Meromorphic functions 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^γ_d(C) be a subvariety which consists of the image of effective divisors of degree d and dimension γ 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^γ_d(C) 【less than or equal】 d - 3γ holds for any d 【less than or equal】 g - 1. In 1996, Martens gave a characterization of C and W^γ_d(C) in case dim W^γ_d(C) = d - 3γ held for d 【less than or equal】 g - 2. In 1999, Kato-Keem gave a characterization of C and W^γ_d(C) in case dim W^γ_d(C) = d - 3γ - 1 held for 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 (B)(2), (2000-2001) entitled "A study on meromorphic functions and Weierstrass points" #10440051. In the present study, first, we remark that even in … More the case gon(C) is even, if C doesn't have an involution, dim W^γ_d(C) 【less than or equal】 d - 3γ holds, too. Then, we give a chracterization of C and W^γ_d(C) in case dim W^γ_d(C) = d - 3γ, d 【less than or equal】 g - 2 and dim W^γ_d(C) = d - 3γ - 1, d 【less than or equal】 g - 4.2. We study projective systems concerned with the error-correcting coding theory, in particular the algebraic geometry codes. Let S be a set in the projective space over a finite field with q elements. Then, we prove that S is a union of 2 (resp. 3) subspaces provided that the numbers of points which intersect with hyperplanes satisfy some conditions. It is an improvement of a result of Homma-Kim-Yoo.We study a characterization of the Fermat curve in terms of total inflection points on smooth plane curve. The Fermat curve of degree d has 3d total inflection points. We consider the converse and prove that if a smooth plane curve of degree 5 has an automorphism of order 5 and 15 total inflection points, then it is birationally equivalent to the Fermat curve. Less