# zbMATH — the first resource for mathematics

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
##### Keywords:
hyperelliptic curves; Jacobian varieties; osculating covers
Full Text: