On the uniqueness of elliptic K3 surfaces with maximal singular fibre

Matthias Schütt; Andreas Schweizer

Displaying similar documents to “On the uniqueness of elliptic K3 surfaces with maximal singular fibre”

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.

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...

Translation surfaces of finite type in ${Sol}_{3}$

Bendehiba Senoussi, Hassan Al-Zoubi (2020)

Commentationes Mathematicae Universitatis Carolinae

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In the homogeneous space Sol $_{3}$ , a translation surface is parametrized by $r (s, t) = γ_{1} (s) * γ_{2} (t)$ , where $γ_{1}$ and $γ_{2}$ are curves contained in coordinate planes. In this article, we study translation invariant surfaces in ${Sol}_{3}$ , which has finite type immersion.

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.

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...

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...

Maximal non $λ$ -subrings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with unity. The notion of maximal non $λ$ -subrings is introduced and studied. A ring $R$ is called a maximal non $λ$ -subring of a ring $T$ if $R \subset T$ is not a $λ$ -extension, and for any ring $S$ such that $R \subset S \subseteq T$ , $S \subseteq T$ is a $λ$ -extension. We show that a maximal non $λ$ -subring $R$ of a field has at most two maximal ideals, and exactly two if $R$ is integrally closed in the given field. A determination of when the classical $D + M$ construction is a maximal non $λ$ -domain is given. A necessary condition...

Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

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We study the chemotaxis system with singular sensitivity and logistic-type source: $u_{t} = Δ u - χ \nabla \cdot (u \nabla v / v) + r u - μ u^{k}$ , $0 = Δ v - v + u$ under the non-flux boundary conditions in a smooth bounded domain $Ω \subset ℝ^{n}$ , $χ, r, μ > 0$ , $k > 1$ and $n \geq 1$ . It is shown with $k \in (1, 2)$ that the system possesses a global generalized solution for $n \geq 2$ which is bounded when $χ > 0$ is suitably small related to $r > 0$ and the initial datum is properly small, and a global bounded classical solution for $n = 1$ .

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

Thomas E. Cecil (1975)

Colloquium Mathematicae

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On annealed elliptic Green's function estimates

Daniel Marahrens, Felix Otto (2015)

Mathematica Bohemica

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We consider a random, uniformly elliptic coefficient field $a$ on the lattice $ℤ^{d}$ . The distribution $〈 \cdot 〉$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G (t, x, y)$ satisfy optimal annealed estimates which are $L^{2}$ and $L^{1}$ , respectively, in probability, i.e., they obtained bounds on $〈 | \nabla_{x} G (t, x, y) {|^{2} 〉}^{1 / 2}$ and $〈 | \nabla_{x} \nabla_{y} G (t, x, y) | 〉$ . In particular, the elliptic Green’s function $G (x, y)$ satisfies optimal annealed bounds. In their recent work,...

Solutions to a class of singular quasilinear elliptic equations

Lin Wei, Zuodong Yang (2010)

Annales Polonici Mathematici

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We study the existence of positive solutions to ⎧ ${d i v (| \nabla u |}^{p - 2} \nabla u) + q (x) u^{- γ} = 0$ on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is $ℝ^{N}$ or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).

Maximal non-pseudovaluation subrings of an integral domain

Rahul Kumar (2024)

Czechoslovak Mathematical Journal

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The notion of maximal non-pseudovaluation subring of an integral domain is introduced and studied. Let $R \subset S$ be an extension of domains. Then $R$ is called a maximal non-pseudovaluation subring of $S$ if $R$ is not a pseudovaluation subring of $S$ , and for any ring $T$ such that $R \subset T \subset S$ , $T$ is a pseudovaluation subring of $S$ . We show that if $S$ is not local, then there no such $T$ exists between $R$ and $S$ . We also characterize maximal non-pseudovaluation subrings of a local integral domain.

Partially elliptic differential equations having distributions as their right members

H. Marcinkowska

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ContentsIntroduction.............................................................................................................................31. Definitions, notations and some auxiliary lemmas...................................................42. The definition of the spaces $H_{p, q; Y} (Ω, ℬ)$ ..........................................................73. Some properties of the spaces $H_{p, q; Y} (Ω, ℬ)$ ...................................................104. Some examples of the spaces $H_{p, q; Y} (Ω, ℬ)$ ....................................................155....