On the structure of connected components of the moduli spaces of compact Riemann surfaces

Abstract

AbstractUsing the Broughton’s equisymmetric stratification of the moduli space of compact Riemann surfaces of genus $$g \ge 2$$ g ≥ 2 , Bartolini, Costa and Izquierdo have shown that its singular locus $$S_{g}$$ S g is disconnected except for $$g = 3, 4, 7, 13, 17, 19, 59$$ g = 3 , 4 , 7 , 13 , 17 , 19 , 59 . One of a lot of problems motivated by this result concerns the study of its connected components, both from the quantitative and qualitative points of view. In 2012, Bartolini and Izquierdo have shown that all strata corresponding to the actions of $$Z_{2}$$ Z 2 and $$Z_{3}$$ Z 3 are contained in a single connected component $$C^{2,3}_g$$ C g 2 , 3 . The other components which were known to these authors were those composed by single strata and they asked whether there are components different from $$C^{2,3}_ g $$ C g 2 , 3 being the union at least two equisymmetric strata. The aim of this paper is to give an affirmative answer to this question, by showing that there are connected components composed of precisely two strata in $$S_{g}$$ S g for $$g = rp(p -1)/2$$ g = r p ( p - 1 ) / 2 where $$r \ge 6$$ r ≥ 6 and $$p > 5r + 3$$ p > 5 r + 3 is a prime for which $$pr + 2$$ p r + 2 is also a prime.