Authors:Patricio Gallardo; Jesus Martinez-Garcia; Zheng Zhang Abstract: We study the moduli space of triples \((C, L_1, L_2)\) consisting of quartic curves C and lines \(L_1\) and \(L_2\) . Specifically, we construct and compactify the moduli space in two ways: via geometric invariant theory (GIT) and by using the period map of certain lattice polarized K3 surfaces. The GIT construction depends on two parameters \(t_1\) and \(t_2\) which correspond to the choice of a linearization. For \(t_1=t_2=1\) we describe the GIT moduli explicitly and relate it to the construction via K3 surfaces. PubDate: 2018-04-24 DOI: 10.1007/s40879-018-0248-7

Authors:Ivan B. Fesenko; Sergei V. Vostokov; Seok Ho Yoon Abstract: We propose and study a generalised Kawada–Satake method for Mackey functors in the class field theory of positive characteristic. The root of this method is in the use of explicit pairings, such as the Artin–Schreier–Witt pairing, for groups describing abelian extensions. We separate and simplify the algebraic component of the method and discuss a relation between the existence theorem in class field theory and topological reflexivity with respect to the explicit pairing. We apply this method to derive higher local class field theory of positive characteristic, using advanced properties of topological Milnor K-groups of such fields. PubDate: 2018-04-23 DOI: 10.1007/s40879-018-0245-x

Authors:Florin Ambro Abstract: We construct an explicit Deligne–Du Bois complex for algebraic varieties which are locally analytically isomorphic to the spectrum of a toric face ring. PubDate: 2018-04-23 DOI: 10.1007/s40879-018-0249-6

Authors:Ivan Cheltsov; Jihun Park; Constantin Shramov Abstract: We prove that the sum of the \(\alpha \) -invariants of two different Kollár components of a Kawamata log terminal singularity is less than 1. PubDate: 2018-04-13 DOI: 10.1007/s40879-018-0237-x

Authors:Lorelei Koss Abstract: A survey of ordinary differential equation models investigating environmental and sustainability issues in the history of Easter Island appeared in 2011. One of the results discussed was a model by Basener et al. which investigated the relationship between humans, the forest stock, and the non-native Polynesian rat that was introduced by early settlers. This paper surveys recent developments in research using differential equation models to understand the interactions between people, rats, and forest stock. PubDate: 2018-04-10 DOI: 10.1007/s40879-018-0242-0

Authors:Alessandro Andretta; Riccardo Camerlo Abstract: We study the density function of measurable subsets of the Cantor space. Among other things, we identify a universal set for \(\varvec{\varSigma }^{1}_{1}\) subsets of in terms of the density function; specifically is the set of all pairs (K, r) with K compact and being the density of some point with respect to K. This result yields that the set of all K such that the range of their density function is , for some fixed uncountable analytic set \(S\subseteq (0;1)\) , is \(\varvec{\varPi }^{1}_{2}\) -complete. PubDate: 2018-04-10 DOI: 10.1007/s40879-018-0238-9

Authors:Grażyna Kwiecińska Abstract: We generalize the notion of strong quasi-continuity for real functions of real variable, given by Grande (Real Anal Exchange 21(1):236–243, 1995/1996), to the case of multifunctions in topological spaces. We introduce a differentiation base in a metric space and show that a strong quasi-continuous multifunction with respect to this differentiation base is almost everywhere continuous. PubDate: 2018-04-09 DOI: 10.1007/s40879-018-0239-8

Authors:DongSeon Hwang Abstract: We present an intersection-theoretic formula concerning curves on projective surfaces in terms of lattices with special emphasis on minimal resolutions of \({\mathbb {Q}}\) -homology projective planes. This formula can be used to detect the existence/nonexistence of curves with given intersection properties. PubDate: 2018-04-09 DOI: 10.1007/s40879-018-0240-2

Authors:Yuri A. Farkov Abstract: We describe three types of compactly supported wavelet frames associated with Walsh functions: (1) MRA-based tight frames, (2) frames obtained from the Daubechies-type “admissible condition”, and (3) frames based on the Walsh–Parseval type kernels. Parametric wavelet sets for Vilenkin groups and some related results are also discussed. PubDate: 2018-04-09 DOI: 10.1007/s40879-018-0220-6

Authors:Asher Auel; Christian Böhning; Alena Pirutka Abstract: We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the specialization method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2, 3) in is not stably rational. Via projections onto the two factors, is a cubic surface bundle and is a conic bundle, and we analyze the stable rationality problem from both these points of view. Also, we introduce, for any \(n\geqslant 4\) , new quadric surface bundle fourfolds with discriminant curve of degree 2n, such that \(X_n\) has nontrivial unramified Brauer group and admits a universally \(\mathrm {CH}_0\) -trivial resolution. PubDate: 2018-04-04 DOI: 10.1007/s40879-018-0233-1

Authors:Edoardo Ballico; Philippe Ellia; Claudio Fontanari Abstract: We prove the following statement, which has been conjectured since 1985: There exists a constant K such that for all natural numbers d, g with \(g\leqslant Kd^{3/2}\) there exists an irreducible component of the Hilbert scheme of \(\mathbb {P}^3\) whose general element is a smooth, connected curve of degree d and genus g of maximal rank. PubDate: 2018-04-02 DOI: 10.1007/s40879-018-0235-z

Authors:Igor Dolgachev; Benson Farb; Eduard Looijenga Abstract: In 1981 William L. Edge discovered and studied a pencil \(\mathscr {C}\) of highly symmetric genus 6 projective curves with remarkable properties. Edge’s work was based on an 1895 paper of Anders Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel Geometry of the Wiman–Edge pencil, II: hyperbolic, conformal and modular aspects (in preparation), we consider \(\mathscr {C}\) from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects. PubDate: 2018-04-02 DOI: 10.1007/s40879-018-0231-3

Authors:Igor Protasov; Ksenia Protasova Abstract: A ballean (or coarse structure) is a set endowed with some family of subsets, the balls, in such a way that balleans with corresponding morphisms can be considered as asymptotic counterparts of uniform topological spaces. For a ballean \({{\mathscr {B}}}\) on a set X, the hyperballean \({{\mathscr {B}}}^{\flat }\) is a ballean naturally defined on the set \(X^{\flat }\) of all bounded subsets of X. We describe all balleans with hyperballeans of bounded geometry and analyze the structure of these hyperballeans. PubDate: 2018-03-29 DOI: 10.1007/s40879-018-0236-y

Authors:Marat Gizatullin Abstract: The aim is to describe two homogeneous affine open sets in some projective spaces. The sets are well known, their groups of automorphisms contain simple exceptional groups of types \(E_6\) or \(E_7\) , although the total groups of automorphisms are infinite-dimensional and both the sets are flexible. We prove existence of automorphisms not belonging to the connected components of unity, construct extended Weyl groups including these non-tame automorphisms. Our methods are based on classical combinatorics associated with 27 lines on non-singular cubic surfaces and with 56 exceptional curves on Del Pezzo surfaces of degree 2. Some traditional applications of Jordan algebras are used. PubDate: 2018-03-28 DOI: 10.1007/s40879-018-0228-y

Authors:Lisa Marquand; Joonyeong Won Abstract: Let S be a smooth rational surface with \(K^2_S\geqslant 3\) . We show that there exist A-polar cylinders for a polarized pair (S, A) except when S is a smooth cubic surface and A is an anticanonical divisor. PubDate: 2018-03-26 DOI: 10.1007/s40879-018-0229-x

Authors:Timothy H. Steele Abstract: Let with \(bB_{1}\) the set of Baire-1 self-maps of I. For , let be the collection of \(\omega \) -limit sets generated by f, and \(\mathrm{\Lambda } (f)=\bigcup _{x\in I}\omega (x,f)\) be the set of \(\omega \) -limit points of f. There exists S a residual subset of \(bB_{1}\) such that for any , the following hold: For any \(x\in I\) , the \(\omega \) -limit set \(\omega (x,f)\) is contained in the set of points at which f is continuous, and \(\omega (x,f)\) is an \( \infty \) -adic adding machine. For any \(\varepsilon >0\) , there exists a natural number M such that \( f^{m}(I)\subset B_{\varepsilon }(\mathrm{\Lambda } (f))\) whenever \(m>M\) . Moreover, \( f:\mathrm{\Lambda } (f)\rightarrow \mathrm{\Lambda } (f)\) is a bijection, and \(\mathrm{\Lambda } (f)\) is closed. The Hausdorff s-dimensional measure of \(\mathrm{\Lambda } (f)\) is zero for all \(s>0\) . The collection of \(\omega \) -limit sets \(\mathrm{\Omega } (f)\) is closed in the Hausdorff metric space. If x is a point at which f is continuous, then (x, f) is a point at which the map given by \( (x,f)\mapsto \omega (x,f)\) is continuous. The n-fold iterate \(f^{n}\) is an element of \(bB_{1}\) for all natural numbers n. The function f is non-chaotic in the senses of Devaney and Li–Yorke. PubDate: 2018-03-26 DOI: 10.1007/s40879-018-0234-0

Authors:Ádám Gyenge; András Némethi; Balázs Szendrői Abstract: We study the geometry and topology of Hilbert schemes of points on the orbifold surface , respectively the singular quotient surface , where is a finite subgroup of type A or D. We give a decomposition of the (equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a certain combinatorial set, the set of Young walls. The generating series of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D is computed in terms of an explicit formula involving a specialized character of the basic representation of the corresponding affine Lie algebra; we conjecture that the same result holds also in type E. Our results are consistent with known results in type A, and are new for type D. PubDate: 2018-03-26 DOI: 10.1007/s40879-018-0222-4

Authors:Juan B. Frías-Medina; Alexis G. Zamora Abstract: We discuss William L. Edge’s approach to Humbert’s curves as a canonical genus 5 curve that is a complete intersection of diagonal quadrics. Moreover, the contribution of Edge to the study of projective curves that are complete intersections of \(n-1\) quadrics is explained and some results, complementary to Edge’s exposition, are proved. PubDate: 2018-03-22 DOI: 10.1007/s40879-018-0232-2

Authors:Sergey Volosivets Abstract: We give necessary and sufficient conditions for a function odd in each variable to belong to Nikol’skii classes defined via mixed modulus of smoothness and mixed derivative (both have arbitrary integer orders). These conditions are given in terms of growth of partial sums of Fourier sine coefficients with power weights or rate of decreasing to zero of these coefficients. A similar problem for generalized “small” Lipschitz classes is also treated. PubDate: 2018-03-21 DOI: 10.1007/s40879-018-0226-0