Syzygies and logarithmic vector fields along plane curves

Alexandru Dimca; Edoardo Sernesi

Displaying similar documents to “Syzygies and logarithmic vector fields along plane curves”

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...

Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

Nazar Arakelian, Herivelto Borges (2015)

Acta Arithmetica

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For each integer s ≥ 1, we present a family of curves that are $_{q}$ -Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be $_{q}$ -Frobenius nonclassical with respect to the linear system of conics. In the $_{q}$ -Frobenius nonclassical cases, we determine the exact number of $_{q}$ -rational points. In the remaining cases, an upper bound for the number of $_{q}$ -rational points will follow from Stöhr-Voloch...

Semistability of Frobenius direct images over curves

Vikram B. Mehta, Christian Pauly (2007)

Bulletin de la Société Mathématique de France

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Let $X$ be a smooth projective curve of genus $g \geq 2$ defined over an algebraically closed field $k$ of characteristic $p > 0$ . Given a semistable vector bundle $E$ over $X$ , we show that its direct image $F_{*} E$ under the Frobenius map $F$ of $X$ is again semistable. We deduce a numerical characterization of the stable rank- $p$ vector bundles $F_{*} L$ , where $L$ is a line bundle over $X$ .

A curve of genus $q$ with a Half-Canonical embedding in $\mathbf{P}^{3}$

Sevin Recillas (1984)

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

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Si costruiscono curve di genere $g=4n—3$ , $n\geq 3$ che hanno $2^{n-3}(2^{n-2}-1)$ fasci semicanonici $L$ tali che $h^{0}(L)=4$ . Per $n+3$ si dimostra che gli $L$ sono molto ampi.

An $a b c d$ theorem over function fields and applications

Pietro Corvaja, Umberto Zannier (2011)

Bulletin de la Société Mathématique de France

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We provide a lower bound for the number of distinct zeros of a sum $1 + u + v$ for two rational functions $u, v$ , in term of the degree of $u, v$ , which is sharp whenever $u, v$ have few distinct zeros and poles compared to their degree. This sharpens the “ $a b c d$ -theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface $x^{a} + y^{a} + z^{c} = 1$ contains only finitely many rational or elliptic...

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.

Self-intersection of the relative dualizing sheaf on modular curves $X_{1} (N)$

Hartwig Mayer (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $N$ be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than $4$ . Our main theorem is an asymptotic formula solely in terms of $N$ for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves $X_{1} (N) / ℚ$ . From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian $J_{1} (N) / ℚ$ of $X_{1} (N) / ℚ$ , and, for sufficiently large $N$ , an effective version of Bogomolov’s conjecture for $X_{1} (N) / ℚ$ . ...

Singularities of $2 Θ$ -divisors in the jacobian

Christian Pauly, Emma Previato (2001)

Bulletin de la Société Mathématique de France

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We consider the linear system $| 2 Θ_{0} |$ of second order theta functions over the Jacobian $J C$ of a non-hyperelliptic curve $C$ . A result by J.Fay says that a divisor $D \in | 2 Θ_{0} |$ contains the origin $𝒪 \in J C$ with multiplicity $4$ if and only if $D$ contains the surface $C - C = {𝒪 (p - q) ∣ p, q \in C} \subset J C$ . In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing $𝒪$ with multiplicity $6$ , divisors containing the fourfold $C_{2} - C_{2} = {𝒪 (p + q - r - s) ∣ p, q, r, s \in C}$ , and divisors singular along $C - C$ , using...

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...

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 monogenity of certain pure number fields of degrees $2^{r} \cdot 3^{k} \cdot 7^{s}$

Hamid Ben Yakkou, Jalal Didi (2024)

Mathematica Bohemica

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Let $K = ℚ (α)$ be a pure number field generated by a complex root $α$ of a monic irreducible polynomial $F (x) = x^{2^{r} \cdot 3^{k} \cdot 7^{s}} - m \in ℤ [x]$ , where $r$ , $k$ , $s$ are three positive natural integers. The purpose of this paper is to study the monogenity of $K$ . Our results are illustrated by some examples.

On some imaginary triquadratic number fields $k$ with ${Cl}_{2} (k) ≃ (2, 4)$ or $(2, 2, 2)$

Abdelmalek Azizi, Mohamed Mahmoud Chems-Eddin, Abdelkader Zekhnini (2021)

Commentationes Mathematicae Universitatis Carolinae

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Let $d$ be a square free integer and $L_{d} : = ℚ (ζ_{8}, \sqrt{d})$ . In the present work we determine all the fields $L_{d}$ such that the $2$ -class group, ${Cl}_{2} (L_{d})$ , of $L_{d}$ is of type $(2, 4)$ or $(2, 2, 2)$ .

An effective proof of the hyperelliptic Shafarevich conjecture

Rafael von Känel (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $C$ be a hyperelliptic curve of genus $g \geq 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$ . We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$ , $S$ and $K$ . In particular, we obtain that for any given number field $K$ , finite set of places $S$ of $K$ and integer $g \geq 1$ one can in principle determine the set of $K$ -isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside...