Theta-characteristics on algebraic curves
Theta-characteristics on algebraic curves
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代数曲线上的 Theta 特性
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发表时间：
1982
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通讯作者：
J. Harris
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作者：
J. Harris
θ特性理论是通过代数发展的，因此它可以应用于可能的奇异和/或可简化的代数曲线。曲线上 θ 特性的配置是根据其奇点来描述的，并应用于平面四次曲线的几何和阿波罗尼乌斯问题。附上 Gorenstein 局部环的一些结果。 0. 简介。人们从不同的角度对 theta 特性理论进行了研究，而且常常是无意的（例如，参见 19 世纪关于平滑平面四次曲线的双切线的文献，例如，[4 和 10]）。然而，直到最近，几乎所有对 theta 特性本身的处理都涉及与紧致黎曼曲面相关的 theta 函数的超越理论。然后，在1971年，Mumford在[8]中对该主题进行了完全代数的处理，在没有参考先验理论的情况下制定了主要结果并证明了它们。这反过来表明，θ 特性可以在更一般的背景下进行研究，例如，在可能的奇异和/或可约代数曲线上。在本文中，我们进行这样的分析。在 ?1 中，遵循 Mumford，我们用代数方法建立了有关 θ 特性的一些基本事实，以便应用于更一般的曲线。 （事实上，（1.10（i））和（1.13）目前的普遍性是由Mumford提出的：虽然他在整个[8]中假设他的曲线X是平滑的，但他的证明似乎按照给定的方式适用。这里（1.13）的论点有些不同，以便使证明更加基本和自足。）一个额外的好处是模空间9Thg的子变体的结果（1.11），该子变种由具有大的曲线组成半正则线性级数；这个事实似乎只能通过代数设置来证明。在 ?2 中，我们研究了奇异曲线上奇数和偶数 θ 特性的数量和配置。结果用两个主要定理（2.12）和（2.22）表示，根据曲线的奇异性或多或少地给出了这些问题的完整答案。有趣的是，这些争论中反复出现的主题是 Mumford 在 [8] 中引入的主题：偶维向量空间上非简并二次形式的最大各向同性子空间的行为。这里，定理 (1.13)、(2.12) 和 (2.22) 的主要动力是下面的断言 (1.4)：任何三个这样的子空间的成对交集的维数之和与 n mod 2 全等。编辑于 1980 年 7 月 11 日收到，并于 1981 年 10 月 22 日修订。1980 年数学学科分类。小学 14H99、14H30；次要 14H20、14H40、14K25。研究由 NSF Grant #MCS78-04008 AOL 部分支持。 （Dl 982 美国数学会 0002-9947/8 1/0000-0371/10.25 美元
The theory of theta-characteristics is developed algebraically, so that it may be applied to possibly singular and/or reducible algebraic curves. The configuration of theta-characteristics on a curve is described in terms of its singularities, with applications to the geometry of plane quartic curves and the problem of Appolonius. Some results on Gorenstein local rings are appended. 0. Introduction. The theory of theta-characteristics has been studied from different points of view, and often inadvertently (cf. for example the 19th century literature on bitangents to smooth plane quartic curves, e.g., [4 and 10]). Until recently, however, virtually all treatments of theta-characteristics per se involved the transcendental theory of the theta-function associated to a compact Riemann surface. Then, in 1971, Mumford in [8] gave a completely algebraic treatment of the subject, formulating the main results and proving them without reference to the transcendental theory. This in turn suggested that theta-characteristics could be studied in a more general context-for example, on possibly singular and/or reducible algebraic curves. In this paper, we undertake such an analysis. In ?1, following Mumford, we establish some of the basic facts about theta-characteristics algebraically, so as to apply to more general curves. (In fact, (1.10(i)) and (1.13) in their present generality are due to Mumford: while he assumes throughout [8] that his curve X is smooth, his proof seems to apply as given. The argument for (1.13) here is somewhat different in order to make the proof more elementary and selfcontained.) One bonus is the result (1.11) on the subvariety of the moduli space 9Thg consisting of curves with large semicanonical linear series; this fact seemingly can be proved only via the algebraic set-up. In ?2, we investigate the number and configuration of odd and even theta-characteristics on a singular curve. The results are expressed in the two main Theorems (2.12) and (2.22), giving a more or less complete answer to these questions in terms of the singularities of the curve. Interestingly, a recurrent motif in these arguments is one introduced by Mumford in [8]: the behavior of the maximal isotropic subspaces for a nondegenerate quadratic form on an even-dimensional vector space. Here, the mainspring of Theorems (1.13), (2.12) and (2.22) is assertion (1.4) below: that the sum of the dimensions of the pairwise intersections of any three such subspaces is congruent to n mod 2. Received by the editors July 11, 1980 and, in revised form, October 22, 1981. 1980 Mathematics Subject Classification. Primary 14H99, 14H30; Secondary 14H20, 14H40, 14K25. Research partially supported by NSF Grant #MCS78-04008 AOl. (Dl 982 American Mathematical Society 0002-9947/8 1/0000-0371/$10.25