Normal forms for Kummer surfaces

TL;DR

This paper derives explicit normal forms for Kummer surfaces linked to specific abelian surfaces, providing formulas in terms of Theta functions and exploring their connection to Riemann identities.

Contribution

It introduces explicit normal forms for Kummer surfaces of various polarization types and relates them to Theta functions and Riemann identities.

Findings

Explicit formulas for Kummer surface coordinates and moduli parameters.

Normal forms are connected to generalized Riemann identities for Theta functions.

Provides a comprehensive framework for understanding Kummer surfaces in algebraic geometry.

Abstract

We determine normal forms for the Kummer surfaces associated with abelian surfaces of polarization of type $(1,1)$, $(1,2)$, $(2,2)$, $(2,4)$, and $(1,4)$. Explicit formulas for coordinates and moduli parameters in terms of Theta functions of genus two are also given. The normal forms in question are closely connected to the generalized Riemann identities for Theta functions of Mumford's.