Complementary Modules of Weierstrass Canonical Forms

IF 0.9 3区 物理与天体物理 Q2 MATHEMATICS
J. Komeda, S. Matsutani, E. Previato
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引用次数: 4

Abstract

The Weierstrass curve is a pointed curve $(X,\infty)$ with a numerical semigroup $H_X$, which is a normalization of the curve given by the Weierstrass canonical form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\dots + A_{r-1}(x) y + A_{r}(x)=0$ where each $A_j$ is a polynomial in $x$ of degree $\leq j s/r$ for certain coprime positive integers $r$ and $s$, $r$<$s$, such that the generators of the Weierstrass non-gap sequence $H_X$ at $\infty$ include $r$ and $s$. The Weierstrass curve has the projection $\varpi_r\colon X \to {\mathbb P}$, $(x,y)\mapsto x$, as a covering space. Let $R_X := {\mathbf H}^0(X, {\mathcal O}_X(*\infty))$ and $R_{\mathbb P} := {\mathbf H}^0({\mathbb P}, {\mathcal O}_{\mathbb P}(*\infty))$ whose affine part is ${\mathbb C}[x]$. In this paper, for every Weierstrass curve $X$, we show the explicit expression of the complementary module $R_X^{\mathfrak c}$ of $R_{\mathbb P}$-module $R_X$ as an extension of the expression of the plane Weierstrass curves by Kunz. The extension naturally leads the explicit expressions of the holomorphic one form except $\infty$, ${\mathbf H}^0({\mathbb P}, {\mathcal A}_{\mathbb P}(*\infty))$ in terms of $R_X$. Since for every compact Riemann surface, we find a Weierstrass curve that is bi-rational to the surface, we also comment that the explicit expression of $R_X^{\mathfrak c}$ naturally leads the algebraic construction of generalized Weierstrass' sigma functions for every compact Riemann surface and is also connected with the data on how the Riemann surface is embedded into the universal Grassmannian manifolds.
Weierstrass规范形式的互补模
Weierstrass曲线是具有数值半群$H_X$的尖曲线$(X,\infty)$,它是由Weierstras正则形式$y^r+a_{1}(X)y^{r-1}+a_{2}(X,使得在$\infty$处的Weierstrass非间隙序列$H_X$的生成器包括$r$和$s$。Weierstrass曲线具有投影$\varpi_r\colon X\ to{\mathbb P}$,$(X,y)\mapsto X$作为覆盖空间。设$R_X:={\mathbf H}^0(X,{\mathcal O}_X(*\infty))$和$R_。在本文中,对于每个Weierstrass曲线$X$,我们给出了$R_{\mathbb P}$-模$R_X$的互补模$R_X ^{\math frak c}$的显式表达式,作为Kunz平面Weierstras曲线表达式的推广。该扩展自然地导致全纯一形式的显式表达式,除了$\infty$,${\mathbf H}^0({\mathbb P},{\mathcal A}_{\mathbbP}(*\infty))$就$R_X$而言。由于对于每一个紧致Riemann曲面,我们都找到了一条对该曲面是双有理的Weierstrass曲线,我们还评论了$R_X^{\mathfrak c}$的显式表达式自然地导致了每个紧致黎曼曲面的广义Weierstrass西格玛函数的代数构造,并且还与关于黎曼曲面如何嵌入到通用Grassmann流形中的数据相联系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.80
自引率
0.00%
发文量
87
审稿时长
4-8 weeks
期刊介绍: Scope Geometrical methods in mathematical physics Lie theory and differential equations Classical and quantum integrable systems Algebraic methods in dynamical systems and chaos Exactly and quasi-exactly solvable models Lie groups and algebras, representation theory Orthogonal polynomials and special functions Integrable probability and stochastic processes Quantum algebras, quantum groups and their representations Symplectic, Poisson and noncommutative geometry Algebraic geometry and its applications Quantum field theories and string/gauge theories Statistical physics and condensed matter physics Quantum gravity and cosmology.
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