Special linear systems on compact Riemann surfaces

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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.