On a stratification of the moduli of K3 surfaces

Gerard van der Geer; T. Katsura

Displaying similar documents to “On a stratification of the moduli of K3 surfaces”

On the completeness of flat surfaces in $S^{3}$

Thomas E. Cecil (1975)

Colloquium Mathematicae

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Some surfaces with maximal Picard number

Arnaud Beauville (2014)

Journal de l’École polytechnique — Mathématiques

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For a smooth complex projective variety, the rank $ρ$ of the Néron-Severi group is bounded by the Hodge number $h^{1, 1}$ . Varieties with $ρ = h^{1, 1}$ have interesting properties, but are rather sparse, particularly in dimension $2$ . We discuss in this note a number of examples, in particular those constructed from curves with special Jacobians.

The KSBA compactification for the moduli space of degree two $K 3$ pairs

Radu Laza (2016)

Journal of the European Mathematical Society

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Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs $(X, H)$ consisting of a degree two $K 3$ surface $X$ and an ample divisor $H$ . Specifically, we construct and describe explicitly a geometric compactification ${\bar{P}}_{2}$ for the moduli of degree two $K 3$ pairs. This compactification...

Counting lines on surfaces

Samuel Boissière, Alessandra Sarti (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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This paper deals with surfaces with many lines. It is well-known that a cubic contains $27$ of them and that the maximal number for a quartic is $64$ . In higher degree the question remains open. Here we study classical and new constructions of surfaces with high number of lines. We obtain a symmetric octic with $352$ lines, and give examples of surfaces of degree $d$ containing a sequence of $d (d - 2) + 4$ skew lines.

Ramification of the Gauss map of complete minimal surfaces in $ℝ^{m}$ on annular ends

Gerd Dethloff, Pham Hoang Ha, Pham Duc Thoan (2016)

Colloquium Mathematicae

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We study the ramification of the Gauss map of complete minimal surfaces in $ℝ^{m}$ on annular ends. This is a continuation of previous work of Dethloff-Ha (2014), which we extend here to targets of higher dimension.

Numerical Campedelli surfaces with fundamental group of order 9

Margarida Mendes Lopes, Rita Pardini (2008)

Journal of the European Mathematical Society

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We give explicit constructions of all the numerical Campedelli surfaces, i.e. the minimal surfaces of general type with $K^{2} = 2$ and $p_{g} = 0$ , whose fundamental group has order 9. There are three families, one with $π_{1}^{alg} = ℤ_{9}$ and two with $π_{1}^{alg} = ℤ_{3}^{2}$ . We also determine the base locus of the bicanonical system of these surfaces. It turns out that for the surfaces with $π_{1}^{alg} = ℤ_{9}$ and for one of the families of surfaces with $π_{1}^{alg} = ℤ_{3}^{2}$ the base locus consists of two points. To our knowlegde, these are the only known examples of surfaces...

On the uniqueness of elliptic K3 surfaces with maximal singular fibre

Matthias Schütt, Andreas Schweizer (2013)

Annales de l’institut Fourier

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We explicitly determine the elliptic $K 3$ surfaces with section and maximal singular fibre. If the characteristic of the ground field is different from $2$ , for each of the two possible maximal fibre types, $I_{19}$ and $I_{14}^{*}$ , the surface is unique. In characteristic $2$ the maximal fibre types are $I_{18}$ and $I_{13}^{*}$ , and there exist two (resp. one) one-parameter families of such surfaces.

Even sets of nodes on sextic surfaces

Fabrizio Catanese, Fabio Tonoli (2007)

Journal of the European Mathematical Society

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We determine the possible even sets of nodes on sextic surfaces in $ℙ^{3}$ , showing in particular that their cardinalities are exactly the numbers in the set ${24, 32, 40, 56}$ . We also show that all the possible cases admit an explicit description. The methods that we use are an interplay of coding theory and projective geometry on one hand, and of homological and computer algebra on the other. We give a detailed geometric construction for the new case of an even set of 56 nodes, but the ultimate verification...

Ramification of the Gauss map of complete minimal surfaces in $ℝ^{3}$ and $ℝ^{4}$ on annular ends

Gerd Dethloff, Pham Hoang Ha (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this article, we study the ramification of the Gauss map of complete minimal surfaces in $ℝ^{3}$ and $ℝ^{4}$ on annular ends. We obtain results which are similar to the ones obtained by Fujimoto ([4], [5]) and Ru ([13], [14]) for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give...

A Note on Surfaces in $\mathbb{H}^{2}\times\mathbb{R}$

Stefano Montaldo, Irene I. Onnis (2007)

Bollettino dell'Unione Matematica Italiana

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In this article we consider surfaces in the product space $\mathbb{H}^{2}\times\mathbb{R}$ of the hyperbolic plane $\mathbb{H}^{2}$ with the real line. The main results are: a description of some geometric properties of minimal graphs; new examples of complete minimal graphs; the local classification of totally umbilical surfaces.

Dehn twists on nonorientable surfaces

Michał Stukow (2006)

Fundamenta Mathematicae

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Let $t_{a}$ be the Dehn twist about a circle a on an orientable surface. It is well known that for each circle b and an integer n, $I (t ⁿ_{a} (b), b) = | n | I (a, b) ²$ , where I(·,·) is the geometric intersection number. We prove a similar formula for circles on nonorientable surfaces. As a corollary we prove some algebraic properties of twists on nonorientable surfaces. We also prove that if ℳ(N) is the mapping class group of a nonorientable surface N, then up to a finite number of exceptions, the centraliser of the subgroup...

Systole growth for finite area hyperbolic surfaces

Florent Balacheff, Eran Makover, Hugo Parlier (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this note, we observe that the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g, n)$ is greater than a function that grows logarithmically in terms of the ratio $g / n$ .

A Characterization of $\omega$ -Limit Sets for Continuous Flows on Surfaces

Víctor Jiménez López, Gabriel Soler López (2006)

Bollettino dell'Unione Matematica Italiana

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An explicit topological description of ω-limit sets of continuous flows on compact surfaces without boundary is given. Some of the results can be extended to manifolds of larger dimensions.

Errata-Corrige : “Canonical surfaces with $p_{g} = p_{a} = 5$ and $K^{2} = 10$ ”

Ciro Ciliberto (1983)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Euler characteristics of moduli spaces of curves

Gilberto Bini, John Harer (2011)

Journal of the European Mathematical Society

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Let $M_{g}^{n}$ be the moduli space of $n$ -pointed Riemann surfaces of genus $g$ . Denote by $\bar{M_{g}^{n}}$ the Deligne-Mumford compactification of $M_{g}^{n}$ . In the present paper, we calculate the orbifold and the ordinary Euler characteristic of $\bar{M_{g}^{n}}$ for any $g$ and $n$ such that $n > 2 - 2 g$ .

Explicit Teichmüller curves with complementary series

Carlos Matheus, Gabriela Weitze-Schmithüsen (2013)

Bulletin de la Société Mathématique de France

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We construct an explicit family of arithmetic Teichmüller curves $𝒞_{2 k}$ , $k \in ℕ$ , supporting $SL (2, ℝ)$ -invariant probabilities $μ_{2 k}$ such that the associated $SL (2, ℝ)$ -representation on $L^{2} (𝒞_{2 k}, μ_{2 k})$ has complementary series for every $k \geq 3$ . Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichmüller geodesic flow restricted to these explicit arithmetic Teichmüller curves $𝒞_{2 k}$ has arbitrarily slow rate of exponential mixing.

Zariski surfaces

Piotr Blass

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CONTENTSAcknowledgements...................................................................................................5Introduction..............................................................................................................6Notations..................................................................................................................8Chapter I. Zariski surfaces: definition and general properties................................10Chapter II. The theory...

Dimension vs. genus: A surface realization of the little k-cubes and an $E_{\infty}$ operad

Ralph M. Kaufmann (2009)

Banach Center Publications

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We define a new $E_{\infty}$ operad based on surfaces with foliations which contains $E_{k}$ suboperads. We construct CW models for these operads and provide applications of these models by giving actions on Hochschild complexes (thus making contact with string topology), by giving explicit cell representatives for the Dyer-Lashof-Cohen operations for the 2-cubes and by constructing new Ω spectra. The underlying novel principle is that we can trade genus in the surface representation vs. the dimension...

Hyperbolic geometry and moduli of real cubic surfaces

Daniel Allcock, James A. Carlson, Domingo Toledo (2010)

Annales scientifiques de l'École Normale Supérieure

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Let $ℳ_{0}^{ℝ}$ be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space $H^{4}$ and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in $PO (4, 1)$ . We also derive several new and several old results on the topology...

On Zariski's theorem in positive characteristic

Ilya Tyomkin (2013)

Journal of the European Mathematical Society

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In the current paper we show that the dimension of a family $V$ of irreducible reduced curves in a given ample linear system on a toric surface $S$ over an algebraically closed field is bounded from above by $- K_{S} . C + p_{g} (C) - 1$ , where $C$ denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality $𝚍𝚒𝚖 (V) = - K_{S} . C + p_{g} (C) - 1$ does not imply the nodality of $C$ even if $C$ belongs...

Natural pseudodistances between closed surfaces

Pietro Donatini, Patrizio Frosini (2007)

Journal of the European Mathematical Society

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Let us consider two closed surfaces $ℳ$ , $𝒩$ of class $C^{1}$ and two functions $ϕ : ℳ \to ℝ$ , $ψ : 𝒩 \to ℝ$ of class $C^{1}$ , called measuring functions. The natural pseudodistance $d$ between the pairs $(ℳ,)$ , $(𝒩, ψ)$ is defined as the infimum of $Θ (f) : = {max}_{P \in ℳ} | ϕ (P) - ψ (f (P)) |$ as $f$ varies in the set of all homeomorphisms from $ℳ$ onto $𝒩$ . In this paper we prove that the natural pseudodistance equals either $| c_{1} - c_{2} |$ , $\frac{1}{2} | c_{1} - c_{2} |$ , or $\frac{1}{3} | c_{1} - c_{2} |$ , where $c_{1}$ and $c_{2}$ are two suitable critical values of the measuring functions. This shows that a previous relation between the natural pseudodistance and...