Special linear systems on compact Riemann surfaces

1.设C是亏格为g的紧黎曼曲面，W^r_d(C)是由雅可比簇J(C)中d次有效因子和维度r的像组成的子簇。1992年，Coppens-Kim-Marten证明了如果C的平方数(C)是奇数，则dim W^r_d(C)[小于等于]d-3r对任意d[小于或等于]g-1成立。1996年，Marten给出了当dim W^r_d(C)=d-3r且d[小于或等于]g-2时C和W^r_d(C)的特征。Kato-Keem给出了DIMW^Rd(C)=d-3r-1且d[小于等于]g-4时C和W^Rd(C)的刻划。2001年，Kato指出，即使在Gon(C)为偶数的情况下，如果C没有对合，则DimW^Rd(C)[小于或等于]d-3r也成立。然后，在d[小于或等于]g-2时，得到了dim W^r_d(C)=d-3r且d[小于或等于]g-2时C和W^r_d(C)=d-3r，以及d[小于或等于]g-4时C和W^r_d(C)=d-3r-1的特征。这是我们在科学研究助学金(C)(2)，(2000年-2001年)支持的题为“紧黎曼曲面上的亚纯函数的研究”#12640180的研究中的主要结果之一。在这个项目中，我们在dim W^r_d(C)=d-3r-2.2的情形下几乎成功地刻画了C和W^r_d(C)。设F_q是具有q个元素的有限域，C⊂F^n_q是线性[n，k，d]_q码。设n_q(k，d)是固定k，d的最小码长，n_q(k，d)有一个上界，称为Griesmer界。在这个项目中，我们证明了对于d‘s的某个范围，n_q(k，d)等于Griesmer界-1。

We study classification problems for compact Riemann surfaces through the existence of meromorphic functions on them and conformal invariants.1.Let C be a compact Riemann surface of genus g and W^r_d(C) be a subvariety which consists of the image of effective divisors of degree d and dimension r in the Jacobian variety J(C). In 1992 Coppens-Kim-Martens proved that if the gonality gon(C) of C is odd, then dim W^r_d(C) 【less than or equal】 d - 3r holds for any d 【less than or equal】 g - 1. In 1996, Martens gave a characterization of C and W^r_d(C) in case dim W^r_d(C) = d - 3r with d 【less than or equal】 g - 2. In 1999, Kato-Keem gave a characterization of C and W^r_d(C) in case dim W^r_d(C) = d - 3r - 1 with d 【less than or equal】 g - 4. In 2001, Kato remarked that even in the case gon(C) is even, if C doesn't have an involution, dim W^r_d(C) 【less than or equal】 d - 3r holds, too. Then, one has a characterization of C and W^r_d(C) in cases dim W^r_d(C) = d - 3r with d 【less than or equal】 g - 2 and dim W^r_d(C) = d - 3r - 1 with d 【less than or equal】 g - 4. It is one of the main results in our study supported by Grant-in-Aid for Scientific Research (C)(2), (2000-2001) entitled "A study on meromorphic functions on compact Riemann surfaces" #12640180. In this project, we almost succeeded a characterization of C and W^r_d(C) in cases dim W^r_d(C) = d - 3r - 2.2.Let F_q be a finite fields with q elements and C ⊂ F^n_q be a linear [n,k,d]_q code. Let n_q(k,d) be the minimum of the code lengths for fixed k,d. There is an upper bound of n_q(k,d) known as the Griesmer bound. In this project, we show that for some range of d's, n_q (k,d) is equal to the Griesmer bound minus 1.