Covers of the Sphere and Moduli of Curves

University Of Florida, Gainesville FL

Investigators

Abstract

Abstract for proposal # 0200225, Helmut Voelklein: Hurwitz spaces parametrize covers of the sphere of given ramification type and monodromy group. Each Hurwitz space has a natural map to the moduli space of genus g curves (for suitable g). This has been used in algebraic geometry for a long time, but only in the case of simple covers (which in particular have a symmetric group as monodromy group). Hurwitz spaces of covers with arbitrary monodromy group were constructed by Fried and Voelklein, and applied to the Inverse Galois problem. Proposed research explores how the group-theoretic methods associated with Hurwitz spaces can be applied in the study of the moduli of curves. This includes algorithmic methods of computational group theory, which are especially useful in the study of the braid group action on generating systems of a finite group. Generalizations replace the braid group by the mapping class group of a punctured surface of non-zero genus. Applications to the Inverse Galois Problem are to be expected. The project is partially in cooperation with G. Frey, K. Magaard and S. Shpectorov. Group Theory is the abstract study of symmetry patterns (of any object). Many algorithms of Group Theory are implemented in modern computer algebra systems. Proposed research uses these computer algebra systems to find and study families of highly symmetric algebraic curves. Algebraic curves are basic objects in mathematics, physics and applications, e.g., cryptography. Elliptic curve cryptography provides encryption schemes used for data security on the internet.