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.

Reviewer: Kazushi Ahara (Kawasaki)

##### 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 |