具有非恒等自同构代数曲线的两个双有理不变量的研究

This research aims to consider the algebraic curves with non-trivial automorphism group and study the relationship between the automorphism structure and the birational invariants gonality and Clifford index. Gonality and Clifford index play very important roles in the field of algebraic curve, but the research on them are quite difficult. On the other hand, the relationship between non-trivial automorphisms and birational invariants on the algebraic curves is attracting more and more interest in recent years. So our research not only focuses on some difficult problems in our field but also conforms to the current tend. Based on our earlier results, we plan to develop three methods: (1), we want to generalize a theory developed by M. Coppens around 2000 so that it can be used in more cases; (2), we need to consider the automorphisms on the space W_d^r induced from the automorphisms on the original curve; (3) we try to apply the methods in the research on the automorphism of Jacobian variety to our research. By developing these methods, we can solve (at least partly) four interesting questions on algebraic curves (stated in the main body of this application). And we wish that our work can provide a clearer understanding on the birational classification and the moduli space of algebraic curves with non-trivial automorphisms.

本项目旨在研究具有非恒等自同构代数曲线的gonality与Clifford指标这两个双有理不变量以及与之相关的若干问题。Gonality与Clifford指标在代数曲线研究中有着极其重要的地位，同时也是该领域中比较困难的研究课题，因而长期备受关注；而代数曲线的自同构与代数曲线双有理不变量之间的关系则是近年来正变得越来越热门的研究方向。因此，我们的研究既注重难点问题的探索，又契合目前的研究趋势。在前期研究的基础上，我们针对四个比较具体的课题（见正文），提出了三个较为可行的方案：1，对M. Coppens在2000年左右发展出的一套理论进行扩展，使之能适用于更广泛的情况；2，研究由曲线自同构诱导出的W_d^r空间上的自同构；3，借鉴曲线雅可比簇自同构的研究方法。我们希望通过这三个方法的发展以及四个课题的研究，能够对具有非恒等自同构的代数曲线的双有理分类以及其模空间构造有一个更深刻的认识。

本项目旨在研究具有非平凡自同构群的代数曲线的若干双有理不变量。在代数曲线的研究领域中，双有理不变量是十分关键也十分困难的课题，而具有非平凡自同构的代数曲线则一直以来是该领域的重要研究对象。..我们的研究以双有理不变量gonality为出发点，并同时考虑代数曲线的Clifford index、曲线上点的Weierstrass gap sequence等。本项目的主要成果包括两项：首先，我们完成了自同构作用下商曲线为有理曲线情况下，gonality为4的曲线的分类。其次，我们发展了del Pozo于2006年关于自同构作用固定点处Weierstrass gap sequence的研究，给出了此类点处Weierstrass weight取得最小值的充分必要条件。..前项成果是此类曲线中首个非素数gonality的完整分类，为今后此类曲线分类问题的研究打下了基础；而后项成果则不仅提供了一个估计曲线自同构固定点处Weierstrass weight的方法，而且展示了Weierstrass gap sequence领域中一些非传统方法的巨大作用。..本项目执行期间，我们撰写论文4篇：截止提交本报告时，正式发表一篇、被接收两篇、投稿中一篇。