Hyperelliptic d-osculating covers and rational surfaces

Abstract

1.1. Let P and (X, q) denote, respectively, the projective line and a fixed elliptic curve marked at its origin, both defined over an algebraically closed field K of arbitrary characteristic p 6= 2. We will study all finite separable marked morphisms π : (Γ, p) → (X, q), called hereafter hyperelliptic covers, such that Γ is a degree-2 cover of P, ramified at the smooth point p ∈ Γ. Canonically associated to π there is the Abel (rational) embedding of Γ into its generalized Jacobian, Ap : Γ → JacΓ, and {0} ( V 1 Γ,p . . . ( V g Γ,p, the flag of hyperosculating planes to Ap(Γ) at Ap(p) ∈ JacΓ (cf. 2.1. & 2.2.). On the other hand, we also have the homomorphism ιπ : X → Jac Γ, obtained by dualizing π. There is a smallest positive integer d such that the tangent line to ιπ(X) is contained in the d-dimensional osculating plane V d Γ,p. We call it the osculating order of π, and π a hyperelliptic d-osculating cover (2.4.(2)). If π factors through another hyperelliptic cover, the arithmetic genus increases, while the osculating order can not decrease (2.8.). Studying, characterizing and constructing those with given osculating order d but maximal possible arithmetic genus, so-called minimal-hyperelliptic d-osculating covers, will be one of the main issues of this article. The other one, to which the first issue reduces, is the construction of all rational curves in a particular anticanonical rational surface associated to X (i.e.: a rational surface with an effective anticanonical divisor). Both problems are interesting on their own and in any characteristic p 6= 2. They were first considered, however, over the complex numbers and through their link with solutions of the Korteweg-deVries hierarchy, doubly periodic with respect to the d-th KdV flow (cf. [1], [3], [8], [9], [14] for d = 1 and [11], [2], [4], [5] for d = 2). We sketch hereafter the structure and main results of our article. (1) We start defining in section 2. the Abel rational embeddingAp : Γ→ JacΓ, and construct the flag {0} ( V 1 Γ,p . . . ( V g Γ,p = H (Γ, OΓ), of hyperosculating planes at the image of any smooth point p ∈ Γ. We then define the homomorphism ιπ : X → JacΓ, canonically associated to the hyperelliptic cover π, and its osculating order (2.4.(2)). Regardless of the osculating order, we prove that any degree-n hyperelliptic cover has odd ramification index at the marked point, say ρ, and factors through a unique one of maximal arithmetic genus 2n ρ+1 2 (2.6.). We finish characterizing the osculating order by the existence of a particular projection κ : Γ→ P (2.6.).