Theta Surfaces

Bernd Sturmfels MPI Leipzig and UC Berkeley

Joint Work with Daniele Agostini, T¨urk¨u Ozl¨um¨ C¸elik and Julia Struwe A second parametrization: u+v X = arctan(u)+arctan(v)=arctan 1 uv 5u+5v Y = arctan(5u)+arctan(5v)=arctan 1 25uv 1 1+(5u)2 1 1+(5v)2 Z = 2 log 5(1+u2) + 2 log 5(1+v 2) This theta surface is derived from the following quartic curve in the complex projective plane P2: xy(x2 + y 2 + z2)=0.

Scherk’s Minimal Surface Our running example is the surface in R3 given by sin(X ) sin(Y ) exp(Z)=0. · This theta surface has the parametric representation (X , Y , Z)= arctan(s), 0 , log(s) log(s2+1)/2 + 0 , arctan(t) , log(t)+log(t2+1)/2 . Scherk’s Minimal Surface Our running example is the surface in R3 given by sin(X ) sin(Y ) exp(Z)=0. · This theta surface has the parametric representation (X , Y , Z)= arctan(s), 0 , log(s) log(s2+1)/2 + 0 , arctan(t) , log(t)+log(t2+1)/2 . A second parametrization: u+v X = arctan(u)+arctan(v)=arctan 1 uv 5u+5v Y = arctan(5u)+arctan(5v)=arctan 1 25uv 1 1+(5u)2 1 1+(5v)2 Z = 2 log 5(1+u2) + 2 log 5(1+v 2) This theta surface is derived from the following quartic curve in the complex projective plane P2: xy(x2 + y 2 + z2)=0. Scherk’s Minimal Surface

sin(X )=sin(Y ) exp(Z) ·

(X , Y , Z)= arctan(s), 0 , log(s) log(s2+1)/2 + 0 , arctan(t) , log(t)+log(t2+1)/2 . Double Translation Surfaces An analytic surface in R3 or C3 is a double translation surface S if it has two distinct representations as Minkowski sum of curves: Bilder Druckqualität - Files - ownCloud 11.02.20, 10'00

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Sophus Lie: Bestimmung aller Fl¨achendie in mehrfacher Weise durch Translationsbewegung einer Curve erzeugt werden, Archiv Math. (1882).

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https://oc.mis.mpg.de/s/lkLFiv2kfnD6rWS?path=%2FObjekt%2003-04%2…ruckqualität#/Objekt%2003-04/Bilder%20Druckqualität/Bild_02.jpg Page 1 of 1 https://oc.mis.mpg.de/s/lkLFiv2kfnD6rWS?path=%2FObjekt%2005%2FBi…20Druckqualität#/Objekt%2005/Bilder%20Druckqualität/Bild_02.jpg Page 1 of 1 In 1892 Lie published an article explaining how these surfaces can be parametrized by abelian integrals. In 1895 Henri Poincar´e gave his proof of the relationship to Abel’s Theorem. This led to the idea that these surfaces are theta divisors of 3-dim’l Jacobians.

Lie supervised two doctoral dissertations on this topic at Leipzig. His students constructed a series of plaster models that visualize the diverse shapes exhibited by theta surfaces. It is surprising that the models were commissioned by Lie, who, unlike Klein, had not been known so far for an engagement in popularizing geometry.

Leipzig Felix Klein held the professorship for geometry at Leipzig until 1886 when he moved to G¨ottingen. Sophus Lie was appointed to be Klein’s successor and he moved from Christiania (Oslo) to Leipzig.

In his first years, Lie was busy with completing his major work Theory of Transformation Groups.Thereafter,double translation surfaces moved back in the focus of his teaching and research. Lie supervised two doctoral dissertations on this topic at Leipzig. His students constructed a series of plaster models that visualize the diverse shapes exhibited by theta surfaces. It is surprising that the models were commissioned by Lie, who, unlike Klein, had not been known so far for an engagement in popularizing geometry.

Leipzig Felix Klein held the professorship for geometry at Leipzig until 1886 when he moved to G¨ottingen. Sophus Lie was appointed to be Klein’s successor and he moved from Christiania (Oslo) to Leipzig.

In his first years, Lie was busy with completing his major work Theory of Transformation Groups.Thereafter,double translation surfaces moved back in the focus of his teaching and research.

In 1892 Lie published an article explaining how these surfaces can be parametrized by abelian integrals. In 1895 Henri Poincar´e gave his proof of the relationship to Abel’s Theorem. This led to the idea that these surfaces are theta divisors of 3-dim’l Jacobians. Leipzig Felix Klein held the professorship for geometry at Leipzig until 1886 when he moved to G¨ottingen. Sophus Lie was appointed to be Klein’s successor and he moved from Christiania (Oslo) to Leipzig.

In his first years, Lie was busy with completing his major work Theory of Transformation Groups.Thereafter,double translation surfaces moved back in the focus of his teaching and research.

In 1892 Lie published an article explaining how these surfaces can be parametrized by abelian integrals. In 1895 Henri Poincar´e gave his proof of the relationship to Abel’s Theorem. This led to the idea that these surfaces are theta divisors of 3-dim’l Jacobians.

Lie supervised two doctoral dissertations on this topic at Leipzig. His students constructed a series of plaster models that visualize the diverse shapes exhibited by theta surfaces. It is surprising that the models were commissioned by Lie, who, unlike Klein, had not been known so far for an engagement in popularizing geometry. Bilder Druckqualität - Files - ownCloud 11.02.20, 09(59

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Richard Kummer: Die Fl¨achen mit unendlichvielen Erzeugungen durchImprint | Privacy Translation Policy von Kurven, Dissertation,Imprint Universit¨at | Privacy Policy Leipzig, 1894.

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Georg Wiegner: Uber¨ eine besondere Klasse von Translationsfl¨achen, Dissertation, Universit¨at Leipzig, 1893.

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John Arndt Eiesland Johan "John" Arndt Eiesland (27 January 1867, Ny-Hellesund, Norway – 11 March 1950, West Virginia Morgantown, West Virginia) was a Norwegian-American mathematician, specializing in differential geometry.

Eiesland immigrated to the USA in 1888 after completing his secondary education in Christiansand. He received his bachelor's degree from the University of South Dakota in 1891. He was a mathematics professor from 1895 to 1903 at Thiel College in Pennsylvania. On a leave of absence, he studied mathematics as a Johns Hopkins Scholar (1897–1898) at Johns Hopkins University, where he received his Ph.D. in 1898.[1] From 1903 to 1907 he was a mathematics instructor at the United States Naval Academy. In 1907 he became a professor of mathematics at West Virginia University and the chair of the mathematics department from 1907 to 1938, when he retired as professor emeritus.[2]

Eiesland was elected a Fellow of the American Association for the Advancement of Sciences in 1903.[3] He was an Invited Speaker of the ICM in 1924 in Toronto.

He married in 1904.

At West Virginia University, Eiesland Hall is named in his honor, and in 1994 his heirs established the John Arndt Eiesland Visiting Professorship of Mathematics.[4]

Selected publications

On Nullsystems in space of five dimensions and their relation to ordinary space. (https://bab el.hathitrust.org/cgi/pt?id=njp.32101043987906;view=1up;seq=119) American Journal of Mathematics 26, no. 2 (1904): 103-148. On a certain system of conjugate lines on a surface connected with Euler's transformation. Trans. Amer. Math. Soc. 6 (1905) 450–471. doi:10.1090/S0002-9947-1905-1500721-9 (http s://doi.org/10.1090%2FS0002-9947-1905-1500721-9) John Eiesland (1908-09)On a certain class completed of algebraic translation-surfaces. the classification (https://babel.hathitrust.org/cgi/pt?id=hv initiated d.32044102902897;view=1up;seq=391) American Journal of Mathematics 29, no. 4 (1907): by Kummer and Wiegner.363–386. He also constructed plaster models. On Translation-Surfaces Connected with a Unicursal Quartic. (https://babel.hathitrust.org/cgi These were donated/pt?id=uc1.c2723078;view=1up;seq=182) to the collection atJohn American Hopkins Journal of Mathematics University. 30, no. 2 (1908): 170–208. On minimal lines and congruences in four-dimensional space. Trans. Amer. Math. Soc. 12 (1911) 403–428. doi:10.1090/S0002-9947-1911-1500898-8 (https://doi.org/10.1090%2FS00 02-9947-1911-1500898-8) The group of motions of an Einstein space. Trans. Amer. Math. Soc. 27 (1925), 213–245. doi:10.1090/S0002-9947-1925-1501308-7 (https://doi.org/10.1090%2FS0002-9947-1925-15 01308-7) 4 The ruled V 4 in S5 associated with a Schläfli hexad. Trans. Amer. Math. Soc. 36 (1934)

https://en.wikipedia.org/wiki/John_Arndt_Eiesland#cite_note-4 Page 1 of 2 Abelian Integrals Consider a quartic curve in the projective plane P2 ,definedby Q C i j q(x, y)= i+j 4 cij x y .  The space of holomorphic di↵erentialsP H0( , ⌦1 )hasthebasis Q Q x y 1 !1 = dx ,!2 = dx ,!3 = dx. qy qy qy

Note: is the canonical220px-Trott_bitangents.png embedding 220×220 pixelsof the underlying Riemann surface. 29/08/15 09:10 Q

Triple_torus_illustration.png 2.204×1.550 pixels 29/08/15 09:08

r We obtain a complex number p !i if we integrate along a path on with end points p and r.Ifthepathisrealthen r ! is real. Q R p i R

https://upload.wikimedia.org/wikipedia/commons/f/f0/Triple_torus_illustration.png Page 1 of 1

https://upload.wikimedia.org/wikipedia/commons/thumb/8/81/Trott_bitangents.png/220px-Trott_bitangents.png Page 1 of 1 Two Norwegians 2 Fix a line (0) in P that intersects in four points p1(0), p2(0), L Q p (0), p (0). For p close to p (0), we obtain the space curve 3 4 j 2Q j pj pj pj ⌦j (pj )=(⌦1j , ⌦2j , ⌦3j )(pj )= !1 , !2 , !3 . pj (0) pj (0) pj (0) R R R Theorem (Abel)

Suppose that the points p1, p2, p3, p4 are collinear. Then

⌦1(p1)+⌦2(p2)+⌦3(p3)+⌦4(p4)=0.

The theta surface of is defined by a local parametrization Q 3 C , (p1, p2) ⌦1(p1)+⌦2(p2). U! 7! Theorem (Lie)

Given any double translation surface 1 + 2 = 3 + 4,thearcs ˙1, ˙2, ˙ ˙ C 2 C C C C C 3, 4 lie on a common quartic in P ,and is the theta surface of . C C Q S Q Bernie Sanders banner – Google-Suche 04.03.20, 06)26

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Suppose that is smooth and let B be its RiemannCalifornia matrix.Thisis Q Die Bilder sind eventuell urheberrechtlich geschützt. Weitere Infos a complex symmetric 3 3 matrix with positive-definite real part. Ähnliche Bilder Mehr ansehen Amazon.com: Bernie Sanders 2020 Bumper Sticker: Automotive amazon.com⇥ The Riemann theta function is a holomorphic function in x C3: 2 https://www.google.com/search?hl=de&tbm=isch&source=hp&bi…nSgoHoAhX7HzQIHca7DOMQ4dUDCAU&uact=5#imgrc=rKR97iZnOXHPXM Page 1 of 13 t t ✓(x, B)= n Z3 exp ( ⇡n Bn) cos(2⇡n x), 2 · Note:P this takes on real values if B and x are real.

Daniele Agostini and Lynn Chua: Computing theta functions with Julia, arXiv:1906.06507

3 The theta divisor is the surface ⇥B := x C ✓(x, B)=0 . { 2 | } Theorem (Riemann)

The theta surface and the theta divisor ⇥B coincide up to an S 3 ane change of coordinates on C .Wehave =⇧↵ ⇥B + c. S · Bilder Druckqualität - Files - ownCloud 11.02.20, 09(59

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1. Introduction 2. Parametrization by Abelian Integrals 3. Symbolic Computations for SpecialImprint Quartics | Privacy Policy 4. Degenerations of Theta Functions 5. Algebraic Theta Surfaces 6. A Numerical Approach for Smooth Quartics 7. Sophus Lie in Leipzig https://oc.mis.mpg.de/s/lkLFiv2kfnD6rWS?path=%2FObjekt%2001%2FBi…%20Druckqualität#/Objekt%2001/Bilder%20Druckqualität/Bild_03.jpg Page 1 of 1 Symbolic Computation Algorithm 1: Computing the parametrized theta surface from its plane quartic Input: The inhomogeneous equation q(x, y) of the plane quartic curve. Output: The parametrization (8) of the theta surface in ane 3-space. S Step 1: Specify two points p1 and p2 on the quartic. Step 2: Fix local parameters (x1,y1)and(x2,y2)aroundp1 and p2 respectively. Step 3: Write yj as an algebraic function in xj on its branch. Step 4: Compute the partial derivative qy on the two branches. Step 5: Substitute x1 = s and x2 = t into the dierential forms 1, 2, 3 in (5). Step 6: By integrating these dierential forms, compute the vectors

(p (s)) = ( s ds, y1(s) ds, 1 ds), 1 1 qy(s,y1(s)) qy(s,y1(s)) qy(s,y1(s)) t y2(t) 1 2(p2(t)) = ( dt, dt, dt). R qy(t,y2(t)) R qy(t,y2(t)) R qy(t,y2(t)) R R R Step 7: Output the sum 1(p1(s)) + 2(p2(t)) of the generating curves. The crux of Algorithm 1 is Step 3, where y is represented as an algebraic function in x . Input: q =(y + x 1)(y x 1)(j y + x + 1)(y x + 1). j This function has algebraic degree at most four, so it can be written in radicals. After all, theOutput: steps aboveX are= meant1 (log( ass a symbolic1) + log( algorithm.t + 1)), Y However,= 1 ( welog( founds) +the log( representationt)), in radicals to be infeasible8 for practical computations unless8 the quartic is very special. Z = 1 (log(s 1) log(s) log(t + 1) + log(t)). In what follows we focus on instances8 where the quartic q(x, y) is reducible. A reducible plane quartic is of the following: four straight lines, a conic and two straight lines, two conics,Equation: or a cubicSetting and a line.u = Inexp the(8 sequel,X ), v we= showexp(8 theY computations), w = exp(8 ofZ such), we theta get surfaces usingu2vw Algorithm2 2u2vw 1. Theuv 2w first+uvw three2+ casesu2v were4uvw workedv 2 outw + byuv Richarduw Kummer2vw w [15]=0 in. his thesis, and the latter case was studied by Georg Wiegner [31]. We here present three further examples of non-algebraic theta surfaces. The algebraic ones will be discussed in Section 5. Consider the first three of the four cases above. Then q(x, y)factorsintotwoconics.The two conics meet in four points in P2. We consider the pencil of conics through these points. Remark 3.2. A result due to Lie states that the theta surface for a product of two conics S only depends on their pencil, provided p1 and p2 lie on the same conic (cf. [15, Section 3]). Hence has infinitely many distinct representations 1 + 2 as a translation surface. We shall returnS to this topic in Theorem 7.3, where it is shownC howC to compute these representations. For instance, for Scherk’s surface in Example 3.1, the four points in P2 are (i :0:1), ( i : 0:1), (0 : i :1), (0 : i : 1), and the pencil is generated by xy and x2 + y2 + z2. We can replace these two quadrics by any other pair in the pencil and obtain two generating curves. We now examine another case which is similar. It will lead to the theta surface in (20). Example 3.3. Fix the four points (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1), (1 : 1 : 1) in P2. Their pencil of conics is generated by y(x z)and(x y)z. We multiply the first conic with a general member of the pencil to get the quartic Q= y(x z) ((1 + )xy xz yz). Here is a parameter, which we included in order to illustrate Remark· 3.2. Dehomogenizing Q, we get q = y(x 1)((1 + )xy x y). We compute , , from the partial derivatives 1 2 3 q = y((1 + )xy x y)+y(x 1)((1 + )y 1)), x q =(x 1)((1 + )xy x y)+y(x 1)((1 + )x ). y

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2 2 Start with q = xy(x + y + 1) and compute derivatives qx , qy and Imprint | Privacy Policy di↵erential forms !1,!2,!3. The integrals over the line (s, 0) are s ⌦11(p1(s)) = s(s2+1) ds =arctan(s),

https://oc.mis.mpg.de/s/lkLFiv2kfnD6rWS?path=%2FObjekt%2002%2FBi…20Druckqualität#/Objekt%2002/Bilder%20Druckqualität/Bild_02.jpg0 Page 1 of 1 ⌦21(p1(s)) = R s(s2+1) ds =0, ⌦ (p (s)) = 1 ds = log(s) 1 log(s2 + 1). 31 1 R s(s2+1) 2 The abelian integral overR the line (0, t)are ⌦ (p (t)) = 0 dt =0, 12 2 t(t2+1) ⌦ (p (t)) = t dt = arctan(t), 22 2 R t(t2+1) ⌦ (p (t)) = 1 dt = log(t)+ 1 log(t2 + 1). 32 2 R t(t2+1) 2 The Minkowski sum of theseR two curves is Scherk’s minimal surface sin(X )=sin(Y ) exp(Z). · Rational Nodal Quartics

Figure 2: The five types of rational nodal quartics Figure 2: The five types of rational nodal quartics

quartics appear in five types: an irreducible quartic with three nodes, a nodal cubic together quartics appear in five types: an irreducible quartic with three nodes, a nodal cubic together withLet aB line,be two the smoothtropical conics, Riemann a smooth matrix conic togetherof the with dual two graph. lines, andWe four lines. with a line, two smooth conics, a smooth conic together with two lines, and four lines. considerTo each of a these one-parameter curves we associate family its of dual symmetric graph =( 3V,E3matrices). This has a vertex for each To each of these curves we associate its dual graph =(V,E). This has a vertex for each irreducible component and one edge for each intersection point⇥ between two components. A irreducible component and one edge for each intersection point between two components. A node on an irreducibleB component:= tB + countsB as anfor intersectiont + of that. component with itself. node on an irreducible componentt counts0 as an intersection! of1 that component with itself.

FigureFigure 3: 3: The The dual dual graphs graphs corresponding corresponding to to the the five five types types of of rational rational nodal nodal quartics quartics

TheThe dual dual graphs graphs play play a akey key role role in in tropical tropical geometry, geometry, namely namely in in the the tropicalization tropicalization of of curvescurves and and their their Jacobians. Jacobians. We We follow follow the the combinatorial combinatorial construction construction in in [3, [3, Section Section 5]. 5]. Fix Fix oneone of of the the graphs graphs inin Figure Figure 3 with3 with an an orientation orientation for for each each edge. edge. The The first first homology homology group group E V H1(, Z)=ker : EZ VZ H1(, Z)=ker : Z Z !! is free abelian of rank 3. Let us fix a -basis 1, 2, 3 for H1(, ). Each j is a directed is free abelian of rank 3. Let us fix a Z-basisZ 1, 2, 3 for H1(, Z).Z Each j is a directed cyclecycle in in.Weencodeourbasisina3.Weencodeourbasisina3 E Ematrixmatrix. The. The entries entries in in the thej-thj-th row row of of areare ⇥ | | E the coordinates of j with respect to⇥ the| standard| basis ofE . Our five matrices are the coordinates of j with respect to the standard basis of Z Z. Our five matrices are 100100 1 1 1001001 1 1010 0 0 1010010100 1 1 1010010100 010010, , 01011010, , 01011010, , 1101101010, , 1010 10 101010. . 001001 0000 01 01 0000 1 11 1 0 0 1010 0 01 1 01011010 0 0 1 1 We define the Riemann matrix of to be the positive definite symmetric 3 3matrix We define the Riemann matrix of to be the positive definite symmetric 3 ⇥3matrix T ⇥ BB:=:= T. . (31)(31) · · IfIf we we change change the the orientations orientations and and cycle cycle bases bases then then the the matrix matrixB Btransformstransforms under under the the action action ofof GL(3 GL(3, Z,)Z by) by conjugation. conjugation. The The Riemann Riemann matrices matrices of of our our five five graphs graphs are are the the matrices matrices in in the the fourthfourth column column in in Figure Figure 4. 4. The The label labelFormFormrefersrefers to to the the quadratic quadratic form form represented represented by byB.B. WeWe now now explain explain how how a Riemanna Riemann matrix matrixBBofof the the dual dual graph graph as as in in (31) (31) induces induces a degener- a degener- ation to a singular curve with dual graph as in Figure 3. Let B be a fixed (real or complex) ation to a singular curve with dual graph as in Figure 3. Let B0 be0 a fixed (real or complex) symmetricsymmetric 3 3 3 matrix3 matrix and and consider consider the the one-parameter one-parameter family family of of classical classical Riemann Riemann matrices matrices ⇥⇥ BBt:=:=tBtB++BB0 forfort t + +. . (32)(32) t 0 !! 11 1111 4.3 Dimension 3 49

Voronoi Cells d Delone Graph Polytope Form Name 3 1 1 13 1 TRUNCATED 01 13 1 OCTAHEDRON 6 @ A K4 2 10 HEXA-RHOMBIC 5 13 1 00 12 1 DODECAHEDRON @ A K4 1 2 10 RHOMBIC 4 12 1 00 12 1 DODECAHEDRON @ A C4 2 10 HEXAGONAL 4 120 00011 PRISM @ A K3 +1 100 3 010 CUBE 0 1 001 1+1+1 @ A We adopted the way of drawing the five parallelohedra from [CS1992]. The procedure of edge deletion becomes visible. First, the Russian crystallographer FEDEROV gave a complete classification of three-dimensional parallelohedra. He and his work was extremely influential for VORONO¨I’s memoir [Vor1908]. Tropical Degeneration Theorem Fix a vertex a of the Voronoi cell for the tropical degeneration. The theta surface is defined by the following finite sum over the vertex set of the Delaunay polytope dual to a: Da,B exp ⇡nT B n +2⇡i nT x . 0 · n 2XDa,B ⇣ ⌘ The number of summands equals 8 for a rational quartic, 6 for a nodal cubic plus line, 4 or 6 for two conics, 4 or 5 for a conic plus two lines, and 4 for four lines. This recovers Eiesland’s formulas.

X Y Z 10 + 10 + 10 =1 B. Bolognese, M. Brandt and L. Chua: From curves to tropical Jacobians and back, in Combinatorial Algebraic Geometry, Fields Inst. Comm. 80, Springer, 2017. coincides with its theta surface, up to an ane transformation. We here validate that result computationally using current tools from numerical algebraic geometry. We sample points on Smooththe theta surfaceQuartics using Algorithm 2 below, and we then check that Riemann’s theta function vanishes at these points using the Julia package Theta.jl due to Agostini and Chua [1]. Algorithm 2: Sampling from a theta surface given its plane quartic Input: The inhomogeneous equation q(x, y) of a smooth plane quartic Output: A point on the corresponding theta surface in R3 or C3 Step 1: Specify two points p1 and p2 on the quartic. Step 2: Take two other points p1 and p2 nearby p1 and p2 respectively. Step 3: Compute the following triples of integrals numerically:

p1 p1 p1 c1 =( 1, 2, 3), p1 p1 p1 (45) c =(p2 , p2 , p2 ). 2 Rp2 1 Rp2 2 Rp2 3 R R R Step 4: Output the sum c1 + c2. This algorithm diers from that in Algorithm 1 in that the computation is now done numericallyOur engine and for not Step symbolically. 3 is the Sage Indeed,package whenRiemannSurfaces the polynomial fromq(x, y)definesasmooth quartic, it is often impractical to recover a symbolic expression for y in terms of x, so we employN. Bruin, numerical J. Sijsling methods and A. even Zotine: for StepsNumerical 1 and computation 2 above. Of of course, when such an expression is available,endomorphism it can rings be usedof Jacobians to strengthen, Proceedings the numerical 13th Algorithmic computations, as we will see later. NumberThe central Theory pointSymposium of Algorithm (ANTS, 2 2019) is computing the abelian integrals in Step 3. Such integrals appear throughout mathematics, from algebraic geometry to number theory and integrable systems, and there is extensive work in evaluating them numerically. Notable implementations are the library abelfunctions in SageMath [28], the package algcurves in Maple [9], and the MATLAB code presented in [13]. The software we used for our exper- iments is the package RiemannSurfaces in SageMath due to Bruin, Sijsling and Zotine [4]. The underlying algorithm views a plane algebraic curve as a ramified cover P1 of Q Q ! the Riemann sphere P1 via the projection (x, y) x. The package lifts paths from P1 to paths on the Riemann surface and integrates the7! abelian dierentials in (7) along these paths via certified homotopy continuation.Q In order to carry this out, it is essential to avoid the ramification points of the projection to P1. This is done by computing the Voronoi decomposition of the Riemann sphere P1 given by the branch points of P1. The integration paths are obtained from edges of the Voronoi cells. Avoidance of theQ ! ramification points is also a feature in the other packages such as algcurves and abelfunctions. It is important to note that avoiding the ramification points is in conflict with our desire to create real theta surfaces and to work with Riemann matrices and theta equations over R. The cycle basis that is desirable for revealing the real structure, as seen in [27], forces us to compute integrals near the ramification points. We had to tweak the method in [4] for our purposes, and that tweaking is responsible for the numerical diculties described below. In what follows we present a case study that illustrates Algorithm 2. Our instance is the Trott curve. This is a smooth plane quartic , defined by the inhomogeneous polynomial Q q(x, y)=144(x4 + y4) 225(x2 + y2)+350x2y2 +81. (46) 20 The Trott Curve q = 144(x4 + y 4) 225(x2 + y 2) + 350x2y 2 + 81

β1

β2

s3 s1 s2 s4

β3

0.926246 0.553994 0.372252 B = 0.553994 1.10799 0.553994 0 1 0.372252 0.553994 0.926246 @ A Algebraic Theta Surfaces Theorem (Eiesland) Every algebraic theta surface is rational and has degree 3, 4, 5 or 6. Its quartic has rational components and no singularities are nodes. Example Consider the toric curve x4 yz3 in P2. This rational quartic has a triple point. The abelian integrals evaluate to

1 2 2 X = sds + tdt = 2 (s + t ), 4 4 1 5 5 Y = Rs ds + R t dt = 5 (s + t ), Z = R ds + R dt = s + t. Elimination of the parametersR sR and t reveals the theta surface

Z 5 20X 2Z + 20Y =0. John Little: Translation manifolds and the converse to Abel’s theorem Compositio Mathematica (1983) The Cardiod Surface and The Deltoid Surface

8X 2Z +8Y 2Z 4YZ X 4XYZ+4XY +2XZ 2YZ+3X Y 512px-Ile_de_ré.JPG 512×384 pixels 12.02.20, 05(10

Integrable Systems

We derived the quintic Z 5 20X 2Z + 20Y from x4 yz3. This is a Schur-Weierstrass polynomial, arising from a degeneration of the sigma function for (3, 4)-curves.

V.M. Buchstaber, V.Z. Enolski and D.V. Leykin: Rational analogs of abelian functions, Functional Analysis and its Applications (1999). A. Nakayashiki: Tau function approach to theta functions, International Mathematics Research Notices (2016).

Plan: Apply our tools to thehttps://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Ile_de_ré.JPG/512px-Ile_de_ré.JPGKP equation for shallow water wavesPage 1 of 1 B. Dubrovin, R. Flickinger and H. Segur: Three-phase solutions of the Kadomtsev-Petviashvili equation, Studies in Applied Math (1997). Sophus Lie and Tropical Geometry

Already in 1869, Lie had shown howFigure to 2: The construct five types of rational infinitely nodal quartics many representations = 1quartics+ appear2 for in fivethe types:tetrahedral an irreducible quartic theta with threesurface nodes, a nodal cubic together S C with aC line, two smooth conics, a smooth conic together with two lines, and four lines. To each of these curves we associate its dual graph =(V,E). This has a vertex for each = ↵ exp(Xirreducible)+ componentexp( andY one)+ edge for eachexp( intersectionZ)= point between. two components. A S · node on an irreducible· component counts· as an intersection of that component with itself. Lie introduces coordinates X = log(U), Y = log(V ), Z = log(W ) and he studies Hadamard product decompositions = of P D1 ·D2 Figure 3: The dual graphs corresponding to the five types of rational nodal quartics = ↵ U + V + W = . P ·The dual graphs· play a key role· in tropical geometry, namely in the tropicalization of curves and their Jacobians. We follow the combinatorial construction in [3, Section 5]. Fix View this plane in 3 andone of thetranslate graphs in Figure it to 3 with contain an orientation1 = for(1 each: edge. 1 : The 1 first: 1). homology group P E V H1(, Z)=ker : Z Z Intersecting with the coordinate planes gives lines !1, 2, 3, 4. is free abelian of rank 3. Let us fix a Z-basis 1, 2, 3 for H1(, Z). Each j is a directed P cycle in .Weencodeourbasisina3 E matrix L. TheL entriesL in theLj-th row of are ⇥ | | E Theorem (Lie) the coordinates of j with respect to the standard basis of Z . Our five matrices are 100 1 100 1 10 0 10100 1 10100 010, 01 10, 01 10, 110 10, 10 10 10. Let 1, 2 be lines through001100in 01.Then00 1=1 10 102 0if and1 0110 0 1 D D P P D ·D only if the six lines We, defineare the Riemann tangent matrix toof to a be common the positive definite conic symmetric in 3 . 3matrix i j ⇥ D L B := T . P (31) · If we change the orientations and cycle bases then the matrix B transforms under the action of GL(3, Z) by conjugation. The Riemann matrices of our five graphs are the matrices in the fourth column in Figure 4. The label Form refers to the quadratic form represented by B. We now explain how a Riemann matrix B of the dual graph as in (31) induces a degener- ation to a singular curve with dual graph as in Figure 3. Let B0 be a fixed (real or complex) symmetric 3 3 matrix and consider the one-parameter family of classical Riemann matrices ⇥ B := tB + B for t + . (32) t 0 ! 1 11 4.3 Dimension 3 49

d Delone Graph Polytope Form Name

Sophus Lie and Amoebas in 1869 3 1 1 13 1 TRUNCATED 01 13 1 OCTAHEDRON 6 @ A K4 Bilder Druckqualität - Files - ownCloud 11.02.20, 09(59 Figure 2: The five types of rational nodal quarticsAdd to your ownCloud Download = ↵ exp(X )+ exp(Y )+ exp(Z)= S Objekt 01 Bilder· Druckqualität · · quartics appear in five types: an irreducible quartic with three nodes, a nodal cubic together 2 10 with a line, two smooth conics,Name a smooth conic together with two lines, and fourSize lines.Modified HEXA-RHOMBIC 5 13 1 To each of these curves we associate its dual graph =(V,E). This has a vertex for each 00 12 1 DODECAHEDRON irreducible component and oneBild_01 edge for each intersection point between two components.4.7 MB 7 days ago A @ A node on an irreducible component counts as an intersection of that component with itself.K4 1 Bild_02 4 MB 7 days ago Bild_03 3.2 MB 7 days ago Bild_04 3.1 MB 7 days ago 2 10 RHOMBIC 4 12 1 0 1 DODECAHEDRON Bild_05 2.8 MB 7 days ago 0 12 Figure 3: The dual graphs corresponding to the five types of rational nodal quartics @ A C4 Bild_06 2.4 MB 7 days ago The dual graphs play a key role in tropical geometry, namely in the tropicalization of curves and their Jacobians. We6 files follow the combinatorial construction in [3,20.3 Section MB 5]. Fix 2 10 one of the graphs in Figure 3 with an orientation for each edge. The first homology group HEXAGONAL 4 120 E V 0 1 PRISM H1(, Z)=ker : Z Z 001 ! @ A is free abelian of rank 3. Let us fix a Z-basis 1, 2, 3 for H1(, Z). Each j is a directedK3 +1 cycle in .Weencodeourbasisina3 E matrix . The entries in the j-th row of are ⇥ | | E the coordinates of j with respect to the standard basis of Z . Our five matrices are 100 1 100 1 10 0 10100 1 10100 100 010, 01 10, 01 10, 110Imprint | Privacy10 Policy, 10 103 10. 010 CUBE 0 1 001 00 01 00 1 1 0 10 0 1 0110 0 1 001 1+1+1 @ A We define the Riemann matrix of to be the positive definite symmetric 3 3matrix ⇥ B := T . We adopted(31) the way of drawing the five parallelohedra from [CS1992]. The procedure of · edge deletion becomes visible. First, the Russian crystallographer FEDEROV gave a complete If we change the orientations and cycle bases then the matrix B transforms under the action classification of three-dimensional parallelohedra. He and his work was extremely influential for of GL(3, Z) by conjugation. The Riemann matrices of our five graphs are the matrices in the fourth column in Figure 4. The label Form refers to the quadratic form representedVORONO by¨I’sB memoir. [Vor1908]. We now explain how ahttps://oc.mis.mpg.de/s/lkLFiv2kfnD6rWS?path=%2FObjekt%2001%2FBi…%20Druckqualität#/Objekt%2001/Bilder%20Druckqualität/Bild_03.jpg Riemann matrix B of the dual graph as in (31) induces a degener-Page 1 of 1 ation to a singular curve with dual graph as in Figure 3. Let B0 be a fixed (real or complex) symmetric 3 3 matrix and consider the one-parameter family of classical Riemann matrices ⇥ B := tB + B for t + . (32) t 0 ! 1 11 Bilder Druckqualität - Files - ownCloud 11.02.20, 09(59

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6 files 13.2 MB quartics appear in five types: an irreducible quartic with three nodes, a nodal cubic together with a line, two smooth conics, a smooth conic together with two lines, and four lines. To each of these curves we associate its dual graph =(V,E). This has a vertex for each irreducible component and one edge for eachImprint | intersection Privacy Policy point between two components. A Nielsnode on Henrik an irreducible Abel componentvisited countsFreiberg as an intersection(also in Saxony) of that component in 1825-1826 with itself.

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FromFigure Wikipedia: 3: The dual graphs corresponding to the five types of rational nodal quartics The dual graphs play a key role in tropical geometry, namely in the tropicalization of “....curves From and theirBerlin Jacobians.Abel Wealso follow followed the combinatorial his friends construction to the in Alps. [3, Section 5]. Fix Heone ofwent the graphs to Leipzig in Figureand 3 withFreiberg an orientationto visit for Georg each edge. Naumann The first homology group E V and his brother the mathematicianH1(, Z)=ker August : Z Z Naumann. ! Inis freeFreiberg abelian ofAbel rankdid 3. Let research us fix a Z in-basis the 1 theory, 2, 3 for ofH functions,1(, Z). Each j is a directed cycle in .Weencodeourbasisina3 E matrix . The entries in the j-th row of are particularly, elliptic, hyperelliptic,⇥ | | and a new classE now the coordinates of j with respect to the standard basis of Z . Our five matrices are known as abelian functions. 100 1 100 1 10 0 10100 1 10100 010, 01 10, 01 10, 110 10, 10 10 10. 001 00 01 00 1 1 0 10 0 1 0110 0 1 We define the Riemann matrix of to be the positive definite symmetric 3 3matrix ⇥ B := T . (31) · If we change the orientations and cycle bases then the matrix B transforms under the action of GL(3, Z) by conjugation. The Riemann matrices of our five graphs are the matrices in the fourth column in Figure 4. The label Form refers to the quadratic form represented by B. We now explain how a Riemann matrix B of the dual graph as in (31) induces a degener- ation to a singular curve with dual graph as in Figure 3. Let B0 be a fixed (real or complex) symmetric 3 3 matrix and consider the one-parameter family of classical Riemann matrices ⇥ B := tB + B for t + . (32) t 0 ! 1 11 Bilder Druckqualität - Files - ownCloud 11.02.20, 10'00

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