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

Rationality of the moduli spaces of plane curves of sufficiently large degree

We prove that the moduli space of plane curves of degree d is rational for all sufficiently large d.Comment: 18 pages; 1 figure; Macaulay2 scripts used can be found at http://www.uni-math.gwdg.de/bothmer/rationality/ or at the end of the latex source fil