Geometry and Integrability in Mathematical Physics GIMP'06

May 15 - 19, 2006, Moscow, Russia

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Armando Treibich

Solutions of the KdV hierarchy, doubly periodic with respect to the $d$-th flow ($d > 1$), and their spectral curves.

We study the hyperelliptic curves defined over an arbitrary algebraically closed field $K$, naturally embedded in their jacobians, which osculate to order $d$ ($d > 0$) at a Weierstrass point, to a given elliptic curve $X$. The corresponding marked projections onto $X$ are then called hyperelliptic $d$-osculating covers. When $K = C$ they give rise to exact solutions of the Korteweg-deVries hierarchy, doubly periodic with respect to $d$-th KdV flow. We tackle this problem, through an algebraic surface approach. The main characters will be played by :

1) a particular ruled surface over $X$, $\pi_S: X \to S$, equipped with its canonical involution $\tau : S \to S$, over the multiplication by $-1$ of $X$ (i.e.: such that $[-1]\circ\tau=\tau \circ\pi_S$ ),

2) the blow up of $S, e : S^\bot \to S$, at the eight fixed points of $\tau$, equipped with the natural involution over $\tau,\tau^\bot: S^\tau \to S^\tau$ ( i.e.: such that $e\circ\tau^\bot=\tau\circ e$),

3) the quotient of $S^\bot$ by $\tau^\bot,\varphi: S^\bot\to\widetilde S: = S^\bot/\tau^\bot$.

We prove that any hyperelliptic osculating cover $\pi:\Gamma\to X$ lifts to $S^\bot, \tau^\bot:\Gamma\to\Gamma^\bot\subset S^\bot$ and projects onto a rational curve of $\widetilde S, \widetilde\Gamma:\varphi(\Gamma^\bot)\subset\widetilde S$. We then define the type of $\pi$, by intersecting $\Gamma^\bot$ with $4$ particular exceptional divisors of $S^\bot$.

Conversely, we find rational curves of $\widetilde S$ and construct $(d-1)$-dimensional families of hyperelliptic osculating covers, of arbitrary high genus, naturally embedded in $S^\bot$.