Let $ℳ$ be the moduli space of principal ${\mathrm{SO}}_r$-bundles on a curve $C$, and $ℒ$ the determinant bundle on $ℳ$. We define an isomorphism of $H^0(ℳ,ℒ)$ onto the dual of the space of $r$-th order theta functions on the Jacobian of $C$. This isomorphism identifies the rational map ${ℳ⇢\left|ℒ\right|}^{*}$ defined by the linear system $\left|ℒ\right|$ with the map $ℳ⇢\left|r\Theta \right|$ which associates to a quadratic bundle $(E,q)$ the theta divisor ${\Theta }_E$. The two components ${ℳ}^{+}$ and ${ℳ}^{-}$ of $ℳ$ are mapped into the subspaces of even and odd theta functions respectively. Finally we discuss the analogous...

Let $C$ be an algebraic projective smooth and trigonal curve of genus $g\ge 5$. In this paper we define, in a way equivalent to that followed by A. Maroni in [1], an integer $m$, called the species of $C$, which is a birational invariant having the property that $0\le m\le \frac{g+2}{3}$ and $m\mathrm{—}g\equiv 0$ mod(2). In section 1 we prove that for every $g(\ge 5)$ and every $m$, as before, there are trigonal curves of genus $g$ and species $m$. In section 2 we define the space ${ℳ}_{g,3;m}^1$ of moduli of trigonal curves of genus $g$ and species $m$. We note that ${ℳ}_{g,3;m}^1$ is irreducible...

Nous étudions les espaces analytiques rigides de dimension 1, réguliers, de genre fini sur un corps valué complet $k$. Nous montrons qu’un tel espace $X$ admet une réduction préstable. Si $k$ est maximalement complet, $X$ se plonge dans une courbe algébrique (analytifiée). On donne aussi une caractérisation des espaces analytiques qui sont le complémentaire d’une partie compacte dans une courbe algébrique.