Meromorphic functions on Riemann surfaces, Weierstrass points

在这个项目中，我们研究了一个紧凑型Riemann表面的分类问题，通过它们的存在的美化函数。它是理论紧凑型里曼表面上的中心主题之一。我们得到的主要结果是：（a）让k成为一个素数。Assume C是一个紧凑的Riemann表面的genus g，它是一个紧凑的Riemann表面的k-sheeted覆盖的genus> 0。在这种情况下，我们研究了Jacobian variety J（C）的结构。Then，we have：If g and d are large enough comparable to k and h，then WイイD31（/）dイエD3（C）is reduced and irreducible "。（作为事实的一部分，我们对G和D有定量估计。）（b）用于一个紧凑的Riemann表面，G，H. Martens proved that dim W D3r（\）D3（C）[less than or equal] d-2 r for d [less than or equal] g + r-1.然后，Mumford（回答。Keem）gave a characterization of W-D3r（\）d-D3（C）which sarisfies dim W-D3r（\）d-2 r（resp. d-2r-1代表d（少于或等于） ... more g + r-3（resp. g + r-5）。如果C是odd gonality，那么它是已知的dim W D3r（/）d D3（C）[less than or equal] d-3 r. 1996年，G。Martens gave a？terization of W？D3r（/）d？D3（C）which satisfies dim W？D3r（/）d？D3（C）= d-3 r.由于这个结果的扩展，我们得到了W-D3r（/）D-D3（C）which satidfies dim W-D3r（/）D-3r-1。（c）让L在Genus g的紧凑的Riemann表面上做一个非常好的例子线捆绑。我们研究了L的正常生成情况，D的情况接近G。作为一个结果，我们成功地描述了L在d = 2g-2，2g-3的情况下无法正常生成。我们向你展示，如果L是特殊的，它通常在D = 2g-5的情况下生成。在情况d = 2g-6中，有一个例外情况，在正常情况下失败的情况下。在我们的讨论中，对L的批评是正常产生的，而绿色-Lazarsfeld扮演了一个重要的角色。最后，我们考虑到L与克利福德指数相关的情况，而格林-拉扎斯菲尔德没有处理此案。Less（低）

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