Hyperbolic geometry and moduli of real cubic surfaces

Daniel Allcock; James A. Carlson; Domingo Toledo

Displaying similar documents to “Hyperbolic geometry and moduli of real cubic 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...

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...

Moduli of smoothness of functions and their derivatives

Z. Ditzian, S. Tikhonov (2007)

Studia Mathematica

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Relations between moduli of smoothness of the derivatives of a function and those of the function itself are investigated. The results are for $L_{p} (T)$ and $L_{p} [- 1, 1]$ for 0 < p < ∞ using the moduli of smoothness $ω^{r} {(f, t)}_{p}$ and $ω_{φ}^{r} {(f, t)}_{p}$ respectively.

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...

Bounds on the denominators in the canonical bundle formula

Enrica Floris (2013)

Annales de l’institut Fourier

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In this work we study the moduli part in the canonical bundle formula of an lc-trivial fibration whose general fibre is a rational curve. If $r$ is the Cartier index of the fibre, it was expected that $12 r$ would provide a bound on the denominators of the moduli part. Here we prove that such a bound cannot even be polynomial in $r$ , we provide a bound $N (r)$ and an example where the smallest integer that clears the denominators of the moduli part is $N (r) / r$ . Moreover we prove that even locally the denominators...

Fundamental solution $E_{n}$ of the operator $\partial^{2} / \partial t^{2} - Δ_{n}$ for n>3

Zofia Szmydt, Bogdan Ziemian (1979)

Annales Polonici Mathematici

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Multiple end solutions to the Allen-Cahn equation in $ℝ^{2}$

Michał Kowalczyk, Yong Liu, Frank Pacard (2013-2014)

Séminaire Laurent Schwartz — EDP et applications

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An entire solution of the Allen-Cahn equation $Δ u = f (u)$ , where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$ , e.g. $f (u) = u (u^{2} - 1)$ , is called a $2 k$ end solution if its nodal set is asymptotic to $2 k$ half lines, and if along each of these half lines the function $u$ looks (up to a multiplication by $- 1$ ) like the one dimensional, odd, heteroclinic solution $H$ , of $H^{''} = f (H)$ . In this paper we present some recent advances in the theory of the multiple end solutions. We begin with the description of the moduli space...

Hyperideal polyhedra in hyperbolic 3-space

Xiliang Bao, Francis Bonahon (2002)

Bulletin de la Société Mathématique de France

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A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic $3$ -space $ℍ^{3}$ which, in the projective model for $ℍ^{3} \subset {ℝℙ}^{3}$ , is just the intersection of $ℍ^{3}$ with a projective polyhedron whose vertices are all outside $ℍ^{3}$ and whose edges all meet $ℍ^{3}$ . We classify hyperideal polyhedra, up to isometries of $ℍ^{3}$ , in terms of their combinatorial type and of their dihedral angles.

Sharp bounds for the intersection of nodal lines with certain curves

Junehyuk Jung (2014)

Journal of the European Mathematical Society

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Let $Y$ be a hyperbolic surface and let $φ$ be a Laplacian eigenfunction having eigenvalue $- 1 / 4 - τ^{2}$ with $τ > 0$ . Let $N (φ)$ be the set of nodal lines of $φ$ . For a fixed analytic curve $γ$ of finite length, we study the number of intersections between $N (φ)$ and $γ$ in terms of $τ$ . When $Y$ is compact and $γ$ a geodesic circle, or when $Y$ has finite volume and $γ$ is a closed horocycle, we prove that $γ$ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between $N (φ)$ and $γ$ is $O (τ)$ . This bound is...

A multiparameter variant of the Salem-Zygmund central limit theorem on lacunary trigonometric series

Mordechay B. Levin (2013)

Colloquium Mathematicae

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We prove the central limit theorem for the multisequence $\sum_{1 \leq n ₁ \leq N ₁} \dots \sum_{1 \leq n_{d} \leq N_{d}} a_{n ₁, . . ., n_{d}} c o s (⟨ 2 π m, A ₁^{n ₁} . . . A_{d}^{n_{d}} x ⟩)$ where $m \in ℤ^{s}$ , $a_{n ₁, . . ., n_{d}}$ are reals, $A ₁, . . ., A_{d}$ are partially hyperbolic commuting s × s matrices, and x is a uniformly distributed random variable in ${[0, 1]}^{s}$ . The main tool is the S-unit theorem.

Soluble Groups with Many Černikov Quotients

Silvana Franciosi, Francesco de Giovanni (1985)

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

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Si studiano i gruppi risolubili non di Černikov a quozienti propri di Černikov. Nel caso periodico tali gruppi sono tutti e soli i prodotti semidiretti $H\ltimes N$ con $N$ $p$ -gruppo abeliano elementare infinito e $H$ gruppo irriducibile di automorfismi di $N$ che sia infinito e di Černikov. Nel caso non periodico invece si riconduce tale studio a quello dei moduli a quozienti propri artiniani su un gruppo risolubile finito, e si fornisce una caratterizzazione di tali moduli.

The existence of Carathéodory solutions of hyperbolic functional differential equations

Adrian Karpowicz (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We consider the following Darboux problem for the functional differential equation $\partial ² u / \partial x \partial y (x, y) = f (x, y, u_{(x, y)}, \partial u / \partial x (x, y), \partial u / \partial y (x, y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b] $0, a] \times (0, b],$ where the function $u_{(x, y)} : [- a ₀, 0] \times [- b ₀, 0] \to ℝ^{k}$ is defined by $u_{(x, y)} (s, t) = u (s + x, t + y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.

Fields of moduli of three-point $G$ -covers with cyclic $p$ -Sylow, II

Andrew Obus (2013)

Journal de Théorie des Nombres de Bordeaux

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We continue the examination of the stable reduction and fields of moduli of $G$ -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic $(0, p)$ , where $G$ has a $p$ -Sylow subgroup $P$ of order $p^{n}$ . Suppose further that the normalizer of $P$ acts on $P$ via an involution. Under mild assumptions, if $f : Y \to ℙ^{1}$ is a three-point $G$ -Galois cover defined over $\bar{ℚ}$ , then the $n$ th higher ramification groups above $p$ for the upper numbering of the (Galois closure of...

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.

Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps

Martine Babillot, Marc Peigné (2006)

Bulletin de la Société Mathématique de France

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We consider a large class of non compact hyperbolic manifolds $M = ℍ^{n} / Γ$ with cusps and we prove that the winding process $(Y_{t})$ generated by a closed $1$ -form supported on a neighborhood of a cusp $𝒞$ , satisfies a limit theorem, with an asymptotic stable law and a renormalising factor depending only on the rank of the cusp $𝒞$ and the Poincaré exponent $δ$ of $Γ$ . No assumption on the value of $δ$ is required and this theorem generalises previous results due to Y. Guivarc’h, Y. Le Jan, J. Franchi and N. Enriquez. ...

Hyperbolic spaces in Teichmüller spaces

Christopher J. Leininger, Saul Schleimer (2014)

Journal of the European Mathematical Society

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We prove, for any $n$ , that there is a closed connected orientable surface $S$ so that the hyperbolic space $ℍ^{n}$ almost-isometrically embeds into the Teichmüller space of $S$ , with quasi-convex image lying in the thick part. As a consequence, $ℍ^{n}$ quasi-isometrically embeds in the curve complex of $S$ .

Essential dimension of moduli of curves and other algebraic stacks

Patrick Brosnan, Zinovy Reichstein, Angelo Vistoli (2011)

Journal of the European Mathematical Society

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In this paper we consider questions of the following type. Let $k$ be a base field and $K / k$ be a field extension. Given a geometric object $X$ over a field $K$ (e.g. a smooth curve of genus $g$ ), what is the least transcendence degree of a field of definition of $X$ over the base field $k$ ? In other words, how many independent parameters are needed to define $X$ ? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete...

The Kodaira dimension of the moduli space of Prym varieties

Gavril Farkas, Katharina Ludwig (2010)

Journal of the European Mathematical Society

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We study the enumerative geometry of the moduli space $ℛ_{g}$ of Prym varieties of dimension $g - 1$ . Our main result is that the compactication of $ℛ_{g}$ is of general type as soon as $g > 13$ and $g$ is different from 15. We achieve this by computing the class of two types of cycles on $ℛ_{g}$ : one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym–Green conjecture on syzygies of Prym-canonical...

An alternative description of the Drinfeld $p$ -adic half-plane

Stephen Kudla, Michael Rapoport (2014)

Annales de l’institut Fourier

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We show that the Deligne formal model of the Drinfeld $p$ -adic half-plane relative to a local field $F$ represents a moduli problem of polarized $O_{F}$ -modules with an action of the ring of integers in a quadratic extension $E$ of $F$ . The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of ${SL}_{2} (F)$ and $SU (C) (F)$ for a two-dimensional split hermitian space $C$ for $E / F$ .

On the structural theory of ${II}_{1}$ factors of negatively curved groups

Ionut Chifan, Thomas Sinclair (2013)

Annales scientifiques de l'École Normale Supérieure

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Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor $L Γ$ is solid. Developing a new approach that combines some methods of Peterson [29], Ozawa and Popa [27, 28], and Ozawa [25], we strengthen this result by showing that $L Γ$ is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in $Sp (n, 1)$ , $n \geq 2$ , are virtually $W^{*}$ -superrigid.

Nonexistence results for the Cauchy problem of some systems of hyperbolic equations

Mokhtar Kirane, Salim Messaoudi (2002)

Annales Polonici Mathematici

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We consider the systems of hyperbolic equations ⎧ $u ₜ ₜ = Δ (a (t, x) u) + Δ (b (t, x) v) + {h (t, x) | v |}^{p}$ , t > 0, $x \in ℝ^{N}$ , (S1) ⎨ ⎩ $v ₜ ₜ = Δ (c (t, x) v) + {k (t, x) | u |}^{q}$ , t > 0, $x \in ℝ^{N}$ ⎧ $u ₜ ₜ = Δ (a (t, x) u) + {h (t, x) | v |}^{p}$ , t > 0, $x \in ℝ^{N}$ , (S2) ⎨ ⎩ $v ₜ ₜ = Δ (c (t, x) v) + {l (t, x) | v |}^{m} + k (t, x) {| u |}^{q}$ , t > 0, $x \in ℝ^{N}$ , (S3) ⎧ $u ₜ ₜ = Δ (a (t, x) u) + Δ (b (t, x) v) + {h (t, x) | u |}^{p}$ , t > 0, $x \in ℝ^{N}$ , ⎨ ⎩ $v ₜ ₜ = Δ (c (t, x) v) + {k (t, x) | v |}^{q}$ , t > 0, $x \in ℝ^{N}$ , in $(0, \infty) \times ℝ^{N}$ with u(0,x) = u₀(x), v(0,x) = v₀(x), uₜ(0,x) = u₁(x), vₜ(0,x) = v₁(x). We show that, in each case, there exists a bound B on N such that for 1 ≤ N ≤ B solutions to the systems blow up in finite time.

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...