Types of connected components of the singular locus of the moduli space of compact Riemann surfaces
Year: 2024
Type: article
Abstract: In 2012 G. Bartolini and M. Izquierdo showed that all equisymmetric strata of Broughton’s stratification, corresponding to the actions of groups of order 2 and 3 on Riemann surfaces of genus g are contained in the same connected component $$\mathcal {C}_g^{2,3}$$ C g 2 , 3 of the singular locus $$\mathcal {S}_g$$ S g . The other components that were known to the literature were those composed of points of certain single strata. This motivated the question of whether the singular locus can contain connected components different from $$\mathcal {C}_g^{2,3}$$ C g 2 , 3 and being the union of at least two strata. The existence of such components was proved for infinitely many genera by G. Gromadzki and the author. In this paper, we will describe and classify all types of connected components of the singular locus. In particular, we prove that every connected component different than $$\mathcal {C}_g^{2,3}$$ C g 2 , 3 coincides with some Cornalba component, and we give necessary and sufficient conditions for a cyclic action to define such a component.
Source: Revista Matemática Complutense
Author: Jakub Szmelter
Institution: University of Gdańsk
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4403703894