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Hyperelliptic \(d\)-osculating covers and rational surfaces. (Projections hyperelliptiques \(d\)-osculantes et surfaces rationnelles.) (English. French summary) Zbl 1310.14032
Let \(k\) be an algebraically closed field of characteristic \(p\neq 2\), and \((X,q)\) a fixed elliptic curve over \(k\) marked at its origin. Let \(\pi:(\Gamma,p)\rightarrow (X,q)\) be a finite separable marked morphism of degree \(n\) such that \(\Gamma\) is a degree-2 cover of \(\mathbb{P}^1\), ramified at the smooth point \(p\in\Gamma\). For any positive integer \(j\), let \(\delta:H^0(\Gamma,\mathcal{O}_{jp}(jp))\rightarrow H^1(\Gamma,\mathcal{O}_{\Gamma})\) denote the coboundary map of the long exact cohomology sequence associated with the exact sequence of \(\mathcal{O}_{\Gamma}\)-modules \(0\rightarrow \mathcal{O}_{\Gamma}\rightarrow \mathcal{O}_{\Gamma}(jp)\rightarrow \mathcal{O}_{jp}(jp)\rightarrow 0\). By the Weierstrass gap theorem, for any \(d\in\{1,\dots, g\}\), there exists \(0<j<2g\) such that \(\delta(H^0(\Gamma,\mathcal{O}_{jp}(jp)))\) is a \(d\)-dimensional subspace, denoted hereafter by \(V_{\Gamma,p}^d\). The filtration \(\{0\}\subset V_{\Gamma,p}^1\subset\cdots\subset V_{\Gamma,p}^g=H^1(\Gamma,\mathcal{O}_{\Gamma})\) is called the flag of hyperosculating spaces to \(A_p(\Gamma)\) at \(0\), where \(A_p:\Gamma\rightarrow \mathrm{Jac}(\Gamma)\) denote the natural embedding of \(\Gamma\) into its generalized Jacobian. There is a minimal integer \(d\geq 1\), called osculating order of \(\pi\), such that the tangent to \(A_p(\pi^*(X))\) at \(0\) is contained in \(V_{\Gamma,p}^d\), and the cover \(\pi\) is called a hyperelliptic \(d\)-osculating cover. In this paper the author studies relations between the degree \(n\), the arithmetic genus \(g\), and the osculating order \(d\) of such covers. He proves 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 \(2g<p(2d+1)\) if \(p>2\).

MSC:
14H40 Jacobians, Prym varieties
14H42 Theta functions and curves; Schottky problem
14H52 Elliptic curves
14H81 Relationships between algebraic curves and physics
14J26 Rational and ruled surfaces
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