Kummer Surfaces: 200 Years of Study Igor Dolgachev The fascinating story about the Kummer surface starts from material where double refraction occurs: a ray of light the discovery by Augustin-Jean Fresnel in 1822 of the equa- splits into two, traveling at the same speed along different tion describing the propagation of light in an optically paths. The speed of light may depend on the coordinates biaxial crystal [Fre]. Biaxial crystals are an example of 푥 = (푥1, 푥2, 푥3) of a point and the unit direction vector 휉 = (휉1, 휉2, 휉3). The propagation of light is described by Igor Dolgachev is a professor of mathematics, emeritus, at the University of the speed 푣(푥, 휉) at 푥 in the direction 휉. We say that the Michigan. His email address is [email protected]. matter is homogeneous if 푣 does not depend on 푥 and we The article is based on the author’s Oliver Club talk at Cornell University de- livered on October 10, 2019, exactly 121 years since the first meeting of the say that it is isotropic if it does not depend on 휉. For exam- club, at which then the faculty member John Hutchinson spoke. As we will see, ple, while a student, James Maxwell described a lens that Hutchinson contributed significantly to the theory of Kummer surfaces. reminded him of the eyes of a fish. Through his fish eye, Communicated by Notices Associate Editor Angela Gibney. he found that light is inhomogeneous but isotropic, bend- For permission to reprint this article, please contact: ing in arcs whose shape depends on where they start and [email protected]. converging to a single point. DOI: https://doi.org/10.1090/noti2168 NOVEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1527 Figure 1. Augustin-Jean Fresnel. A biaxial crystal gives an example of homogeneous but Figure 2. Sir William Hamilton. anisotropic propagation. Fresnel found the equation of the propagation of light in such a crystal to be of the form: planes forming a tetrahedron such that each plane cuts out 휉2 휉2 휉2 1 + 2 + 3 = 0, (1) the surface along a pair of conics. The three vertices of the −2 −2 −2 −2 −2 −2 푎1 − 푣 푎2 − 푣 푎3 − 푣 tetrahedron are conjugate with respect to the two conics where 푎1, 푎2, 푎3 are constants describing the property of lying in the face of the tetrahedron they span. Moreover the crystal (principal refraction indices). By substituting 16 intersection points of four pairs of conics are singular (푧1, 푧2, 푧3) = 푣(휉1, 휉2, 휉3), one can rewrite the previous points of the surface. Cayley also discovered an important equation as property of tetraedroid quartic surfaces: they are projec- tively self-dual (or reciprocal). Wave surfaces were the sub- 푎2푧2 푎 푧2 1 1 + 2 2 ject of study for many famous mathematicians of the 19th 2 2 2 2 2 2 2 2 푧1 + 푧2 + 푧3 − 푎1 푧1 + 푧2 + 푧 − 푎2 century. 푎 푧2 Among them were A. Cauchy, J. Darboux, J. MacCul- + 3 3 = 0. 2 2 2 2 lagh, J. Sylvester, and W. Hamilton (see [Lor96, pp. 114– 푧1 + 푧2 + 푧3 − 푎3 115]). A nice modern exposition of the theory of Fresnel’s After clearing the denominators and homogenizing, we wave surfaces can be found in [Knö86]. find an equation of a quartic surface in ℙ3. In 1833 Sir Projective equivalence classes of Fresnel’s wave surfaces William Hamilton discovered that the surface has four real depend on two parameters and as we shall see momen- singular points tarily, these are examples of Kummer surfaces which are 2 2 2 2 determined by three parameters. 푎1 − 푎2 푎2 − 푎3 (±푎3 , 0, ±푎1 , 1) In 1847 Adolph Göpel, using the transcendental theory of √푎2 − 푎2 √푎2 − 푎2 1 3 1 3 theta functions, had found a relation of order four between and also four real planes 훼푥 + 훽푦 + 훾푧 + 푤 = 0 that cut out theta functions of second order in two variables that ex- the surface along a conic (trope-conics), where presses an equation of a general Kummer surface [Göp47]. 2 2 2 2 To give Göpel’s equation, we next briefly discuss theta func- 푎3 푎1 − 푎2 푎1 푎2 − 푎3 (훼, 훽, 훾, 1) = (± 2 2 2 , 0, ± 2 2 2 , 1) tions. 푎2 √푎1 − 푎3 푎2 √푎1 − 푎3 Let 푇 = ℂ푔/Λ be a compact 푔-dimensional torus, the [Ham37, p. 134]. In fact, as we shall see later, over ℂ, it quotient of ℂ푔 by the group of translations Λ isomor- has additionally 12 nodes and 12 trope-conics. phic to ℤ2푔. There are no nonconstant holomorphic func- In 1849 Arthur Cayley proved that Fresnel’s wave surface tions on 푇 because it is a compact complex manifold; in- is a special kind of a tetraedroid quartic surface. The latter stead one considers nonzero holomorphic sections of a is characterized by the following property. There are four holomorphic line bundle 퐿 on 푇. For general Λ no line 1528 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 10 Figure 4. Ernst Kummer. of zeros in ℙ3 of a quartic polynomial Figure 3. Arthur Cayley. 퐹 = 퐴(푥4 + 푦4 + 푧4 + 푤4) + 2퐵(푥2푦2 + 푧2푤2) bundle has enough sections to embed 푇 into a projective + 2퐶(푥2푧2 + 푦2푤2) + 2퐷(푥2푤2 + 푦2푧2) + 4퐸푥푦푧푤, space, i.e., 푇 does not admit a structure of a projective alge- braic variety. However, in 1857 Bernard Riemann found where the coefficients (퐴, 퐵, 퐶, 퐷, 퐸) satisfy a certain ex- a condition on Λ such that 푇 admits a line bundle 퐿 with plicit equation in terms of theta constants, the values of dim Γ(푇, 퐿푛) = 푛푔 and whose sections embed 푇 onto a pro- theta functions at 0. The images of 16 two-torsion points 1 jective space. Such complex tori are now called principally 휖 ∈ Λ/Λ ∈ ℂ2/Λ are singular points of 푋. The Abel-Jacobi 2 1 푥 푥 polarized abelian varieties. They depend on 푔(푔 + 1) com- map 퐶 → Jac(퐶), 푥 ↦ (∫ 휔 , ∫ 휔 ) mod Λ embeds 퐶 2 푥0 1 푥0 2 plex parameters. Holomorphic sections of 퐿푛 can be lifted into Jac(퐶) and the images of the curves 퐶+휖 = {푐+휖, 푐 ∈ 퐶} to holomorphic functions on ℂ푔 that are Λ-invariant up are the 16 trope-conics of 푋. to some multiplicative factor. They are called theta func- In 1864 Ernst Kummer had shown that 16-nodal quar- tions of order 푛. For 푛 = 1, such a holomorphic func- tic surfaces depend on three complex parameters and Fres- tion is the famous Riemann theta function Θ(푧, Λ). One nel’s wave surface represents only a special case of these can modify Riemann’s expression for Θ(푧, Λ) to include quartic surfaces. Kummer proved that such surfaces con- some parameters which are elements (푚, 푚′) of the group tain 16 trope-conics which together with 16 nodes form 푔 푔 (ℤ/푛ℤ) ⊕ (ℤ/푛ℤ) , called theta characteristics. They gen- an abstract incidence configuration (166) (this means that erate the linear space Γ(푇, 퐿푛). each node lies on six trope-conics and each trope-conic An example of a principally polarized abelian variety is contains six nodes) [Kum64]. Kummer shows that any 16- the Jacobian variety Jac(퐶) of a Riemann surface 퐶 of genus nodal quartic surface has a tetrahedron with conic-tropes 푔. Here Λ is spanned by vectors 푣 = (∫ 휔 ,…, ∫ 휔 ) ∈ in the faces intersecting at two points on the edges. More- 푖 훾푖 1 훾푖 푔 푔 ℂ , 푖 = 1, … , 2푔, where (휔1, … , 휔푔) is a basis of the linear over, he proved that no vertex can be a node. From this he space of holomorphic differential 1-forms and (훾1, … , 훾2푔) deduced that there exists a quadric surface that contains is a basis of 퐻1(푇, ℤ). the four trope-conics. Using this observation, he found an Göpel was able to find a special basis (휃0, 휃1, 휃2, 휃3) in equation of a general Kummer surface of the form the space Γ(Jac(퐶), 퐿2) ≅ ℂ4 such that the map (푥2 + 푦2 + 푧2 + 푤2 + 푎(푥푦 + 푧푤) + 푏(푥푧 + 푦푤) Φ ∶ 푇 → ℙ3, (푧 , 푧 ) ↦ (휃 (푧) ∶ 휃 (푧) ∶ 휃 (푧) ∶ 휃 (푧)) 1 2 0 1 2 3 +푐(푥푤 + 푦푧))2 + 퐾푥푦푧푤 = 0, satisfies Φ(−푧1, −푧2) = Φ(푧1, 푧2) and its image 푋 is the set where 푥푦푧푤 = 0 is the equation of a chosen tetrahedron NOVEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1529 Figure 5. Carl Borchardt. Figure 6. Felix Klein. and 퐾 = 푎2 + 푏2 + 푐2 − 2푎푏푐 − 1. Since then any 16-nodal quartic surface became known as a Kummer quartic surface. The first book entirely devoted to Kummer surfaces It took almost 30 years since Göpel’s discovery to real- that combines geometric, algebraic, and transcendental ap- ize that the Göpel equation, after a linear change of vari- proaches to their study was published by Ronald Hudson in ables, can be reduced to a Kummer equation. This was 1905 [Hud05]. done by Carl Borchardt in 1877 [Bor77]. In fact, Göpel’s As I understand, Kummer’s interest in 16-nodal quar- discovery leads to a modern definition of the Kummer sur- tic surfaces arose from his pioneering study of two- face and its higher-dimensional version, the Kummer va- dimensional families (congruences) of lines in ℙ3. They riety. One considers any 푔-dimensional complex torus 푇 are naturally parameterized by irreducible surfaces in the 3 3 and divides it by the involution 푧 ↦ −푧; the orbit space is Grassmann variety 퐺1(ℙ ) of lines in ℙ .