Authors:David Eklund Abstract: This paper is about the family of smooth quartic surfaces that are invariant under the Heisenberg group \(H_{2,2}\) . For a very generic X, we show that the Picard number of X is 16 and determine its Picard lattice. It turns out that a very generic X contains 320 irreducible conics which generate the Picard group of X. PubDate: 2018-02-06 DOI: 10.1007/s40879-018-0216-2

Authors:Friedrich Götze; Andrei Yu. Zaitsev Abstract: Let \(X_1,\ldots ,X_n\) be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums with respect to the arithmetic structure of coefficients \(a_k\) in the context of the Littlewood–Offord problem. In recent papers of Eliseeva, Götze and Zaitsev, we discussed the relations between the inverse principles stated by Nguyen, Tao and Vu and similar principles formulated by Arak in his papers from the 1980’s. In this paper, we will derive some more general and more precise consequences of Arak’s inequalities providing new results in the Littlewood–Offord problem. PubDate: 2018-02-02 DOI: 10.1007/s40879-018-0215-3

Authors:Kamel Mazhouda; Lejla Smajlović Abstract: We derive several formulas for evaluation of the nth Li coefficient associated to a function field K of genus g with a finite field of constants \({\mathbb F}_{q}\) and deduce some consequences of the obtained formulas and the Riemann hypothesis in the function fields case, which is equivalent to the statement for all positive integers n. For example, we deduce sharp upper and lower bounds for the sum \(\sum _{\widetilde{D}} q^{N(\widetilde{D})}\) over all divisor classes \(\widetilde{D}\) of degree between 0 and \(2g-2\) , where \(N(\widetilde{D})\) denotes the maximal number of linearly independent positive divisors in \(\widetilde{D}\) . We also show that for large q and fixed genus, the special values \(L'(1)\) and \(L''(1)\) of the L-function associated to K are approximately equal to and respectively, where denotes the class number of K. PubDate: 2018-01-18 DOI: 10.1007/s40879-018-0212-6

Authors:Alessandra Bertapelle; Cristian D. González-Avilés Abstract: We extend Greenberg’s original construction to arbitrary schemes over (certain types of) local artinian rings. We then establish a number of properties of the extended functor and determine, for example, its behavior under Weil restriction. We also discuss a formal analog of the functor. PubDate: 2018-01-08 DOI: 10.1007/s40879-017-0210-0

Authors:Ilya Karzhemanov; Ilya Zhdanovskiy Abstract: We consider the so-called surjective rational maps. We study how the surjectivity property behaves in families of rational maps. Some (counter) examples are provided and a general result is proved. PubDate: 2018-01-05 DOI: 10.1007/s40879-017-0205-x

Authors:Alexander Kharazishvili Abstract: A translation invariant measure on the real line R is constructed, which extends the Lebesgue measure on R and for which the Steinhaus property fails in a strong form. PubDate: 2018-01-02 DOI: 10.1007/s40879-017-0211-z

Authors:Pavel Andreev Pages: 767 - 787 Abstract: We present an approach leading to Finsler geometry without differential calculus of tensors. Several natural examples of such singular Finsler spaces are studied. One class of such examples contains Busemann G-spaces with non-positive curvature. Starting with a singular version of the axiomatics, some simplest properties known in the smooth Finsler geometry are interpreted. PubDate: 2017-12-01 DOI: 10.1007/s40879-017-0169-x Issue No:Vol. 3, No. 4 (2017)

Authors:Valerii N. Berestovskii Pages: 788 - 807 Abstract: The author proved in the late 1980s that any homogeneous manifold with an intrinsic metric is isometric to some homogeneous quotient space of a connected Lie group by its compact subgroup with an invariant Finslerian or sub-Finslerian metric. In the case of a trivial compact subgroup, the invariant Riemannian or sub-Riemannian metrics are singled out from invariant Finslerian or sub-Finslerian metrics by being in one-to-one correspondence with special one-parameter Gaussian convolutions semigroups of absolutely continuous probability measures. Any such semigroup is generated by a second order hypoelliptic operator. In the present paper, in connection with this, the author discusses briefly the operator definition of lower bound for Ricci curvature by Baudoin–Garofalo. Earlier, Agrachev defined a notion of curvature for sub-Riemannian manifolds. As an alternative, the author discusses in some detail old definitions of curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov’ev. To calculate the Solov’ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special riggings of invariant bracket generating distributions on manifolds. As a justification, we find a foliation on the cotangent bundle over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields for the Pontryagin–Hamilton function. This function was applied in the Pontryagin maximum principle for the time-optimal problem. The foliation is entirely described by the coadjoint representation of the Lie group G. We also use the canonical symplectic form on and its values for the above mentioned invariant Hamiltonian vector fields. In particular, the above rigging method is applicable to contact sub-Riemannian manifolds, sub-Riemannian Carnot groups, and homogeneous sub-Riemannian manifolds possessing a submetry onto a Riemannian manifold. In Sects. 5 and 6, some examples are presented. PubDate: 2017-12-01 DOI: 10.1007/s40879-017-0171-3 Issue No:Vol. 3, No. 4 (2017)

Authors:Weixu Su; Youliang Zhong Pages: 1045 - 1057 Abstract: This paper is a brief survey of some results on the Finsler structures of Teichmüller metric. We mainly describe the connection between holomorphic quadratic differentials and extremal length of measured foliations. Some rigidity results about the Teichmüller metric inspired by Royden’s theorem are discussed. Teichmüller theory is probably the best example of an instance where Finsler geometry enters in the field of low dimensional geometry and topology. PubDate: 2017-12-01 DOI: 10.1007/s40879-017-0161-5 Issue No:Vol. 3, No. 4 (2017)

Authors:Wenhua Zhao Abstract: Let K be a field of characteristic zero, \({\mathcal {A}}\) a K-algebra and \(\delta \) a K-derivation of \({\mathcal {A}}\) or K- \({\mathcal {E}}\) -derivation of \({\mathcal {A}}\) (i.e., \(\delta =\mathrm{Id}_{\mathcal {A}}-\phi \) for some K-algebra endomorphism \(\phi \) of \({\mathcal {A}}\) ). Motivated by the Idempotent conjecture proposed in Zhao (Commun Contemp Math, https://doi.org/10.1142/S0219199717500560, arXiv:1701.05992), we first show that for every idempotent e lying in both the kernel \({\mathcal {A}}^\delta \) and the image \(\mathrm{Im}\,\delta {:}{=}\delta ({\mathcal {A}})\) of \(\delta \) , the principal ideal \((e)\subseteq \mathrm{Im}\,\delta \) if \(\delta \) is a locally finite K-derivation or a locally nilpotent K- \({\mathcal {E}}\) -derivation of \({\mathcal {A}}\) ; and \(e{\mathcal {A}}, {\mathcal {A}}e \subseteq \mathrm{Im}\,\delta \) if \(\delta \) is a locally finite K- \({\mathcal {E}}\) -derivation of \({\mathcal {A}}\) . Consequently, the Idempotent conjecture holds for all locally finite K-derivations and all locally nilpotent K- \({\mathcal {E}}\) -derivations of \({\mathcal {A}}\) . We then show that (if and) only if \(\delta \) is surjective, which generalizes the same result due to Nouazé and Gabriel (J Algebra 6(1):77–99, 1967) and Wright (Illinois J Math 25(3):423–440, 1981) for locally nilpotent K-derivations of commutative K-algebras to locally finite K-derivations and K- \({\mathcal {E}}\) -derivations \(\delta \) of all K-algebras \({\mathcal {A}}\) . PubDate: 2017-12-27 DOI: 10.1007/s40879-017-0209-6

Authors:Aiden A. Bruen; James M. McQuillan Abstract: Given two triangles which are in perspective from a vertex V, there are, in general, three other associated pairs of triangles which are also in perspective from V. If the Desargues axes for two of the four pairs are equal, and if the original two triangles form a 6-arc, we show that the Desargues lines for the four pairs are equal. We obtain a complete classification of such pairs of triangles. Then, we examine pairs of triangles in perspective using the inherent polarity of the associated configuration. In particular, we classify all cases when one or more of the 10 points is self-conjugate. Finally, we study a well-known classical result on double perspectivity of pairs of triangles inscribed in a conic. Using new methods, we provide a significant extension to perspectivities of pairs of inscribed n-gons. PubDate: 2017-12-20 DOI: 10.1007/s40879-017-0208-7

Authors:Fedor A. Bogomolov; Hang Fu Abstract: We construct pairs of elliptic curves over number fields with large intersection of projective torsion points. PubDate: 2017-12-15 DOI: 10.1007/s40879-017-0207-8

Authors:Laura P. Schaposnik Abstract: We give a geometric characterisation of the topological invariants associated to -Higgs bundles through KO-theory and the Langlands correspondence between orthogonal and symplectic Hitchin systems. By defining the split orthogonal spectral data, we obtain geometric description of the intersection of the moduli space of those Higgs bundles with the -Hitchin fibration in terms of a collection of compact abelian varieties, and provide a natural stratification of the moduli space of -Higgs bundles. PubDate: 2017-12-14 DOI: 10.1007/s40879-017-0206-9

Authors:Jacob Cable; Hendrik Süß Abstract: We give a classification of all pairs \((X,\xi )\) of Gorenstein del Pezzo surfaces X and vector fields \(\xi \) which are K-stable in the sense of Berman–Witt–Nyström and therefore are expected to admit a Kähler–Ricci solition. Moreover, we provide some new examples of Fano threefolds admitting a Kähler–Ricci soliton. PubDate: 2017-12-11 DOI: 10.1007/s40879-017-0204-y

Authors:Sorin V. Sabau; Kazuhiro Shibuya; Ryozo Yoshikawa Abstract: We study the behavior of the geodesics of strong Kropina spaces. The global and local aspects of geodesics theory are discussed. Our theory is illustrated with several examples. PubDate: 2017-11-02 DOI: 10.1007/s40879-017-0189-6

Authors:Michael Kapovich; Bernhard Leeb; Joan Porti Abstract: We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of rank one Lie groups to higher rank. Our main goal is to give several new equivalent characterizations for this important class of discrete subgroups. Our characterizations capture “rank one behavior” of Anosov subgroups and are direct generalizations of rank one equivalents to convex cocompactness. Along the way, we considerably simplify the original definition, avoiding the geodesic flow. We also show that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group. PubDate: 2017-10-30 DOI: 10.1007/s40879-017-0192-y

Authors:Carlos Eduardo Durán; Henrique Vitório Abstract: We express invariants of Finsler manifolds in a geometrical way by means of using moving planes and their associated Jacobi curves, which are curves in a fixed homogeneous Grassmannian manifold. We also use this language to review some applications. PubDate: 2017-10-24 DOI: 10.1007/s40879-017-0190-0