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 algebraic 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 . In [Kum66] Kum- called the Kummer variety of 푇 and is denoted by Kum(푇). mer gives a classification of quadratic line congruences It was shown by Andre Weil that 푇 has always a structure of (i.e., congruences of lines such that through a general point a Kähler manifold. If 푇 admits an embedding into a pro- 푥 ∈ ℙ3 passes exactly two lines from the congruence). Let jective space, then Kum(푇) is a projective algebraic variety 푛 be the class of the congruence, i.e., the number of lines with 22푔 singular points. Moreover, if 푇 is a principally of the congruence that lie in a general plane. Kummer had polarized abelian variety, for example the Jacobian variety shown that 2 ≤ 푛 ≤ 7 and when 푛 = 2 all the lines are of a genus 푔 algebraic curve, one can embed Kum(푇) into tangent to a Kummer quartic surface at two points. There 푔 the projective space ℙ2 −1 to obtain a self-dual subvariety are six congruences like that whose lines are tangent to the of degree 2푔−1푔! with 22푔 singular points and 22푔 trope hy- same Kummer surface. perplanes. If 푔 ≥ 3, each intersects it with multiplicity 2 In 1870, Felix Klein in his dissertation develops a beau- along a subvariety isomorphic to the Kummer variety of a tiful relationship between the Kummer surfaces and qua- (푔 − 1)-dimensional principally polarized abelian variety. dratic line complexes [Kle70]. A quadratic line complex There is also a generalized Kummer variety of complex di- is the intersection 푋 = 퐺 ∩ 푄 of the Grassmann quadric 3 5 mension 2푟 introduced by Arnaud Beauville in 1983. It is 퐺 = 퐺1(ℙ ) in the Plücker space ℙ with another quadric a nonsingular compact holomorphic symplectic manifold hypersurface. For any point 푥 ∈ ℙ3 the set of lines pass- birationally isomorphic to the kernel of the addition map ing through 푥 is a plane 휎푥 contained in 퐺. Its intersection 퐴(푟+1) → 퐴, where 퐴 is an abelian variety of dimension 2 with 푋 is a conic. The locus of points 푥 such that this conic (푟+1) 푟 and 퐴 = 퐴 /픖푟+1 is its symmetric product. The family becomes reducible is the singular surface of the complex, of such surfaces is one of a few known examples of com- and Klein had shown that it is a Kummer surface if the in- plete families of compact holomorphic symplectic mani- tersection 퐺 ∩ 푄 is transversal. Its 16 nodes correspond to folds of dimension 2푟. points where 휎푥 ∩ 푋 is a double line. In fact, Klein shows

1530 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 10 that the set of lines in 푋 is parameterized by the Jacobian of 퐶, Jac(퐶), where 퐶 is the Riemann surface of genus 2, the double cover of ℙ1 realized as the pencil of quadrics spanned by 퐺 and 푄 ramified along six points corresponding to six singular quadrics in the pencil. The Kummer surface is isomorphic to Kum(Jac(퐶)). By a simultaneous diagonalization of quadrics 퐺 and 푄, Klein shows that the Kummer surface admits a birational nonsingular model as a complete intersection of three quadrics

6 6 6 2 2 2 2 ∑ 푥푖 = ∑ 푎푖푥푖 = ∑ 푎푖 푥푖 = 0, 푖=1 푖=1 푖=1

2 where 푦 = (푥 − 푎1) ⋯ (푥 − 푎6) is the genus 2 Riemann surface 퐶 from above. Instead of 16 nodes and 16 trope-conics we now have two sets of 16 skew lines that form an abstract incidence configuration (166) isomorphic to the Kummer configuration of 16 nodes and 16 trope-conics. Klein also shows that Fresnel’s wave surface is characterized by the condition that 퐶 is bielliptic, i.e., it admits an involution with quotient an elliptic curve. By degenerating the quadratic complex to a tetrahedral quadratic complex one ob- Figure 7. John Hutchinson. tains Cayley’s tetraedroid quartic surface. In Göpel’s equation of a Kummer surface from above this corresponds to He proves that these 60 transformations generate an in- vanishing of the coefficient 퐸. Jessop’s book [Jes03] gives finite discontinuous group. However, Hutchinson did not an exposition of the works of Kummer and Klein on the re- address the question of whether adding these new trans- lationship between line geometry and Kummer surfaces. formations to Klein transformations would generate the Klein’s equations of a Kummer surface exhibit obvi- whole group Bir(푋). ous symmetry defined by changing signs of the variables. In 1850 Thomas Weddle, correcting a mistake of Michel These symmetries generate an elementary abelian 2-group Chasles, noticed that the locus of singular points of 25, the direct sum of five copies of the cyclic group ofor- quadric surfaces passing through a fixed set of six general points in ℙ3 is a quartic surface with singular points at the der 2. The quartic model also admits 16 involutions 푡푖 de- six points [Wed50]. It contains 15 lines joining pairs of fined by the projection from the nodes 푝푖 (take a general the points and the unique twisted cubic curve through the point 푥, join it with the node 푝푖, and then define 푡푖(푥) to be the residual intersection point). They are birational trans- six nodes (it seems that Chasles asserted that there is noth- formations, i.e., defined by an invertible rational change ing else except the twisted cubic). Different equations of of variables which may be not defined on some locus of the Weddle surface were given by Arthur Cayley in 1861, zeros of a finite set of polynomials. In 1886 Klein asked Carl Hierholzer in 1871, and Eugen Hunuady in 1882. The whether the group of birational automorphisms Bir(푋) of simplest equation was given later by F. Caspari [Cas91]: a (general) Kummer surface is generated by the group 25 푎푎′푦푧푤 푥 푎 푎′ ⎛ ⎞ and the projection involutions [Kle86]. In the remainder, 푏푏′푥푧푤 푦 푏 푏′ det ⎜ ⎟ = 0. we will discuss the progress on this problem. ⎜푐푐′푥푦푤 푧 푐 푐′ ⎟ In 1901 John Hutchinson, using the theory of theta func- ⎝푑푑′푥푦푧 푤 푑 푑′⎠ tions, showed that a choice of one of 60 Göpel tetrad of Here the six points are the points (1 ∶ 0 ∶ 0 ∶ 0), (0 ∶ nodes (means the tetrahedron with the tetrad as its vertices 1 ∶ 0 ∶ 0), (0 ∶ 0 ∶ 1 ∶ 0), (0 ∶ 0 ∶ 0 ∶ 1) and two has no trope-conics on its faces ) leads to an equation of points with coordinates given in the last two columns of the Kummer surface of the form the matrix. It is immediate to see that, after change of variables 푥 = 푎푎′푥′, 푦 = 푏푏′푦′, 푧 = 푐푐′푧, 푤 = 푑푑′푤′, the same 푞(푥1푥2 + 푥3푥4, 푥1푥3 + 푥2푥4, 푥1푥4 + 푥2푥3) + 푐푥1푥2푥3푥4 = 0, inversion transformation used by Hutchinson leaves the where 푞 is a quadratic form in three variables [Hut01]. He Weddle surface invariant. observed that the transformation 푥푖 ↦ 1/푥푖 leaves this In 1889 Friedrich Schottky, using the theory of theta func- equation invariant, and hence defines a birational involutions, proved that a Weddle surface is birationally isomor- tion of the Kummer surface. phic to a Kummer surface [Sch89].

NOVEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1531 Hessian surface of a nonsingular cubic surface 퐹3(푥, 푦, 푧, 푤) = 0 (the Hessian surface defined by the Hes- sian matrix of 퐹3(푥, 푦, 푧, 푤)) [Hut00]. The Hessian surface can also be defined as the locus of singular polar quadrics of the surface (quadrics given by linear combinations of partial derivatives of 퐹3). The rational map that assigns to a singular quadric its singular point defines a fixed-point- free involution on a nonsingular model of the Hessian surface with quotient isomorphic to an Enriques surface. There are 192 Weber hexads, each defines a birational involution of a Kummer surface, called the Hutchinson-Weber involution. The automorphism groups as well as complex dynamics of the Hessian quartic surfaces are the subjects of study of several recent papers. It is not surprising that none of the classical geometers could decide whether a given finite set of birational transformations generates the group Bir(푋) of birational automorphisms of 푋. It had to wait until the end of the century for new technical tools to arrive. Figure 8. Friedrich Schottky. A nonsingular birational model of a Kummer surface is an example of a K3 surface. By definition, a K3 surface 푌 This implies that the groups of birational automor- is a compact analytic simply connected surface with trivial phisms of a Weddle surface and a Kummer surface are first Chern class 푐1(푌). Its second Betti number is equal isomorphic. For example, in 1911 Virgil Snyder gave many to 22. By a theorem of John Milnor, its homotopy type is geometric constructions of involutions of the Weddle sur- uniquely determined by the quadratic form expressing the face that give corresponding involutions of the Kummer 2 22 1 cup-product on 퐻 (푌, ℤ) ≅ ℤ . It is a unique unimodular surface. In a paper of 1914, Francis Sharpe and Clide Craig even quadratic lattice of signature (3, 19) isomorphic to the pioneered the new approach to study birational automor- orthogonal sum of two copies of the even rank 8 negative phisms of algebraic surfaces based on Francesco Severi’s definite unimodular lattice 퐸8(−1) and three copies of the Theory of the Base. It consists of representing an automor- integral hyperbolic plane 푈. All K3 surfaces are diffeomor- phism of an algebraic surface as a transformation of the phic and realize the homotopy type defined by this lattice. group of algebraic cycles on the surface. This gave an easy Any complex K3 surface admits a Kähler metric and, by proof of Hutchinson’s result that 60 Göpel-Hutchinson in- Yau’s theorem, it also admits a Ricci-flat metric. It is not volutions generate an infinite group. known how to write it explicitly (a problem of great im- The Hutchinson involutions act freely on a nonsingular portance for physicists). The Kummer surface is flat in this model of the Kummer surface and the quotient by these metric outside the singular points, but still one does not involutions are special Enriques surfaces. Enriques surfaces know how to extend this flat metric to a Ricci-flat metric were discovered by Federigo Enriques in 1894 and, together on 푌. with K3 surfaces, now occupy an important role in the The quadratic lattice of cohomology 퐻2(푌, ℤ) of an alge- theory of algebraic surfaces. In their paper of 2013 un- 2 braic K3 surface contains a sublattice 푆푌 = 퐻 (푌, ℤ)푎푙푔 ≅ der the title “Enriques surfaces of Hutchinson-Göpel type ℤ휌 of algebraic 2-cycles. It contains 푐 (퐿), where 퐿 is an am- and Mathieu automorphisms” Shigeru Mukai and Hasinori 1 ple line bundle. The signature of 푆푌 is equal to (1, 휌 − 1) Ohashi study some finite groups of automorphisms of En- and, in general, it is not a unimodular lattice. The group riques surfaces. Bir(푋) is isomorphic to the group Aut(푌) of biregular au- John Hutchinson also discovered the following amaz- tomorphisms of 푌 and it admits a natural representation ing fact. A tetrad of nodes is called a Rosenhain tetrad if 휌 ∶ Aut(푌) → O(퐻2(푌, ℤ)) in the orthogonal group of any three nodes in the tetrad lie on a trope-conic. A We- 2 퐻 (푌, ℤ) that leaves 푆푌 invariant. ber hexad is the symmetric sum of a Göpel tetrad and a The fundamental Global Torelli Theorem for K3 sur- Rosenhain tetrad. Hutchinson proves that the linear sys- faces of Ilya I. Pyatetsky-Shapiro and Igor R. Shafarevich (its tem of quadric surfaces through a Weber hexad of nodes on original proof relies heavily on the theory of Kummer sur- a Kummer surface defines a birational isomorphism to the faces) allows one to describe the image of homomorphism 휌 as the set of all isometries of 퐻2(푌, ℤ) that leave (af- 1 John Hutchinson, Virgil Snyder, Francis Sharpe, and Clide Craig were on the ter complexification) 퐻2,0(푌, ℂ) invariant and also leave faculty of the Cornell Mathematics Department.

1532 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 10 invariant the semi-group of cohomology classes of holo- [Ham37] W. Hamilton, Third supplement to an essay on the the- morphic curves on 푌. The group of automorphisms Aut(푌) ory of systems of rays, Trans. Royal Irish Acad. 17 (1837), 1–144. acts naturally on the lattice 푆푌 and on the hyperbolic space ℍ휌−1 associated with the linear space 푆 ⊗ ℝ of signature [Hud05] R. W. H. T. Hudson, Kummer’s quartic surface, Cam- 푌 bridge University Press, Cambridge, 1905. Reprinted in (1, 휌 − 1). In this way it is realized as a discrete group of 1990 with a forward by W. Barth. MR1097176 motions of a hyperbolic space. By using an isometric em- [Hut00] J. Hutchinson, The Hessian of the cubic surface. II, Bull. bedding of 푆푌 into the unimodular even lattice 퐼퐼1,25 of Amer. Math. Soc 6 (1900), 328–337. MR1557720 signature (1, 25) isomorphic to the orthogonal sum of the [Hut01] J. Hutchinson, On some birational transformations of Leech lattice and the hyperbolic plane 푈, Richard Borcherds the Kummer surface into itself, Bull. Amer. Math. Soc 7 introduced a method that in several cases allows one to (1901), 211–217. MR1557786 compute the automorphism group of a K3 surface using [Jes03] C. Jessop, A treatise of the line complex, Cambridge Uni- versity Press, Cambridge, 1903. Reprinted by Chelsea Publ. the isometries of 퐼퐼1,25 defined by the reflections into Leech roots. In 1998, based on Borcherds’ ideas, Shigeyuki Kondo Co., New York, 1969. MR0247995 [Kle70] F. Klein, Zur Theorie der Liniencomplexe des ersten und proved that the group of birational automorphisms of a Zwieter Grades, Math. Ann. 2 (1870), 198–226. general Kummer surface (i.e., the Jacobian surface of a gen- 5 [Kle86] F. Klein, Ueber Configurationen, welche der Kummer- eral curve of genus 2) is generated by the group 2 , 16 schen Fläche zuglecih eingeschrieben und umgeschriben sind, projection involutions, 60 Hutchinson-Göpel involutions, Math. Ann. 27 (1886), 106–142. and 192 Hutchinson-Weber involutions.2 The group of bi- [Knö86] H. Knörrer, Die Fresnesche Wellenfläche, Arithmetik rational automorphisms of an arbitrary Kummer surface is und Geometrie. Mathematische Miniaturen, 3, Birkhüser still unknown. Verlag, Basel, 1986. MR0879281 For a modern exposition of the classical theory of Kum- [Kum64] E. Kummer, Ueber die Flächen vierten Grades mit sech- mer surfaces, we refer the reader to [Dol12]. I apologize szehn singulären Punkten, Monatsberichte Berliner Akad. 6 for omitting, because of the brevity of the article, many (1864), 246–260. [Kum66] E. Kummer, Ueber die algebraische Strahlensysteme important contributions to the study of Kummer surfaces insbesondere die erste und zweiten Ordnung, Monatsberichte by various mathematicians in the past and in the present. Berliner Akad. 8 (1866), 1–120. Note that a search of the title “Kummer surface” in the data [Lor96] G. Loria, Il passato ed il presente delle principali theorie base of MathSciNet gives 132 items (72 of them in this cen- geometriche, Carlo Clausen, Torino, 1896. MR0879281 tury). [Sch89] F. Schottky, Ueber die Beziehungen zwischen den sechzehn Thetafuncktionen von zwei Variablen, J. Reine References Angew. Math. 105 (1889), 233–244. [Bor77] C. Borchardt, Ueber die Darstellung der Kummerische [Wed50] T. Weddle, On theorems in space analogous to those fläche vierter Ordnung mit sechzehnen Knoten dürch die of Pascal and Brianchon in a plane, Cambridge and Dublin Göpelsche biquadratische Relation zwischen vier Thetafunktio- Quarterly Math. J. 5 (1850), 58–69. nen mit zwei Variablen, J. Reine Angew. Math. 83 (1877), 234–244. [Cas91] F. Caspari, Sur les deux formes sous lesquelles s’expriment au moyen des fonctions théta de deux arguments, les coor- donnéesde la surface du quatriémedegré, décritepar les som- mets des cones du second ordre qui passéntpar six points donnés, C. R. HébdomadairesAcad. Sci. Paris 112 (1891), 1356– 1359. [Dol12] Igor V. Dolgachev, Classical algebraic geometry. a modern view, Cambridge University Press, Cambridge, 2012. MR2964027 [Fre] A. Fresnel, Oeuvres compl‘etes d’Augustin Fresnel, Im- Igor Dolgachev primerie Impériale, Paris. 1866, par M. M. Henri de San- armont, Emile´ Verdet, et LéonorFresnel. Credits [Göp47] A. Göpel, Theoriae transcendentium abelianarum Opening image is courtesy of Claudio Rocchini. Licensed primi ordinis adumbratio levis, J. Math. Pures Appl. 35 under the Creative Commons Attribution-Share Alike 3.0 (1847), 277–312. Unported license (https://creativecommons.org /licenses/by-sa/3.0/deed.en). Figures 1–6 and 8 are courtesy of Wikimedia Commons. 2In fact, Kondo includes in the set of generators some new automorphisms of in- Figure 7 is courtesy of Internet Archive. finite order discovered by JongHae Keum instead of Hutchinson-Weber involu- Photo of Igor Dolgachev is courtesy of Igor Dolgachev. tions. It was later observed by Ohashi that one can use Hutchinson-Weber involutions instead of Keum’s automorphisms.

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