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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.

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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