Errata-Corrige : “Canonical surfaces with $p_g = p_a = 5$ and $K^2 = 10$”

Ciro Ciliberto

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

Thomas E. Cecil (1975)

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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.

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.

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.

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.

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...

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 $g / n$ .

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...

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...

Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Pavel Etingof, Victor Ginzburg (2010)

Journal of the European Mathematical Society

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The hypersurface in $ℂ^{3}$ with an isolated quasi-homogeneous elliptic singularity of type ${\tilde{E}}_{r}, r = 6, 7, 8$ , has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type $E_{r}$ provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we first deform the polynomial algebra $ℂ [x_{1}, x_{2}, x_{3}]$ to a noncommutative algebra with generators $x_{1}, x_{2}, x_{3}$ and the following 3 relations...

On surfaces with p $_{𝑔}$ = q = 1 and non-ruled bicanonical involution

Carlos Rito (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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This paper classifies surfaces $S$ of general type with $p_{g} = q = 1$ having an involution $i$ such that $S / i$ has non-negative Kodaira dimension and that the bicanonical map of $S$ factors through the double cover induced by $i .$ It is shown that $S / i$ is regular and either: a) the Albanese fibration of $S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S / i$ is birational to a $K 3$ surface. For case a) a list of possibilities and examples are given. An example for case b) with $K^{2} = 6$ is also constructed.

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.

Cylindrical real hyper surfaces in $\mathbf{C}^{n}$

Claudio Rea (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si stabiliscono due condizioni sufficienti per un germe di ipersuperficie reale di classe $c^{\infty}$ in $\mathbf{C}^{n}$ affinchè esistano coordinate olomorfe rispetto alle quali l'ipersuperficie risulti essere il luogo di zeri di una funzione di $k<n$ variabili e $k$ sia minimale rispetto a questa proprietà. In altre parole si vuole che l'ipersuperficie, a meno di una trasformazione bi-olomorfa, sia l’unione di sottovarietà lineari complesse, parallele di dimensione $n — k$ .

Effective bounds for Faltings’s delta function

Jay Jorgenson, Jürg Kramer (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces $X$ , nowadays called Faltings’s delta function and here denoted by $δ_{Fal} (X)$ . For a given compact Riemann surface $X$ of genus $g_{X} = g$ , the invariant $δ_{Fal} (X)$ is roughly given as minus the logarithm of the distance with respect to the Weil-Petersson metric of the point in the moduli space $ℳ_{g}$ of genus $g$ curves determined by $X$ to its boundary $\partial ℳ_{g}$ . In this paper we begin by revisiting a formula derived...

Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

Fabrizio Catanese, Frédéric Mangolte (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $W \to X$ be a real smooth projective 3-fold fibred by rational curves such that $W (ℝ)$ is orientable. J. Kollár proved that a connected component $N$ of $W (ℝ)$ is essentially either Seifert fibred or a connected sum of lens spaces. Answering three questions of Kollár, we give sharp estimates on the number and the multiplicities of the Seifert fibres (resp. the number and the torsions of the lens spaces) when $X$ is a geometrically rational surface. When $N$ is Seifert fibred over a base orbifold $F$ , our...

Displaying similar documents to “Errata-Corrige : “Canonical surfaces with p g = p a = 5 and K 2 = 10 ””

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