The variety of moduli of curves of genus four is rational
Doklady Akademii Nauk, Tome 290 (1986) no. 6, pp. 1292-1294.

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@article{DAN_1986_290_6_a1,
     author = {P. I. Katsylo},
     title = {The variety of moduli of curves of genus four is rational},
     journal = {Doklady Akademii Nauk},
     pages = {1292--1294},
     publisher = {mathdoc},
     volume = {290},
     number = {6},
     year = {1986},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DAN_1986_290_6_a1/}
}
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P. I. Katsylo. The variety of moduli of curves of genus four is rational. Doklady Akademii Nauk, Tome 290 (1986) no. 6, pp. 1292-1294. http://geodesic.mathdoc.fr/item/DAN_1986_290_6_a1/