The rationality of the moduli space of genus-4 curves endowed with an order-3 subgroup of their Jacobian
The rationality of the moduli space of genus-4 curves endowed with an order-3 subgroup of their Jacobian
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赋予雅可比行列式3阶子群的genus-4曲线模空间的合理性
DOI：
10.1307/mmj/1291213953
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发表时间：
2010
影响因子：
0.9
通讯作者：
A. Verra
中科院分区：
文献类型：
--
作者：
I. Bauer;A. Verra
令 C 为属 g 的平滑、不可约复射影曲线，并令 η ∈ Pic0(C) 为平凡丛 OC 的非平凡 n 次方根。由于几个不同的原因，现在和最近，人们特别关注上述对 (C, η) 的模空间 Rg,n 及其可能的紧化（参见例如 [CapCasC]）。例如，它们是 n = 2 情况的推广，即所谓的 Prym 模空间，通常用 Rg 表示。由于它们与 Prym 品种理论相关，因此本案的兴趣占据了突出的位置。特别是，现在关于 Rg 的 Kodaira 维数的许多结果已经可用，而 Rg 的经典几何描述在 g ≤ 7 时存在。更准确地说，让我们提到 Farkas 和 Ludwig [FLu] 证明了 Rg 在 g ≥ 14 和 g = 15 时是一般类型。另一方面，Rg 的非有理参数化在 g ≤ 7 时已知 [Cat, D, Do, ILoS, Ve1, Ve2]。我们还可以考虑对 (C,Z/nZ) 的模空间 Rg,<n>，其中 C 是亏格 g 的平滑、不可约复射影曲线，Z/nZ 是 Pic0(C) 的 n 阶循环子群。由于 Rg,<2> = Rg,2，这些 mouli 空间以（稍微）不同的方式推广 Prym 模空间。与 n = 2 的情况相反，对于 n > 2 时的 Rg,n 和 Rg,<n> 知之甚少。特别是，使得 Rg,n 和 Rg,<n> 具有负 Kodaira 维数的所有对 (g, n) 的（可能很短）列表是未知的。 Rg,n 和 Rg,<n> 的合理性已在一些非常低属的情况下得到了证明：R4 的情况是 Catanese [Cat] 的结果。 R3的合理性由Katsylo在[Ka]中证明。 Catanese 和 Dolgachev 也提供了独立证明；参见 [D]（也适用于 R2）。最近，R3,3和R3,<3>的合理性已被Catanese和第一作者[BCat]证明。为了完成这个图，我们回想一下，R1,n 对于每个素数 n 都是一条不可约曲线，并且它的几何亏格是众所周知的。
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.
Prym 品种模空间的 Kodaira 维数
影响因子：
2.6
作者：
Farkas;Gavril;Ludwig;Katharina
通讯作者：
Katharina