On the moduli space of pointed algebraic curves of low genus: a computational approach
type: Item
instance of: scholarly article
title: On the moduli space of pointed algebraic curves of low genus: a computational approach
zbMATH Open document ID: 1077.14037
author: Tetsuo Nakano
author: Tatsuji Mori
published in: Tokyo Journal of Mathematics
publication date: 2005-01-04T00:00:00Z
review text: Let \(X\) be a nonsingular projective curve of genus \(g\) and \(P\) a point of \(X\). The semigroup \(N_P\) is defined as the subsemigroup of nonnegative integers consisting of those \(n\) such that there exists a rational function \(f\) on \(X\) which is regular on \(X\setminus P\) and has a pole of order exactly \(n\) at \(P\). The difference \({\mathbb N}\setminus N_P=\{j_1,j_2,\dots,j_g\}\) is known to be a finite set of cardinality \(g\). The increasing sequence \(j_1<j_2<\cdots<j_g\) is called the gap sequence at \(P\). A semigroup \(N\) of \({\mathbb N}\) such that \({\mathbb N}\setminus N\) has cardinality \(g\) is called a numerical semigroup of genus \(g\); it can be shown that, for any numerical semigroup \(N\) of genus \(g\), the gap sequence of \(N\) satisfies \(j_i\leq 2i-1\), and the minimal element \(\mu\) of \(N\) satisfies \(2\leq\mu\leq g+1\). In particular, for each genus \(g\) there are only finitely many numerical subgroups \(N_1,N_2,\dots, N_{l_g}\) of genus \(g\). Consider now the moduli space \({\mathcal M}_{g,1}\) of genus \(g\) nonsingular curves with one marked point, with \(g\geq 2\). To a point \((X,P)\) in \({\mathcal M}_{g,1}\) one can associate the genus \(g\) numerical subgroup \(N_P\). Thus, one obtains a direct sum decomposition \({\mathcal M}_{g,1}={\mathcal M}_{g,1}^{N_1}\cup{\mathcal M}_{g,1}^{N_2}\cup\cdots\cup {\mathcal M}_{g,1}^{N_{l_g}}\), where \({\mathcal M}_{g,1}^N\) is the subscheme of \({\mathcal M}_{g,1}\) consisting of points \((X,P)\) such that \(N_P=N\). The moduli space \({\mathcal M}_{g,1}^N\) is known to be nonempty for any numerical subgroup of genus \(g\leq 7\) [\textit{M.~Haure}, Ann. de l'Éc. Norm. (3) 13, 115--196 (1896; JFM 27.0468.02); \textit{J.~Komeda}, J. Pure Appl. Algebra 97, No.1, 51--71 (1994; Zbl 0849.14011)]. Moreover \textit{H~.C.~Pinkham} [Astérisque 20, 1--131 (1974; Zbl 0304.14006)] showed that \({\mathcal M}_{g,1}^N\) can be constructed from the miniversal deformation of the monomial curve …
reviewed by: Domenico Fiorenza
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14Q99
Mathematics Subject Classification ID: 14H55
zbMATH DE Number: 2124832
zbMATH Keywords: moduli spaces of curves
zbMATH Keywords: gap sequences
zbMATH Keywords: SINGULAR
describes a project that uses: SINGULAR
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208488
OpenAlex ID: W1996316103
DOI: 10.3836/TJM/1244208488
description: scientific article; zbMATH DE number 2124832