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Meromorphic functions on Riemann surfaces, Weierstrass points

黎曼曲面、Weierstrass 点上的亚纯函数

基本信息

批准号：
10440051
负责人：
KATO Takao
金额：
$ 3.9万
依托单位：
Yamaguchi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

项目摘要

In this project, we studied classification problems of compact Riemann surfaces through the existence of meromorphic functions on them. It is one of the central subject on the theory compact Riemann surfaces. The main results we obtained are the following :(a) Let k be a prime number. Assume C is a compact Riemann surface of genus g which is a k-sheeted covering of a compact Riemann surface of genus h>0. In this case, we studied the structure of a subvariety WィイD31(/)dィエD3(C) of the Jacobian variety J(C). Then, we have : "If g and d are large enough comparable to k and h, then WィイD31(/)dィエD3(C) is reduced and irreducible". (As a matter of fact, we have quantitative estimates for g and d).(b) For a compact Riemann surface C of genus g, H. Martens proved that dim WィイD3r(/)dィエD3(C)【less than or equal】d-2r for d【less than or equal】g+r-1. Then, Mumford (resp. Keem) gave a characterization of WィイD3r(/)dィエD3(C) which sarisfies dim WィイD3r(/)dィエD3(C)=d-2r (resp. d-2r-1) for d【less than or equal … More 】g+r-3 (resp. g+r-5). If C is of odd gonality, then it is known that dim WィイD3r(/)dィエD3(C)【less than or equal】d-3r. In 1996, G. Martens gave a characterization of WィイD3r(/)dィエD3(C) which satisfies dim WィイD3r(/)dィエD3(C)=d-3r. As an extension of this result, we gave a characterization of WィイD3r(/)dィエD3(C) which satidfies dim WィイD3r(/)dィエD3(C)=d-3r-1.(c) Let L be a very ample line bundle of degree d and dimension r on a compact Riemann surface of genus g. We studied the normally generatedness of L in case d is near to g. As a result, we succeeded to describe L which fails to be normally generated in case d=2g-2, 2g-3. Moreover, we showed that if L is special, it is always normally generated in case d=2g-5. In case d=2g-6, there is one exceptional case, in which L fails to be normally generated. In our discussion, a critetion for L to be normally generated due to Green-Lazarsfeld plays an important role. Finally, we thought about the case that L contributes the Clifford index, while Green-Lazarsfeld did not treated this case. Less

本项目通过亚纯函数的存在性研究了紧致黎曼曲面的分类问题。它是紧致黎曼曲面理论的中心课题之一。我们得到的主要结果如下：（a）设k为素数。设C是亏格为g的紧致黎曼曲面，它是亏格为h>0的紧致黎曼曲面的k-单复盖。在这种情况下，我们研究了雅可比簇J（C）的子簇W <$D31（/）d <$D3（C）的结构。然后，我们有：“如果g和d足够大，可以与k和h相比较，那么W D 31（/）d D 3（C）是约化的和不可约的”。(As事实上，我们对g和d有定量估计。(b)对于亏格为g的紧致Riemann曲面C，H.马滕斯证明了dim W D3 r（/）d D3（C）[小于或等于]d-2 r对于d[小于或等于]g+r-1。然后，芒福德（Mumford）。Keem）给出了W_n_d d-2 r-1）对于d[小于或等于 ...更多信息 ]g+r-3（分别为g+r-5）。如果C是奇性的，则已知dim W D3 r（/）d D3（C）[小于或等于]d-3r。1996年，G. Martens给出了满足dim W <$D3r（/）d <$D3（C）=d-3r的W <$D3r（/）d <$D3（C）的一个刻划.作为这一结果的推广，我们给出了满足dim W_（？）d_（？）D_（？）D_（？）D_（？）（C）=d-3r-1的W_（？）D_（？）D_（？）(c)设L是亏格为g的紧致Riemann曲面上的d次r维非常充分的线丛。研究了当d接近g时L的正规生成性。结果，我们成功地描述了在d=2g-2，2g-3的情况下不能正常生成的L。此外，我们还证明了，如果L是特殊的，它总是正常生成的情况下，d=2g-5。当d=2g-6时，存在一种例外情况，即L无法正常生成。在我们的讨论中，L是由Green-Lazarsfeld正常生成的一个判据起着重要的作用。最后，我们考虑了L贡献Clifford指数的情况，而Green-Lazarsfeld没有处理这种情况。少