Birational Finite Extensions of Mappings from a Smooth Variety

Marek Karaś

Displaying similar documents to “Birational Finite Extensions of Mappings from a Smooth Variety”

$J$ -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 tangent cones to Schubert varieties in type $E$

Mikhail V. Ignatyev, Aleksandr A. Shevchenko (2020)

Communications in Mathematics

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We consider tangent cones to Schubert subvarieties of the flag variety $G / B$ , where $B$ is a Borel subgroup of a reductive complex algebraic group $G$ of type $E_{6}$ , $E_{7}$ or $E_{8}$ . We prove that if $w_{1}$ and $w_{2}$ form a good pair of involutions in the Weyl group $W$ of $G$ then the tangent cones $C_{w_{1}}$ and $C_{w_{2}}$ to the corresponding Schubert subvarieties of $G / B$ do not coincide as subschemes of the tangent space to $G / B$ at the neutral point.

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...

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 $p \neq 3$ .

Equidistribution towards the Green current

Vincent Guedj (2003)

Bulletin de la Société Mathématique de France

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Let $f : ℙ^{k} \to ℙ^{k}$ be a dominating rational mapping of first algebraic degree $λ \geq 2$ . If $S$ is a positive closed current of bidegree $(1, 1)$ on $ℙ^{k}$ with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks $λ^{- n} {(f^{n})}^{*} S$ converge to the Green current $T_{f}$ . For some families of mappings, we get finer convergence results which allow us to characterize all $f^{*}$ -invariant currents.

Elements of large order on varieties over prime finite fields

Mei-Chu Chang, Bryce Kerr, Igor E. Shparlinski, Umberto Zannier (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $𝒱$ be a fixed algebraic variety defined by $m$ polynomials in $n$ variables with integer coefficients. We show that there exists a constant $C (𝒱)$ such that for almost all primes $p$ for all but at most $C (𝒱)$ points on the reduction of $𝒱$ modulo $p$ at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

On varieties of Hilbert type

Lior Bary-Soroker, Arno Fehm, Sebastian Petersen (2014)

Annales de l’institut Fourier

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A variety $X$ over a field $K$ is of Hilbert type if $X (K)$ is not thin. We prove that if $f : X \to S$ is a dominant morphism of $K$ -varieties and both $S$ and all fibers $f^{- 1} (s)$ , $s \in S (K)$ , are of Hilbert type, then so is $X$ . We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.

Purity of level $m$ stratifications

Marc-Hubert Nicole, Adrian Vasiu, Torsten Wedhorn (2010)

Annales scientifiques de l'École Normale Supérieure

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Let $k$ be a field of characteristic $p > 0$ . Let $D_{m}$ be a ${BT}_{m}$ over $k$ (i.e., an $m$ -truncated Barsotti–Tate group over $k$ ). Let $S$ be a $k$ -scheme and let $X$ be a ${BT}_{m}$ over $S$ . Let $S_{D_{m}} (X)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_{m}$ . If $p \geq 5$ , we show that $S_{D_{m}} (X)$ is pure in $S$ , i.e. the immersion $S_{D_{m}} (X) ↪ S$ is affine. For $p \in {2, 3}$ , we prove purity if $D_{m}$ satisfies a certain technical property depending only on its $p$ -torsion $D_{m} [p]$ . For $p \geq 5$ , we apply the developed techniques to show that...

On a result by Clunie and Sheil-Small

Dariusz Partyka, Ken-ichi Sakan (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping $F$ in the unit disk $𝔻$ , if $F (𝔻)$ is a convex domain, then the inequality $| G (z_{2}) - G (z_{1}) | < | H (z_{2}) - H (z_{1}) |$ holds for all distinct points $z_{1}, z_{2} \in 𝔻$ . Here $H$ and $G$ are holomorphic mappings in $𝔻$ determined by $F = H + \bar{G}$ , up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain $Ω$ in $ℂ$ and improve it provided $F$ is additionally a quasiconformal mapping...

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 $ρ_{X} - ρ_{T} = 4$ .

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock, Alexander B. Goncharov (2009)

Annales scientifiques de l'École Normale Supérieure

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A cluster ensemble is a pair $(𝒳, 𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $Γ$ . The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p : 𝒜 \to 𝒳$ . The space $𝒜$ is equipped with a closed $2$ -form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

Degrees of compatible $L$ -subsets and compatible mappings

Fu Gui Shi, Yan Sun (2024)

Kybernetika

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Based on a completely distributive lattice $L$ , degrees of compatible $L$ -subsets and compatible mappings are introduced in an $L$ -approximation space and their characterizations are given by four kinds of cut sets of $L$ -subsets and $L$ -equivalences, respectively. Besides, some characterizations of compatible mappings and compatible degrees of mappings are given by compatible $L$ -subsets and compatible degrees of $L$ -subsets. Finally, the notion of complete $L$ -sublattices is introduced and it is shown...

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...

Construction of Mendelsohn designs by using quasigroups of $(2, q)$ -varieties

Lidija Goračinova-Ilieva, Smile Markovski (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $q$ be a positive integer. An algebra is said to have the property $(2, q)$ if all of its subalgebras generated by two distinct elements have exactly $q$ elements. A variety $𝒱$ of algebras is a variety with the property $(2, q)$ if every member of $𝒱$ has the property $(2, q)$ . Such varieties exist only in the case of $q$ prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property $(2, q)$ , $2 < q$ , blocks of Steiner system of type $(2, q)$ are obtained. The stated correspondence...

Obstruction sets and extensions of groups

Francesca Balestrieri (2016)

Acta Arithmetica

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Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X {(_{k})}^{é t, B r} \subset X {(_{k})}^{B r ₁}$ . In the first part, we apply ideas from the proof of $X {(_{k})}^{é t, B r} = X {(_{k})}^{_{k}}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $\subset \subset_{k}$ are such that $\subset E x t (,_{k})$ , then $X {(_{k})}^{} = X {(_{k})}^{}$ . This allows us to conclude, among other things, that $X {(_{k})}^{é t, B r} = X {(_{k})}^{_{k}}$ and $X {(_{k})}^{S o l, B r ₁} = X {(_{k})}^{S o l_{k}}$ .

On the birational gonalities of smooth curves

E. Ballico (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $C$ be a smooth curve of genus $g$ . For each positive integer $r$ the birational $r$ -gonality $s_{r} (C)$ of $C$ is the minimal integer $t$ such that there is $L \in {Pic}^{t} (C)$ with $h^{0} (C, L) = r + 1$ . Fix an integer $r \geq 3$ . In this paper we prove the existence of an integer $g_{r}$ such that for every integer $g \geq g_{r}$ there is a smooth curve $C$ of genus $g$ with $s_{r + 1} (C) / (r + 1) > s_{r} (C) / r$ , i.e. in the sequence of all birational gonalities of $C$ at least one of the slope inequalities fails.

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 the compositum of all degree $d$ extensions of a number field

Itamar Gal, Robert Grizzard (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the compositum $k^{[d]}$ of all degree $d$ extensions of a number field $k$ in a fixed algebraic closure. We show $k^{[d]}$ contains all subextensions of degree less than $d$ if and only if $d \leq 4$ . We prove that for $d > 2$ there is no bound $c = c (d)$ on the degree of elements required to generate finite subextensions of $k^{[d]} / k$ . Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of $d$ , but that one can take $c = d$ when $d$ is prime. This question was inspired by work of...

Selectors of discrete coarse spaces

Igor Protasov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Given a coarse space $(X, ℰ)$ with the bornology $ℬ$ of bounded subsets, we extend the coarse structure $ℰ$ from $X \times X$ to the natural coarse structure on $(ℬ ∖ {\emptyset}) \times (ℬ ∖ {\emptyset})$ and say that a macro-uniform mapping $f : (ℬ ∖ {\emptyset}) \to X$ (or $f : {[X]}^{2} \to X$ ) is a selector (or 2-selector) of $(X, ℰ)$ if $f (A) \in A$ for each $A \in ℬ ∖ {\emptyset}$ ( $A \in {[X]}^{2}$ , respectively). We prove that a discrete coarse space $(X, ℰ)$ admits a selector if and only if $(X, ℰ)$ admits a 2-selector if and only if there exists a linear order “ $\leq$ " on $X$ such that the family of intervals ${[a, b] : a, b \in X, a \leq b}$ is a base for the bornology $ℬ$ .