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Rationality of the moduli variety of curves of genus 5. (English. Russian original) Zbl 0781.14018

Math. USSR, Sb. 72, No. 2, 439-445 (1992); translation from Mat. Sb. 182, No. 3, 457-464 (1991).
From the introduction: The ground field is taken to be the field of the complex numbers \(\mathbb{C}\). By \(M_ g\) we denote the moduli variety of curves of genus \(g\). As we know, \(M_0\)={point} and \(M_1=\mathbb{C}\). The question of whether \(M_ g\) is rational for \(g\geq 2\) is a classical moduli problem. In 1960, J.-i. Igusa proved the rationality of \(M_ 2\) [Ann. Math. (2) 72, 612–649 (1960; Zbl 0122.39002)]. The question about the rationality of \(M_3\) turned out to be very difficult and does not have an answer yet — the only known fact is that \(M_3\times \mathbb{C}^2\) is rational. Recently, N. I. Shepherd-Barron has proved the rationality of \(M_4\) and \(M_6\) [cf. Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, Part 1, Proc. Symp. Pure Math. 46, No. 1, 165–171 (1987; Zbl 0669.14015) and Compos. Math. 70, No. 1, 13–25 (1989; Zbl 0704.14044)].
In this paper we prove the rationality of \(M_5\).

MSC:

14H10 Families, moduli of curves (algebraic)
14M20 Rational and unirational varieties
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