Theta divisor and Abel map for 4-by-4 matrices

Abstract

We consider a hyperbolic quartic curve F(x,y,z)=0 associated with a 4-by-4 nilpotent matrix which has one ordinary double point satisfying F(x,y,z)=F(x,−y,z). Let R be the Riemann surface determined as a non-singular model of the curve F(x,y,z)=0. In this situation, the genus of R is 2. We express the period matrix of R using explicit line integrals depending on a simple symplectic transformation of the canonical cycles which Maple algcurves package provided, and determine the Riemann constant of the Riemann vanishing theorem. We also provide a pictorial visualization of the real parts of the image of an Abel map and theta divisor.