On the automorphisms of surfaces of general type in positive characteristic, II

Edoardo Ballico

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On the automorphisms of surfaces of general type in positive characteristic

Edoardo Ballico (1993)

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

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Here we give an explicit polynomial bound (in term of $K_{X}^{2}$ and not depending on the prime $p$ ) for the order of the automorphism group of a minimal surface $X$ of general type defined over a field of characteristic $p > 0$ .

A boundedness theorem for morphisms between threefolds

Ekatarina Amerik, Marat Rovinsky, Antonius Van de Ven (1999)

Annales de l'institut Fourier

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The main result of this paper is as follows: let $X, Y$ be smooth projective threefolds (over a field of characteristic zero) such that $b_{2} (X) = b_{2} (Y) = 1$ . If $Y$ is not a projective space, then the degree of a morphism $f : X \to Y$ is bounded in terms of discrete invariants of $X$ and $Y$ . Moreover, suppose that $X$ and $Y$ are smooth projective $n$ -dimensional with cyclic Néron-Severi groups. If $c_{1} (Y) = 0$ , then the degree of $f$ is bounded iff $Y$ is not a flat variety. In particular, to prove our main theorem we show the non-existence of...

The Brauer–Manin obstruction for curves having split Jacobians

Samir Siksek (2004)

Journal de Théorie des Nombres de Bordeaux

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Let $X \to 𝒜$ be a non-constant morphism from a curve $X$ to an abelian variety $𝒜$ , all defined over a number field $k$ . Suppose that $X$ is a counterexample to the Hasse principle. We give sufficient conditions for the failure of the Hasse principle on $X$ to be accounted for by the Brauer–Manin obstruction. These sufficiency conditions are slightly stronger than assuming that $𝒜 (k)$ and $Ш (𝒜 / k)$ are finite.

Geometric linear normality for nodal curves on some projective surfaces

F. Flamini, C. Madonna (2001)

Bollettino dell'Unione Matematica Italiana

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In questo lavoro si generalizzano alcuni risultati di [3] riguardanti la proprietà di alcune curve nodali, su superficie non-singolari in $P^{r}$ , di essere «geometricamente linearmente normali» (concetto che estende la ben nota proprietà di essere linearmente normale). Precisamente, per una data curva $C$ , irriducibile e dotata di soli punti nodali come uniche singolarità, che giace su una superfice $S$ proiettiva, non-singolare e linearmente normale, si determina un limite superiore «sharp»...

Estimates of the number of rational mappings from a fixed variety to varieties of general type

Tanya Bandman, Gerd Dethloff (1997)

Annales de l'institut Fourier

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First we find effective bounds for the number of dominant rational maps $f : X \to Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type ${A \cdot K_{X}^{n}}^{{B \cdot K_{X}^{n}}^{2}}$ , where $n = \dim X$ , $K_{X}$ is the canonical bundle of $X$ and $A, B$ are some constants, depending only on $n$ . Then we show that for any variety $X$ there exist numbers $c (X)$ and $C (X)$ with the following properties: For any threefold $Y$ of general type the number of dominant rational maps $f : X \to Y$ is bounded above by $c (X)$ . ...

A group law on smooth real quartics having at least $3$ real branches

Johan Huisman (2002)

Journal de théorie des nombres de Bordeaux

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Let $C$ be a smooth real quartic curve in $ℙ^{2}$ . Suppose that $C$ has at least $3$ real branches $B_{1}, B_{2}, B_{3}$ . Let $B = B_{1} \times B_{2} \times B_{3}$ and let $O \in B$ . Let $τ_{O}$ be the map from $B$ into the neutral component Jac $(C) {(ℝ)}^{0}$ of the set of real points of the jacobian of $C$ , defined by letting $τ_{O} (P)$ be the divisor class of the divisor $\sum P_{i} - O_{i}$ . Then, $τ_{O}$ is a bijection. We show that this allows an explicit geometric description of the group law on Jac $(C) {(ℝ)}^{0}$ . It generalizes the classical geometric description of the group law on the neutral component of the set of real...

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