We define a class of numerical semigroups S, which we call Castelnuovo semigroups, and study the subvariety ${\mathcal{M}}_{g,1}^S$ of ${\mathcal{M}}_{g,1}$ consisting of marked smooth curves with Weierstrass semigroup S. We determine the number of irreducible components of these loci and determine their dimensions. Curves with these Weierstrass semigroups are always Castelnuovo curves, which provides the basic tool for our argument. This analysis provides examples of numerical semigroups for which ${\mathcal{M}}_{g,1}^S$ is reducible and non-equidimensional.

中文翻译：

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Castelnuovo曲线上的Weierstrass半群

我们定义一类数字半群S，我们将其称为Castelnuovo半群，并研究子变量${\mathcal{中号}}_{G，\mathrm{1个}}^{\mathrm{小号}}$ 的 ${\mathcal{中号}}_{G，\mathrm{1个}}$由带有Weierstrass半群S的明显的光滑曲线组成。我们确定这些基因座的不可还原成分的数量，并确定其尺寸。具有这些Weierstrass半群的曲线始终是Castelnuovo曲线，这为我们的论证提供了基本工具。该分析提供了数值半群的示例，${\mathcal{中号}}_{G，\mathrm{1个}}^{\mathrm{小号}}$ 是可约且非等维的。