Abstract
A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.
Highlights
Our first example of a theta surface is Scherk’s minimal surface (Fig. 1), given by the equation sin(X) − sin(Y ) · exp(Z) = 0. (1)This surface arises from the following quartic curve in the complex projective plane P2: xy(x2 + y2 + z2) = 0. (2)We use X, Y, Z as affine coordinates for R3 and x, y, z as homogeneous coordinates for P2
Building on state-ofthe-art methods for evaluating abelian integrals and theta functions, we develop a numerical algorithm whose input is a smooth quartic curve in P2 and whose output is its theta surface
Theorem 3 (Lie’s Theorem) The arcs C1, C2, C3, C4 lie on a common reduced quartic curve Q in the projective plane P2, and S coincides with the theta surface associated to Q
Summary
Our first example of a theta surface is Scherk’s minimal surface (Fig. 1), given by the equation sin(X) − sin(Y ) · exp(Z) = 0. This surface arises from the following quartic curve in the complex projective plane P2: xy(x2 + y2 + z2) = 0. This leads to the formula in Theorem 4 for the implicit equation of a degenerate theta surface. Building on state-ofthe-art methods for evaluating abelian integrals and theta functions, we develop a numerical algorithm whose input is a smooth quartic curve in P2 and whose output is its theta surface. Double translation surfaces represent invariants in the study of 4-wave interactions in [4]
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