Some surfaces with maximal Picard number

Arnaud Beauville

Displaying similar documents to “Some surfaces with maximal Picard number”

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...

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.

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...

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...

Brill–Noether loci for divisors on irregular varieties

Margarida Mendes Lopes, Rita Pardini, Pietro Pirola (2014)

Journal of the European Mathematical Society

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We take up the study of the Brill-Noether loci $W^{r} (L, X) : = {η \in {Pic}^{0} (X) | h^{0} (L \otimes η) \geq r + 1}$ , where $X$ is a smooth projective variety of dimension $> 1$ , $L \in Pic (X)$ , and $r \geq 0$ is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for $h^{0} (K_{D})$ , where $D$ is a divisor that moves linearly on a smooth projective variety $X$ of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension $> 2$ . In the $2$ -dimensional case...

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.

$J$ -invariant of linear algebraic groups

Viktor Petrov, Nikita Semenov, Kirill Zainoulline (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $G$ be a semisimple linear algebraic group of inner type over a field $F$ , and let $X$ be a projective homogeneous $G$ -variety such that $G$ splits over the function field of $X$ . We introduce the $J$ -invariant of $G$ which characterizes the motivic behavior of $X$ , and generalizes the $J$ -invariant defined by A. Vishik in the context of quadratic forms. We use this $J$ -invariant to provide motivic decompositions of all generically split projective homogeneous $G$ -varieties, e.g. Severi-Brauer varieties,...

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

Thomas E. Cecil (1975)

Colloquium Mathematicae

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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...

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...

On the Picard number of divisors in Fano manifolds

Cinzia Casagrande (2012)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$ . We consider the image $𝒩_{1} (D, X)$ of $𝒩_{1} (D)$ in $𝒩_{1} (X)$ under the natural push-forward of $1$ -cycles. We show that $ρ_{X} - ρ_{D} \leq codim 𝒩_{1} (D, X) \leq 8$ . Moreover if $codim 𝒩_{1} (D, X) \geq 3$ , then either $X ≅ S \times T$ where $S$ is a Del Pezzo surface, or $codim 𝒩_{1} (D, X) = 3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that $ρ_{X} - ρ_{T} = 4$ .

Non-supersingular hyperelliptic jacobians

Yuri G. Zarhin (2004)

Bulletin de la Société Mathématique de France

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Let $K$ be a field of odd characteristic $p$ , let $f (x)$ be an irreducible separable polynomial of degree $n \geq 5$ with big Galois group (the symmetric group or the alternating group). Let $C$ be the hyperelliptic curve $y^{2} = f (x)$ and $J (C)$ its jacobian. We prove that $J (C)$ does not have nontrivial endomorphisms over an algebraic closure of $K$ if either $n \geq 7$ or $p \neq 3$ .

The Brauer group and the Brauer–Manin set of products of varieties

Alexei N. Skorobogatov, Yuri G. Zahrin (2014)

Journal of the European Mathematical Society

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Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $Q$ , and let $\bar{X}$ and $\bar{Y}$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$ , respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br $(\bar{X}) \oplus$ Br( $\bar{Y})$ has finite index in the Galois invariant subgroup of Br $(\bar{X} \times \bar{Y})$ . This implies that the cokernel of the natural map Br $(X) \oplus$ Br $(Y) \to$ Br $(X \times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer–Manin set of the...

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.

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Claire Voisin (2013)

Annales scientifiques de l'École Normale Supérieure

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We prove that Bloch’s conjecture is true for surfaces with $p_{g} = 0$ obtained as $0$ -sets $X_{σ}$ of a section $σ$ of a very ample vector bundle on a variety $X$ with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as $0$ on holomorphic $2$ -forms of $X_{σ}$ , then it acts as $0$ on $0$ -cycles of degree $0$ of $X_{σ}$ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general $X_{σ}$ ...

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...

Varieties of minimal rational tangents of codimension 1

Jun-Muk Hwang (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a uniruled projective manifold and let $x$ be a general point. The main result of [2] says that if the $(- K_{X})$ -degrees (i.e., the degrees with respect to the anti-canonical bundle of $X$ ) of all rational curves through $x$ are at least $dim X + 1$ , then $X$ is a projective space. In this paper, we study the structure of $X$ when the $(- K_{X})$ -degrees of all rational curves through $x$ are at least $dim X$ . Our study uses the projective variety $𝒞_{x} \subset ℙ T_{x} (X)$ , called the VMRT at $x$ , defined as the union of tangent directions to the...

The local lifting problem for actions of finite groups on curves

Ted Chinburg, Robert Guralnick, David Harbater (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $k$ be an algebraically closed field of characteristic $p > 0$ . We study obstructions to lifting to characteristic $0$ the faithful continuous action $φ$ of a finite group $G$ on $k [[t]]$ . To each such $φ$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$ . We say that the KGB obstruction of $φ$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X / H$ and $Y / H$ have the same genus for all subgroups $H \subset G$ . We determine for which $G$ the KGB...

On the real $X$ -ranks of points of $ℙ^{n} (ℝ)$ with respect to a real variety $X \subset ℙ^{n}$

Edoardo Ballico (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $X \subset ℙ^{n}$ be an integral and non-degenerate $m$ -dimensional variety defined over $ℝ$ . For any $P \in ℙ^{n} (ℝ)$ the real $X$ -rank $r_{X, ℝ} (P)$ is the minimal cardinality of $S \subset X (ℝ)$ such that $P \in 〈 S 〉$ . Here we extend to the real case an upper bound for the $X$ -rank due to Landsberg and Teitler.

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...