Computation of Riemann matrices of hyperbolic quartic curves

Abstract

We classify the graphs which describe the monodromy structure of the Riemann surfaces of smooth quartic hyperbolic curves. The combinatorial classification provides a method to construct the homology basis for computing the Riemann period matrix of the Riemann surface of a hyperbolic quartic curve. We give a smooth quartic curve and produce a simpler Riemann matrix than that of the Vinnikov standard form.