{"title":"Complementary Modules of Weierstrass Canonical Forms","authors":"J. Komeda, S. Matsutani, E. Previato","doi":"10.3842/SIGMA.2022.098","DOIUrl":null,"url":null,"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.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Symmetry Integrability and Geometry-Methods and Applications","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.3842/SIGMA.2022.098","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.
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.