On the Castelnuovo-Severi inequality for Riemann surfaces

Abstract

Some consequences of equality in the Castelnuovo-Severi inequality are discussed. In particular, it is shown that if a Riemann surface of genus ten, W₁₀, admits four coverings of tori, each in three sheets, then W₁₀ admits an elementary abelian group of order 27. By previous work this last result is then characterized by the vanishing of certain thetanulls. An elementary discussion of the direct product of monodromy groups is an essential part of the proofs.