The rationality of the moduli space of genus-4 curves endowed with an order-3 subgroup of their Jacobian

Abstract

Let C be a smooth, irreducible complex projective curve of genus g and let η ∈ Pic0(C) be a nontrivial nth root of the trivial bundle OC. For several different reasons, special attention has been paid, now and in the recent past, to the moduli spaces Rg,n of pairs (C, η) as above and to its possible compactifications (see e.g. [CapCasC]). For instance, they are generalizations of the case n = 2, the so-called Prym moduli spaces, usually denoted by Rg. Since they are related to the theory of Prym varieties, the interest in this case occupies a prominent position. In particular, many results on the Kodaira dimension of Rg are now available, while classical geometric descriptions of Rg exist for g ≤ 7. More precisely, let us mention that Farkas and Ludwig [FLu] proved that Rg is of general type for g ≥ 14 and g = 15. On the other hand, unirational parameterizations of Rg are known for g ≤ 7 [Cat, D, Do, ILoS, Ve1, Ve2]. One can also consider the moduli spaces Rg,〈n〉 of pairs (C,Z/nZ), where C is a smooth, irreducible complex projective curve of genus g and Z/nZ is a cyclic subgroup of order n of Pic0(C). As Rg,〈2〉 = Rg,2, these mouli spaces are generalizing the Prym moduli spaces in a (slightly) different way. In contrast to the case n = 2, not very much is known about Rg,n and Rg,〈n〉 for n > 2. In particular, the (probably short) list of all pairs (g, n) such that Rg,n and Rg,〈n〉 have negative Kodaira dimension is not known. The rationality of Rg,n and Rg,〈n〉 has been proved in some cases of very low genus: the case of R4 is a result of Catanese [Cat]. The rationality of R3 was proved by Katsylo in [Ka]. Independent proofs are also due to Catanese and to Dolgachev; see [D] (also for R2). Recently, the rationality of R3,3 and of R3,〈3〉 has been proven by Catanese and the first author [BCat]. To complete the picture, we recall that R1,n is an irreducible curve for every prime n and that its geometric genus is well known.