Rationality of fields of invariants for some representations of SL(2)\timesSL(2)

We prove that the quotient by SL(2)\timesSL(2) of the space of bidegree (a, b) curves on P^1\timesP^1 is rational when ab is even and a\not=b.Comment: added Section 2.1 in which a general method is explained. to appear in Compositio Mat

On the rationality of the moduli space of L\"uroth quartics

We prove that the moduli space M_L of L"uroth quartics in P^2, i.e. the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of PGL_3(CC) is rational, as is the related moduli space of Bateman seven-tuples of points in P^2.Comment: 7 page