Hyperelliptic $d$-osculating covers and rational surfaces

Treibich, Armando

doi:10.24033/bsmf.2669

Hyperelliptic

d

-osculating covers and rational surfaces
[Projections hyperelliptiques

d

-osculantes et surfaces rationnelles]

Treibich, Armando

Bulletin de la Société Mathématique de France, Tome 142 (2014) no. 3, pp. 379-409

Abstract (VO)
Résumé

Let $d$ be a positive integer, $𝕂$ an algebraically closed field of characteristic $𝐩 \neq 2$ and $X$ an elliptic curve defined over $𝕂$ . We consider the hyperelliptic curves equipped with a projection over $X$ , such that the natural image of $X$ in the Jacobian of the curve osculates to order $d$ to the embedding of the curve, at a Weierstrass point. We first study the relations between the degree $n$ , the arithmetic genus $g$ and the osculating degree $d$ of such covers. We prove that they are in a one-to-one correspondence with rational curves of linear systems in a rational surface and deduce ( $d - 1$ )-dimensional families of hyperelliptic d-osculating covers, of arbitrary big genus $g$ if $𝐩 = 0$ or such that $2 g < 𝐩 (2 d + 1)$ if $𝐩 > 2$ . It follows at last, $(g + d - 1)$ -dimensional families of solutions of the $K d V$ hierarchy, doubly periodic with respect to the $d$ -th variable.

Soit $d$ un entier positif, $𝕂$ un corps algébriquement clos de caractéristique $𝐩 \neq 2$ et $X$ une courbe elliptique définie sur $𝕂$ . On étudie les courbes hyperelliptiques munies d'une projection sur $X$ , telles que l'image naturelle de $X$ dans la jacobienne de la courbe, oscule à l'ordre $d$ au plongement de celle-ci, et ce en un point de Weierstrass. On étudie tout d'abord les relations entre le degré $n$ , le genre arithmétique $g$ et l'ordre d'osculation $d$ des ces projections. On prouve qu'elles sont en correspondance biunivoque avec des courbes rationnelles dans des systèmes linéaires d'une surface rationnelle et on en déduit des familles $(d - 1)$ -dimensionnelles de revetements hyperelliptiques $d$ -osculants de genre $g$ , arbitrairement grand si la caractéristique $𝐩 = 0$ , ou $2 g < 𝐩 (2 d + 1)$ si $p 𝐩 > 2$ . Il en résulte des familles $(g + d - 1)$ -dimensionnelles de solutions de la hiérarchie $K d V$ , doublement périodiques par rapport à la $d$ -ième variable.

Publié le : 2014-07-21

MR Zbl

DOI : 10.24033/bsmf.2669

Classification : 14H40, 14H42, 14H52, 14H81, 14J26

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     author = {Treibich, Armando},
     title = {Hyperelliptic $d$-osculating covers and rational surfaces},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {379--409},
     year = {2014},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {142},
     number = {3},
     doi = {10.24033/bsmf.2669},
     mrnumber = {3295718},
     zbl = {1310.14032},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2669/}
}

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Treibich, Armando. Hyperelliptic $d$-osculating covers and rational surfaces. Bulletin de la Société Mathématique de France, Tome 142 (2014) no. 3, pp. 379-409. doi: 10.24033/bsmf.2669

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