Meromorphic functions on Riemann surfaces, Weierstrass points

点击翻译按钮获取中文摘要

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