Geodesic
The rationality of moduli spaces of hyperelliptic curves
Izvestiya. Mathematics , Tome 25 (1985) no. 1, pp. 45-50.

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The rationality of the moduli spaces of hyperelliptic curves of genus $g\ne4$ is proved. Bibliography: 5 titles. 
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P. I. Katsylo. The rationality of moduli spaces of hyperelliptic curves. Izvestiya. Mathematics , Tome 25 (1985) no. 1, pp. 45-50. http://geodesic.mathdoc.fr/item/IM2_1985_25_1_a2/

[1] Springer T., Teoriya invariantov, Mir, M., 1981 | MR | Zbl

[2] Algebraicheskie poverkhnosti, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 75, 1965 | MR | Zbl

[3] Katsylo P. I., “Ratsionalnost prostranstv orbit neprivodimykh predstavlenii gruppy $SL_2$”, Izv. AN SSSR. Ser. matem., 47:1 (1983), 26–36 | MR

[4] Popp H., Moduli theory and classification theory of algebraic varieteies, Lecture Notes in Math., 620, Springer-Verlag, 1977 | MR | Zbl

[5] Gordan P., Vorlesungen über invariantentheorie, II (herausgegeben von G. Kerschensteiner), Teubner, Leipzig, 1887 | MR | Zbl