Theta Surfaces
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Theta Surfaces ¨ um ¨ u¨ Ozl ¨ C¸elik2,3 · Julia Struwe2,3 · Bernd Sturmfels3,4 Daniele Agostini1 · Turk Received: 31 January 2020 / Accepted: 20 June 2020 / © The Author(s) 2020
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 Poincar´e 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. Keywords Translation surface · Abelian integral · Riemann theta function · Theta divisor Mathematics Subject Classification (2010) 14K25 · 14H40 · 01A55 · 37K10
1 Introduction 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.
Dedicated to J¨urgen Jost on the occasion of his 65th birthday. ¨ um C T¨urk¨u Ozl¨ ¸ elik [email protected] Daniele Agostini [email protected] Julia Struwe [email protected] Bernd Sturmfels [email protected] 1
Humboldt-Universit¨at zu Berlin, 10117, Berlin, Germany
2
Universit¨at Leipzig, 04109, Leipzig, Germany
3
Max Planck Institute for Mathematics in the Sciences, 04103, Leipzig, Germany
4
UC Berkeley, Berkeley, CA, USA
(1)
D. Agostini et al.
This surface arises from the following quartic curve in the complex projective plane P2 : xy(x 2 + y 2 + z2 ) = 0.
(2)
We use X, Y , Z as affine coordinates for R3 and x, y, z as homogeneous coordinates for P2 . Scherk’s minimal surface is obtained as the Minkowski sum of two parametric space curves: (X, Y, Z) = arctan(s), 0, log(s) − log(s 2 + 1)/2 (3) + 0, − arctan(t), − log(t) + log(t 2 + 1)/2 . The derivation of (1) from (2) via (3) is given in Example 1. This computation is originally due to Richard Kummer [18, p. 52] whose 1894 dissertation also displays a plaster model. Following the classical literature (cf. [21, 28]), a surface of double translation equals
S = C1 + C2 = C3 + C4 , where C1 , C2 , C3 , C4 are curves in R3 , and the two decompositions are distinct. We note that Scherk’s minimal surface (1) is a surface of double translation. A first representation S = C1 + C2 was given in (3). A second representation S = C3 + C4 equals u+v , X = arctan(u) + arctan(v) = arctan 1−uv 5u+5v , Y = arctan(5u) + arctan(5v) = arctan 1−25uv 2 2 1 1 Z = log 1+(5u) + log 1+(5v) . (4) 5(1+u2 ) 5(1+v 2 ) 2 2 It is instructive to verify that both parametrizations (3) and (4) satisfy the equation (1). A remarkable theorem due to Sophus Lie [20], refined by Henri Poincar´e in [27], states that these are precisely the surface
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