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...

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...

Shells of monotone curves

Josef Mikeš, Karl Strambach (2015)

Czechoslovak Mathematical Journal

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We determine in $ℝ^{n}$ the form of curves $C$ corresponding to strictly monotone functions as well as the components of affine connections $\nabla$ for which any image of $C$ under a compact-free group $Ω$ of affinities containing the translation group is a geodesic with respect to $\nabla$ . Special attention is paid to the case that $Ω$ contains many dilatations or that $C$ is a curve in $ℝ^{3}$ . If $C$ is a curve in $ℝ^{3}$ and $Ω$ is the translation group then we calculate not only the components of the curvature and the Weyl...

Riemannian geometries on spaces of plane curves

Peter W. Michor, David Mumford (2006)

Journal of the European Mathematical Society

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We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from $S^{1}$ to the plane modulo the group of diffeomorphisms of $S^{1}$ , acting as reparametrizations. In particular we investigate the metric, for a constant $A > 0$ , $G_{c}^{A} (h, k) : = \int_{S^{1}} (1 + A κ_{c} {(θ)}^{2}) 〈 h (θ), k (θ) 〉 | c^{'} (θ) | d θ$ where $κ_{c}$ is the curvature of the curve $c$ and $h$ , $k$ are normal vector fields to $c$ . The term $A κ^{2}$ is a sort of geometric Tikhonov regularization because, for $A = 0$ , the geodesic distance between any two distinct curves is 0, while...