Linear Systems and Projective Models on Algebraic Curves

它成立s_c(2)≦g+ 2对于任何代数曲线C有g属，其中。s_c(2)是C的平面模型的最小度。Martens和Keem推测s_c(2)=g-t+ 2对t属曲线有双重覆盖的代数曲线C成立。在本研究中，我们可以得出当t= 0,1和t=2且g≥10时，Martens和Keem的猜想成立，但对于每g≤9时，我们都可以找到一个重要的反例。2009年12月05-06日，我们在横滨国立大学召开了第七届代数曲线及相关主题年会；2010年12月11-12日，我们在埼玉大学召开了第八届代数曲线及相关主题年会；2011年12月10-11日，我们分别在东京都大学召开了第九届代数曲线及相关主题年会。关于代数曲线平面模型的最小次，我们得到了一些很好的结果。

IIt holds s_c(2)≦g+ 2 for any algebraic curve C with genus g where. s_c(2) is the minimal degree of plane models of C. Martens and Keem conjectured that s_c(2)=g-t+ 2holds for an algebraic curve C admitting a double covering to a curve of genus t. In this research, we can conclude that Martens and Keem conjecture is true for t=0, 1 and t=2 with g≧10, however we can find an important counter examples for every g≦9. On 05-06, December, 2009, we held 7th Annual Conference of Algebraic Curves and Related Topics at Yokohama National Universiy, on 11-12, December, 2010, we held 8th Annual Conference of Algebraic Curves and Related Topics at Saitama Universiy and on 10-11, December, 2011, we held 9th Annual Conference of Algebraic Curves and Related Topics at Tokyo Metropolitan Universiy, respectively. We can get some good results about the minimal degree for plane models o f an algebraic curve.