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A combinatorial description of a moduli space of curves and of extremal polynomials. (English. Russian original) Zbl 1097.14021
Sb. Math. 194, No. 10, 1451-1473 (2003); translation from Mat. Sb. 194, No. 10, 27-48 (2003).
Let’s consider the moduli space $$\mathcal{H}$$ of real hyperelliptic curves. A real hyperelliptic curve is a curve $M=\{(x,w)\in C^2\mid w^2=p(x)\},$ where $$p(x)$$ is a real polynomial. In the moduli space, we regard two curves $$w^2=p(x)$$ and $$w^2=p(ax+b)$$ as same curves for real numbers $$a,b$$. This object is deeply related to extremal polynomials, which is one of very old famous problems. Because $$p(x)$$ is a real polynomial, the branch point set e ($$=$$ zero point set of $$p(x)$$) of an hyperelliptic covering has symmetry with respect to the real axis. Hence $$M$$ has another involution $$(w,z)\mapsto({\bar w},{\bar z})$$. Let $$H_1^-(M,)$$ be the $$-1$$ eigenspace of this involution in $$H_1(M;R).$$ $$H_1^-(M,)$$ is generated by loops in $$C\setminus {\mathbf e}$$. Let $$M_0\in\mathcal{H}$$ be a base point, and $$\tilde\mathcal{H}$$ be the universal covering of $$\mathcal{H}$$ with start point $$M_0$$. That is, an element in $$\tilde\mathcal{H}$$ is a path in $$\mathcal{H}$$ with start point $$M_0$$. Here we define the period map $$\Pi:\tilde\mathcal{H}\to H_1^-(M,)^{\ast}$$. For $$\gamma\in\tilde\mathcal{H}$$ and $$C\in H_1^-(M,)$$, $$\Pi(\gamma)(C)$$ is the intersection number of $$C$$ and $$C_\gamma$$, where $$C_\gamma$$ is a parallel translation of $$C$$ along $$\gamma$$. In this paper, the author determines the image of the period map, and using this he gives a cellular decomposition of $$\tilde\mathcal{H}$$. These are very interesting results also for topologists. The reviewer hopes for some topological application of the period map to the mapping class group theory of hyperelliptic curves.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems) 05C10 Planar graphs; geometric and topological aspects of graph theory 14H45 Special algebraic curves and curves of low genus 14P05 Real algebraic sets 14H30 Coverings of curves, fundamental group
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