Filament geometry

About this book

The topic that serves as a springboard of this text is that of classifying equivariant maps from a compact Riemann surface to projective space, each acted on by a finite group, the first conformally and the second linearly. The theory applies group cohomology, invariant divisor calculations, and particularly the Chevalley-Weil theorem. The Chevalley-Weil theorem, proven in 1934, is a sort of equivariant Riemann-Roch theorem that describes how to decompose a complete linear system with a linear group action into isotypic components. The theory described here goes beyond the Chevalley-Weil theorem by studying not just the isotypic components, but how they generate each other and interrelate to one another in complex conditions of dependency. Our principal results are: (1) Chapter 3, Theorem 1.12, which proves that for any equivariant pair of Riemann surface and projective representation, equivariant maps exist. If the projective representation is not trivial, then infinitely many such maps exist. (2) The material in chapter 5, that assigns a fine divisor class invariant to every such equivariant map and explains constructively how to calculate with it. (3) All of the equivariant maps with a particular divisor class are contained in a unique family with a restrictive sort of linear variation. (4) All equivariant maps that are not "ancestral" are contained within a family of the type just mentioned that is "generated" by ancestral maps.