Nodal cubic surfaces and the rationality of the moduli space of curves of genus two

Abstract

Let ~//2 = the moduli space of curves of genus two. It was proved by Igusa in I-2] that ~g2 is a rational variety. Here we propose an easy proof of this fact in the case of a ground field of characteristic zero. Let us give the main step of the proof and an outline of the whole paper. Let S be a "nodal cubic" that is a cubic surface in p3 with a unique ordinary double point P; Ts, e its tangent space at P and TCs, e its tangent cone at P. S c~ T Cs, e is a set of six distinct lines through P. So P(TCs, p) is a conic )7 with six distinguished points P1 . . . . . P6 on it. Let F s be the genus two curve obtained as a double cover of )7 branched at P1 . . . . . P6" Conversely given a genus two curve F its image by the bicanonical map ~p is a conic V C P 2 containing six distinguished points 01 . . . . . 06 which are the branch locus of 02. By blowing up p2 at 01 . . . . . 06 and by blowing down the proper transform of V we obtain a nodal cubic Sr. Then the following facts can be easily checked. (1.1) Any isomorphism ~:$1~$2 , Si nodal cubic in p 3 , i=1 ,2 , is induced by a projective linear transformation in p3.