On the moduli space of pointed algebraic curves of low genus: a computational approach
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 of \(N\), i.e. of the affine curve \(\text{Spec}\,{\mathbb C}[t^{a_1},t^{a_2},\dots,t^{a_n}]\), where \(\{a_1,a_2,\dots,a_n\}\) are generators for the semigroup \(N\). The authors use the algorithms implemented in the computer algebra system SINGULAR by \textit{B.~Martin} [in: Algorithmic algebra and number theory. Sel. pap. conf. Heidelberg 1997, 283--293 (1999; Zbl 0942.14002)], to compute these miniversal deformations up to genus 5. As a result they are able to compute (with a few exceptions) the dimension of \({\mathcal M}_{g,1}^N\) for \(2\leq g\leq 5\), and to prove its irreducibility. More precisely, they show that \({\mathcal M}_{g,1}^N\) is a non-empty Zariski open subset of a projective variety \(\overline{\mathcal M}_{g,1}^N\) which (again, up to few exceptions) turns out to be a weighted projective space; in particular, this implies that \({\mathcal M}_{g,1}^N\) is unirational for most numerical semigroups \(N\) of genus \(g\) with \(2\leq g\leq 5\).
scientific article; zbMATH DE number 2124832