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rat3 - RATIONALITY OF R2 AND R3 IGOR V DOLGACHEV To Fedya...

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RATIONALITY OF R 2 AND R 3 IGOR V. DOLGACHEV To Fedya Bogomolov 1. Introduction Let R g be the moduli space of genus g curves together with a non- trivial 2-torsion divisor class . In this paper we shall prove that the moduli spaces R 2 and R 3 are rational varieties. The rationality of R 4 was proven by F. Catanese [3]. He also claimed the rationality of R 3 but the proof was never published. The first published proof of rationality of R 3 was given by P. Katsylo in [10]. Some years earlier A. Del Centina and S. Recillas [5] constructed a map of degree 3 from R 3 to the moduli space M be 4 of bi-elliptic curves of genus 4 and claimed that it could be used for proving the rationality of R 3 based on the rationality of M be 4 proven by F. Bardelli and Del Centina in [1]. In an unpublished preprint of 1990 I had shown that it is indeed possible. The present paper is based on this old preprint and also includes a proof of rationality of R 2 which I could not find in the literature. The relation between the moduli spaces R 3 and M be 4 is based on an old construction of P. Roth [13] and, independently, A. Coble [4]. Much later it had been rediscovered and generalized by S. Recillas [11], and nowadays is known as the trigonal construction. To each curve C of genus g together with a g 1 4 it associates a curve X of genus g + 1 together with a g 1 3 and a non-trivial 2-torsion divisor class η . The Prym variety of the pair ( X, η ) is isomorphic to the Jacobian variety of C . When g = 3 and g 1 4 = | K C + | , the associated curve X turns out to be a canonical bi-elliptic curve of genus 4, the bi-elliptic involution τ switches the two g 1 3 on X , and the 2-torsion class η is coming from a 2-torsion divisor class on the elliptic quotient X/ ( τ ). To make this paper self-contained we remind the construction following A. Coble. The author is grateful to the referee for some valuable comments on the paper. 2. Rationality of R 2 Let C be a genus 2 curve and x 1 , . . . , x 6 be its six Weierstrass points. A non-trivial 2-torsion divisor class on C is equal to the divisor class 1
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2 IGOR V. DOLGACHEV [ x i - x j ] for some i = j . The hyperelliptic series g 1 2 defines a degree 2 map C P 1 and the images of the Weierstrass points are the zeroes of a binary form of degree 6. This defines a birational isomorphism between the moduli space M 2 of genus 2 curves and the GIT-quotient P ( V (6)) // SL(2), where V ( m ) denotes the space of binary forms of de- gree m . A non-trivial 2-torsion divisor class is defined by choosing a degree 2 factor of the binary sextic. Thus the moduli space R 2 is bi- rationally isomorphic to the GIT-quotient ( P ( V (4)) × P ( V (2))) // SL(2) and the canonical projection R 2 → M 2 corresponds to the multipli- cation map V (4) × V (2) V (6). At this point we may conclude by referring to Katsylo’s result on rationality of fields of invariants of SL(2) in reducible representations [9]. However, we proceed by giving a more explicit proof.
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