Moduli space of semistable pairs on a curve
Moduli space of semistable pairs on a curve
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曲线上半稳定对的模空间
DOI：
10.1112/plms/s3-62.2.275
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发表时间：
1991
影响因子：
1.8
通讯作者：
N. Nitsure
中科院分区：
文献类型：
--
作者：
N. Nitsure
设X为任意特征的代数闭域k上的光滑投影曲线。由Hitchin[2]定义的X上的稳定对（E, <p）是X上的向量束E与（^-模）的态射<j>: E^>QX®E，使得对于E的任意^-不变固有子束F，不等式JU(F) < /i（£）成立（其中JU =度/秩）。在[2]中，Hitchin证明了紧黎曼曲面上所有秩2的稳定对同构类的集合可以给出解析范畴内具有粗模性质的复流形的结构。在本文中，我们构造了一个代数范畴内的粗模格式。我们不再用cf>：£•Qx <8> E取（£，0），而是考虑更一般的情况，即<p在x上的任意固定线束L上取值。我们对E的秩不加任何限制。在下面的§5中，我们完成了对于任意线束L，对于半稳定对（E, <p）的（5-等价类）存在一个粗模格式M（r, d， L）的证明：M（r, d， L）是拟射影格式，并且有一个开放的子格式M'，它是稳定对的模格式。M的构造是普通向量束模的构造的推广；有关后者的说明，请参见[7]或[9]。在§6中，我们证明了从M（r, d， L）到H°（X， L） X的Hitchin态射…x H°（x， U）将一对（E, <j>）映射到它的特征多项式，是一个适当的态射。在§7中，我们研究了格式M（r, d， L）的一些性质。从这些可以特别地得出，当d是奇数时，M（2, d, Qx）是一个非奇异的变异体（不可约且被约）。
Let X be a smooth projective curve over an algebraically closed field k of any characteristic. A stable pair (E, <p) on X, as defined by Hitchin [2], is a vector bundle E on X together with a morphism <j>: E^>QX ® E of (^-modules such that for any ^-invariant proper subbundle F of E, the inequality JU(F) < /i(£) holds (where JU = degree/rank). In [2], Hitchin has proved that the set of all isomorphism classes of stable pairs of rank 2 over a compact Riemann surface can be given the structure of a complex manifold which has the coarse moduli property in the analytic category. In this paper we construct a coarse moduli scheme within the algebraic category in the following more general setup. Instead of taking (£, 0) with cf>: £ • Qx <8> E, we consider the more general situation where <p takes values in any fixed line bundle L on X. We do not place any restriction on the rank of E. In § 5 below, we complete the proof that for any line bundle L, there exists a coarse moduli scheme M(r, d, L) for (5-equivalence classes of) semistable pairs (E, <p: E—»L®£) of rank r, degree d, on X. The scheme M(r, d, L) is quasi-projective, and has an open subscheme M' which is the moduli scheme of stable pairs. Our construction of M is a generalization of the construction of the moduli of ordinary vector bundles; for an exposition of the latter, see [7] or [9]. In § 6, we show that the Hitchin morphism from M(r, d, L) to H°(X, L)x ... x H°(X, U), which maps a pair (E, <j>) to its characteristic polynomial, is a proper morphism. In § 7, we study some properties of the scheme M(r, d, L). From these it follows, in particular, that M(2, d, Qx) is a non-singular variety (irreducible and reduced) when d is odd.