A proof of the crossing number of $K_{3,n}$ in a surface

Pak Tung Ho

Displaying similar documents to “A proof of the crossing number of $K_{3, n}$ in a surface”

Classification of rings with toroidal Jacobson graph

Krishnan Selvakumar, Manoharan Subajini (2016)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with nonzero identity and $J (R)$ the Jacobson radical of $R$ . The Jacobson graph of $R$ , denoted by $𝔍_{R}$ , is defined as the graph with vertex set $R ∖ J (R)$ such that two distinct vertices $x$ and $y$ are adjacent if and only if $1 - x y$ is not a unit of $R$ . The genus of a simple graph $G$ is the smallest nonnegative integer $n$ such that $G$ can be embedded into an orientable surface $S_{n}$ . In this paper, we investigate the genus number of the compact Riemann surface in which $𝔍_{R}$ can be embedded and...

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.

Symmetry of minimizers with a level surface parallel to the boundary

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi (2015)

Journal of the European Mathematical Society

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We consider the functional $ℐ_{Ω} (v) = \int_{Ω} [f (| D v |) - v] d x,$ where $Ω$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$ , Crasta [Cr1] has shown that if $ℐ_{Ω}$ admits a minimizer in $W_{0}^{1, 1} (Ω)$ depending only on the distance from the boundary of $Ω$ , then $Ω$ must be a ball. With some restrictions on $f$ , we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss...

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...

Flexibility of surface groups in classical simple Lie groups

Inkang Kim, Pierre Pansu (2015)

Journal of the European Mathematical Society

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We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is $S U (p, q)$ (resp. $S O^{*} (2 n)$ , $n$ odd) and the surface group is maximal in some $S (U (p, p) \times U (q - p)) \subset S U (p, q)$ (resp. $S O^{*} (2 n - 2) \times S O (2) \subset S O^{*} (2 n)$ ). This is a converse, for classical groups, to a rigidity result of S. Bradlow, O. García-Prada and P. Gothen.

Scaling limit and cube-root fluctuations in SOS surfaces above a wall

Pietro Caputo, Eyal Lubetzky, Fabio Martinelli, Allan Sly, Fabio Lucio Toninelli (2016)

Journal of the European Mathematical Society

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Consider the classical $(2 + 1)$ -dimensional Solid-On-Solid model above a hard wall on an $L \times L$ box of $ℤ^{2}$ . The model describes a crystal surface by assigning a non-negative integer height $η_{x}$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $η$ is proportional to $\exp (- β ℋ (η))$ , where $β$ is the inverse-temperature and $ℋ (η)$ sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough...

A class of non-rational surface singularities with bijective Nash map

Camille Plénat, Patrick Popescu-Pampu (2006)

Bulletin de la Société Mathématique de France

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Let $(𝒮, 0)$ be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor $E$ and its irreducible components $E_{i}$ , $i \in I$ . The Nash map associates to each irreducible component $C_{k}$ of the space of arcs through $0$ on $𝒮$ the unique component of $E$ cut by the strict transform of the generic arc in $C_{k}$ . Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if $E \cdot E_{i} < 0$ for any $i \in I$ . ...

On ramified covers of the projective plane II: Generalizing Segre’s theory

Michael Friedman, Rebecca Lehman, Maxim Leyenson, Mina Teicher (2012)

Journal of the European Mathematical Society

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The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in $ℙ^{3}$ . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, $B$ and $E$ , we give a necessary and sufficient condition for $B$ to be the branch curve of a surface $X$ in $ℙ^{N}$ and $E$ to be the image of the double curve of a $ℙ^{3}$ -model of $X$ . In the classical Segre theory, a...