic net on Mⁿ of length at most 3 diameter(Mⁿ). Moreover, this geodesic net is either a closed geodesic, consists of two geodesic loops emanating from the same point, or consists of three geodesic segments between the same endpoints. Geodesic nets can be viewed as the critical points of the length functional on the space of graphs immersed into a Riemannian manifold. One can also consider other natural functionals on the same space, in particular, the maximal length of an edge. We prove that either there exists a closed geodesic of length ≤ 2 diameter(Mⁿ), or there exists a critical point of this functional on the space of immersed θ-graphs such that the value of the functional does not exceed the diameter of Mⁿ. If n=2, then this critical θ-graph is not only immersed but embedded.Mathematics Subject Classifications (2000). 53C23, 49Q10     

Closed Geodesics on Incomplete Surfaces

We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used The authors acknowledge the support of the Australian Research Council     

Numerical Schottky Uniformizations

A real algebraic curve of algebraic genus g ≥ 2 is a pair (S, τ), where S is a closed Riemann surface of genus g and τ is a reflection on S (anticonformal involution with fixed points). In this note, we discuss a numerical (Burnside) program which permits to obtain a Riemann period matrix of the surface S in terms of an uniformizing real Schottky group. If we denote by Aut₊(S, τ) the group of conformal automorphisms of S commuting with the real structure τ, then it is a well known fact that |Aut₊(S,τ)| ≥ 12(g−1). We say that (S,τ) is maximally symmetric if |Aut₊(S,τ)|=12(g−1). We work explicitly such a numerical program in the case of maximally symmetric real curves of genus two. We construct a real Schottky uniformization for each such real curve and we use the numerical program to obtain a real algebraic curve, a Riemann period matrix and the accessory parameters in terms of the corresponding Schottky uniformization. In particular, we are able to give for Bolza’s curve a Schottky uniformization (at least numerically), providing an example for which the inverse uniformization theorem is numerically solved.Partially supported by Projects Fondecyt 1030252 1030373 and UTFSM 12.03.21     

Polytopal Approximation Bounding the Number of k-Faces

Assume that M is a convex body with C² boundary in ^d. The paper considers polytopal approximation of M with respect to the most commonly used metrics, like the symmetric difference metric δ_S, the L_p metric, 1p∞, or the Banach–Mazur metric. In case of δ_S, the main result states that if P_n is a polytope whose number of k faces is at most n then

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The analogous estimates are proved for all the other metrics. Finally, the optimality of these estimates is verified up to a constant depending on the metric and the dimension.     

Solid-state reaction in Ti/Ni multilayers studied by magneto-optical and optical spectroscopies, and X-ray diffraction

Comparative study of the solid-state reaction (SSR) in a series of Ti/Ni multilayered films (MLF) with a bilayer period of 0.65-22.2 nm and a constant Ti to Ni sublayer thickness ratio has been performed by using the experimental and computer-simulated magneto-optical (MO) and optical spectroscopies as well as X-ray diffraction (XRD). It was shown that alloyed-like regions in an amorphous structure is spontaneously formed near the interfaces between pure elements during the film deposition. The thickness of this region was estimated as 2-3.8 nm on the basis of the MO and optical studies. The SSR in the Ti/Ni MLF caused by an annealing at 580 K for 60 min increases the thickness of these interfacial amorphous regions. It was shown that SSR takes place mainly in the Ti/Ni MLF with relatively “thick” sublayers. The existence of a threshold nominal Ni-sublayer thickness for observing the equatorial Kerr effect of about 3.0 and 4.5 nm for the as-deposited and annealed Ti/Ni MLF, respectively, is explained by formation of the nonmagnetic alloyed regions between pure components during the film deposition as a result of the SSR. For the case of Ti/Ni MLF, the MO and optical approaches turn out to be more sensitive in determining the thickness of the reacted zone, while XRD is more useful for the structural analysis. It was also shown that the very thin nonreacted Ni sublayers have different MO properties (and hence electronic structure) from the bulk. Received 2 May 2001 and Received in final form 21 November 2001     

Noncommutative cyclic covers and maximal orders on surfaces

In this paper, we construct various examples of maximal orders on surfaces, including some del Pezzo orders, some ruled orders and some numerically Calabi-Yau orders. The method of construction is a noncommutative version of the cyclic covering trick. These noncommutative cyclic covers are very computable and we give a formula for their ramification data. This often allows us to determine if a maximal order, described via ramification data, can be constructed as a noncommutative cyclic cover. The construction also has applications to Brauer-Severi varieties and, in the quaternion case, we show how to obtain some Brauer-Severi varieties from G-Hilbert schemes of P¹-bundles.     

Effect of epitaxial strain on the atomic structure of Pd clusters on MgO(100) substrate

Morphology and atomic structure of supported Pd clusters on MgO(100) substrate are investigated theoretically using a mixed approach: a semi-empirical potential for the metal bonding within the cluster and a potential fitted to ab initio calculations for the metal-oxide interaction. We find that the clusters adopt a truncated pyramidal morphology in agreement with experimental results. The detailed study of the epitaxial relation as a function of cluster size shows the existence of a critical size around 3 nm where elastic strain due to the misfit between the substrate and the deposit is released by the introduction of interfacial dislocations.