A stratification of the moduli space of pointed non-singular curves (Q1646624)

Let \(N\) be a numerical semigroup of genus \(g\geq 4\) whose set of gaps \(\mathbb N_0\setminus N=\{\ell_1<\ldots<\ell_g\}\) satisfy \(\ell_2=2\) and \(\ell_g=2g-3\). It is important to notice the existence of two gaps of \(N\), say \(\lambda, \mu\), which are uniquely determined by \(\mu+\lambda=\ell_g\) and \(\mu<\lambda\). In the paper under review, the authors study the moduli space \(\mathcal M\) of genus \(g\) (non-singular, projective, irreducible over an algebraically closed field \(k\)) algebraic pointed curves \((X,P)\) being \(N\) the Weierstrass semigroup at \(P\). [Let us point out that we can assume that \(X\) is contained in the \(g\)-dimensional projective space over \(k\) as a curve of degree \(2g-2\)]. They follow closely \textit{K.-O. Stöhr}'s approach [J. Reine Angew. Math. 441, 189--213 (1993; Zbl 0771.14009)] (case of a symmetric semigroup or \(\ell_g=2g-1\)); cf. [\textit{G. Oliveira} and \textit{K.-O. Stöhr}, Geom. Dedicata 67, No. 1, 65--82 (1997; Zbl 0904.14019)] (case of a pseudo-symmetric semigroup or \(\ell_g=2g-2\)). In particular, Gröbner basis techniques are considered. The main result is the existence of a stratification \(\mathcal M=\mathcal M_0\cup\mathcal M_1\) by means of two disjoint locally closed varieties which are closely related to the aforementioned gaps \(\mu\) and \(\lambda\). To explain this one is led to consider the sequence of positive divisors of the curve \(X\), \(E_{\ell_2}\leq\ldots\leq E_{\ell_g}\), where \(E_{\ell_i}:=X\cdot T^{(i)}-(\ell_i-1)P\) with \(T^{(i)}\) being the \(i\)-th osculating space at \(P\). By results of \textit{F. L. R. Pimentel} [Geom. Dedicata 85, No. 1--3, 125--134 (2001; Zbl 0991.14014)] and \textit{N. Medeiros} [J. Pure Appl. Algebra 170, No. 2--3, 267--285 (2002; Zbl 1039.14015)], \(E_\mu=0\) and \(\deg(E_\lambda)\in\{0,1\}\). Thus \(\mathcal M_0\) (resp. \(\mathcal M_1\)) is made of those pointed curves where \(\deg(E_\lambda)=0\) (resp. \(\deg(E_\lambda)=1\)). As a matter of fact in both sets \(\mathcal M_i\) Stöhr's method can be applied, and the following useful arithmetical criterion: \(\lambda-\mu\) is also a gap at \(P\) and \(2\mu\geq \lambda\) whenever \(E_\lambda)\neq 0\) can be used to decide the emptiness property of the strata.