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IJMMS 2004:44, 2331–2345 PII. S0161171204308112 http://ijmms.hindawi.com © Hindawi Publishing Corp. REAL QUARTIC SURFACES CONTAINING 16 SKEW LINES ISIDRO NIETO Received 12 August 2003 It is well known that there is an open three-dimensional subvariety M s of the Grassmannian of lines in P3 which parametrizes smooth irreducible complex surfaces of degree 4 which are Heisenberg invariant, and each quartic contains 32 lines but only 16 skew lines, being determined by its conﬁguration of lines, are called a double 16. We consider here the problem of visualizing in a computer the real Heisenberg invariant quartic surface and the real double 16. We construct a family of points l ∈ M s parametrized by a two-dimensional semialgebraic variety such that under a change of coordinates of l into its Plüecker, coordinates transform into the real coordinates for a line L in P3 , which is then used to construct a program in Maple 7. The program allows us to draw the quartic surface and the set of transversal lines to L. Additionally, we include a table of a group of examples. For each test example we specify a parameter, the viewing angle of the image, compilation time, and other visual properties of the real surface and its real double 16. We include at the end of the paper an example showing the surface containing the double 16. 2000 Mathematics Subject Classiﬁcation: 14N15, 14P10, 14Q10, 68U05. 1. Introduction. Let H t be the well-known Heisenberg group of level 2 (for the precise deﬁnition, we refer to Section 3). We consider H t -invariant quartic surfaces Xf = {f (z0 , z1 , z2 , z3 ) = 0}, where f is a homogeneous polynomial of degree 4 in the variables z0 , z1 , z2 , z3 , which is H t -invariant over the complex numbers ﬁeld. One of the problems posed in [1] is that if such a quartic surface contains a complex line, then determine the conﬁguration of lines or ﬁnd a characterization of Xf in terms of the conﬁguration of lines contained in it. It is a classical fact of line geometry that the lines in P3 are parametrized by the Grassmannian variety denoted here by Gr. In order to formulate the problem more precisely, we introduce the complex vector space W of polynomials of degree 4 in the variables z0 , . . . , z3 which are H t -invariant. The condition that Xf contains a line is stated as l ⊂ Xf if and only if f |l = 0. It is not diﬃcult to prove (see, e.g., [1, Section 4]) that dimC (W ) = 5 (in [1], one can give an explicit set of generators for W ), and for a “generic” l, the last condition implies that there exist a 5 × 5 matrix M(l) and ν ∈ CP4 such that M(l) · ν = 0, which implies that det(M(l)) = 0. By choosing the so-called K-coordinates associated to a line, l transforms into a point (x0 : · · · : x5 ) such that x02 + · · · + x52 = 0 which is the well-known equation for Gr in the K-coordinates (c.f. Section 3). The equation involving the matrix M(l) is written in the K-coordinates as Σ5i=0 Πj≠i xj2 = 0. The previous arguments are only heuristical and will be made precise in Section 3. Both equations above are the equations which deﬁne the threefold M ⊂ Gr. In this paper, we consider the converse problem. Namely, we are 2332 ISIDRO NIETO interested in ﬁnding points in a subvariety of M s ⊂ M (for which a precise deﬁnition will be given in Section 2) such that each point in M s arises as a real line in P3 (i.e., deﬁned by real coordinates), the quartic Xf determined by l is real (i.e., the points are deﬁned in real three-dimensional projective space), and the coeﬃcients of f are deﬁned over the real numbers ﬁeld. Moreover, if according to the theory of [1, Section 4], a point in M s generates a complex line which determines a smooth, irreducible complex H t -invariant quartic surface containing it and its H t orbit, then an easy extension of the theory to the real case shows that for the special case treated here, the H t -invariant quartic surface Xf is real and its deﬁning polynomial f is real. The next problem is to ﬁnd a special hyperplane H ⊂ RP3 to be able to graph Xf ∩ H and visualize it along with its conﬁguration of lines called here a double 16 on a computer. It is of interest to note that the H t -invariant quartic surfaces containing a complex line form a three-dimensional complex parameter space within the 34-dimensional space of quartic surfaces in CP3 . Such surfaces of degree 4 deﬁned over the complex ﬁeld arise as projective models of linear systems of abelian surfaces of polarization type (1, 3) (c.f. [1, 5]). We give an outline of the paper. In Section 2, we state suﬃcient conditions for an arbitrary point l ∈ M to be in M s (Proposition 2.7). We then construct a subset of M s parametrized by a two-dimensional semialgebraic set (Remark 2.9). In Section 3, after giving a few basic facts of line geometry and the deﬁnition of the Heisenberg group, we show that the H t -invariant quartic surfaces deﬁned by the points of Proposition 2.7 are real quartic surfaces and contain the double 16 with real lines, and that the quartics are determined by the conﬁguration. In Section 4, we give a detailed description of how the Maple program works and is used to visualize these surfaces along with their line conﬁgurations (to the extent to which they can be shown) on the computer. In Section 5, we describe the results given by the program for a group of examples in a table which describes, for the surface drawn with its ten transversals, the following parameters: d (which deﬁnes the quartic surface; this is the value λ of (2.8)), the angle (u, v) of the image surface, compilation time for each surface, and the visual description of the double 16. We printed one test example of a surface for which one can see a line and most of the double 16, at one speciﬁc angle, and it is given at the end of the paper. The program can be used as a guide to produce Heisenberg invariant Kummer quartic surfaces, which contain other types of curves, not necessarily rational ones, and can be useful for other researchers working on similar problems. 2. An elementary proposition and a locus of real solutions. Another more suggestive way of writing the equation for M as deﬁned in [1, Section 4] is Σ5i=0 xi2 = Σ5i=0 1 = 0. xi2 (2.1) In the sequel, the following set of equations, derived from the above deﬁnition, will be more useful to us: REAL QUARTIC SURFACES CONTAINING 16 SKEW LINES 2333 x02 + x12 + x22 = −x32 − x42 − x52 , (2.2) 1 1 1 1 1 1 + + =− 2 − 2 − 2, x02 x22 x42 x3 x5 x1 (2.3) which is of course deﬁned away from {xi = 0} for i = 0, . . . , 5. Otherwise, (2.3) becomes Σ5i=0 Πj≠i xj2 = 0. According to [1, Section 4], there exists an open subvariety of M, a √ threefold denoted in [1] as M s , deﬁned as follows. For this, we let η = −1, x1 = ηy1 , x3 = ηy3 , x5 = ηy5 . (2.4) We also introduce the following change of variables: qi = x 2 i for i even, y 2 i for i odd. (2.5) We ﬁx Qjk = xj = xk = 0 = Σi≠j, i≠k xi2 = 0 , P = 1 : 2 : 3 : η4 : 5 η : 6 η , i ∈ {±1}, i = 1, . . . , 6, ᏼ1 = q0 − q1 = q2 − q3 = q4 − q5 = 0 . (2.6) By deﬁning ᏽ= Σ = σ (P ) | σ ∈ S6 , Qjk , ᏼ= σ ᏼ1 , (2.7) σ ∈S6 j≠k then M s = M − ᏽ − Σ − ᏼ. M s has the property that for each l ∈ M s , there exists an equation of degree 4 invariant under the Heisenberg group H t , irreducible, and uniquely determined by its conﬁguration of lines, and the surface deﬁned by it is smooth (see [1, Proposition 4.4]). We can ﬁnd the solutions to (2.2), (2.3) as a particular case of the following oneparameter set of equations for q0 > 0, q1 > 0, q2 > 0, q3 > 0, q4 > 0, q5 > 0, λ > 0: λ= 1 1 1 1 1 1 + + = + + , q2 q0 q4 q3 q1 q5 (2.8) 1 = q0 + q2 + q4 = q1 + q3 + q5 . To solve the last set of equations is to solve, for real variables x, y, z and a real parameter λ, the system of equations xy + xz + yz = λxyz, x + y + z = 1, x > 0, y > 0, (2.9) z > 0, and solve them for the case (x, y, z) = (q0 , q2 , q4 ) and (q1 , q3 , q5 ). We obtain the following elementary proposition. 2334 ISIDRO NIETO Proposition 2.1. The system of equations (2.9) has a solution if and only if λ > 9. For such λ, set ρ = (λ − 9)(λ − 1)/4λ2 and σ = (λ − 3)/2λ; then √ √ 1 <σ− ρ<σ+ ρ<1 λ (2.10) √ √ and the system of equations has a real solution if and only if z ∈ (σ − ρ, σ + ρ) or z < 1/λ when the solutions for x and y are given by positive 1 Ω x = (1 − z) 1 ± − , 2 N Ω 1 , y = (1 − z) 1 ∓ − 2 N (2.11) where Ω = λz2 − (λ − 3)z + 1, N = (λz − 1)(1 − z). Proof. Fixing λ > 0 and 1/λ < z < 1, we must ﬁnd the intersection points of the line x + y = 1 − z and the hyperbola αxy + x + y = 0 (where α = (1 − λz)/z). Solving these yields the expressions for x and y as stated above. These solutions are real exactly when 1± −Ω/N > 0. But −Ω/N is positive if and only if Ω < 0, N > 0 or Ω > 0, N < 0, which are equivalent to Ω < 0, z > 1/λ or Ω > 0, z < 1/λ. The condition Ω > 0 (resp., Ω < 0) √ √ √ √ is equivalent to saying that z > σ + ρ or z < σ − ρ (resp., z ∈ (σ − ρ, σ + ρ)), hence the claim. Clearly, ρ > 0 if and only if λ > 9 or λ < 1, but (2.10) rules out the second possibility. Equation (2.10) can easily be veriﬁed if λ > 0. This can be applied with (x, y, z) = (q0 , q2 , q4 ) or (q1 , q3 , q5 ). Proposition 2.2. The case λ = 9 in Proposition 2.1 is exceptional. Fix the aﬃne plane H = {g = q0 +q2 +q4 −1 = 0} in the R3 deﬁned by the coordinates q0 , q2 , q4 , and deﬁne, for each λ ∈ R≥0 , fλ = q2 q4 + q0 q4 + q0 q2 − λq0 q2 q4 (2.12) which is a surface in the A3 deﬁned by the coordinates q0 , q2 , q4 . Let Cλ = H ∩ {fλ = 0}. Then the linear system of curves {Cλ = {fλ = g = 0}}λ∈R≥0 is always smooth except for λ = 1, 9 with singularities Sing C9 = 1 1 1 , , 3 3 3 , Sing C1 = Q = (−1, 1, 1) . (2.13) Proof. To simplify the computations, let ∂i fλ = ∂fλ /∂qi . Recall that fλ = q2 q4 + q0 q2 + q0 q4 − λq0 q2 q4 and −g = 1 − q0 − q2 − q4 . Note that ∂i fλ = qj + qk − λqj qk for i, j, k ∈ {0, 2, 4} with i ≠ j ≠ k and ∂i g = 1. The Jacobian matrix is ∂2 fλ 1 ∂0 fλ 1 ∂4 fλ . 1 (2.14) It is of rank less than 1 if and only if 0 = ∂i − ∂j fλ = 0 for i, j ∈ {0, 2, 4}. (2.15) REAL QUARTIC SURFACES CONTAINING 16 SKEW LINES 2335 q0 − q2 1 − λq4 = q2 − q4 1 − λq0 = q4 − q0 1 − λq2 = 0. (2.16) It follows that It is enough to prove the following three cases (the others are derived from these). (I) 1 − λq4 = q0 −q4 = q2 −q4 = 0. Thus q2 = q4 = q0 . q4 = 1/λ. Substituting in g = 0, one obtains g = 1 − 3q0 . Therefore, q0 = 1/3. Substituting this value of q0 in 0 = fλ , one obtains 0 = 2q22 , a contradiction. (II) 0 = q2 −q0 = q0 −q4 = q2 −q4 . It follows that q0 = q4 = q2 . Hence g = 1−3q0 = 0. Substituting q0 = 1/3 in fλ , one obtains fλ = 3/9 − λ/27 = 0, therefore λ = 9. (III) 1 − λq4 = 1 − λq0 = q4 − q0 = 0. Therefore, q0 = q4 = 1/λ. From 0 = g = 1 − 2/λ − q2 , it follows that q2 = 1 − 2/λ and 0 = fλ = q4 (1 − 1/λ), hence λ = 1. Fix once again the R3 deﬁned by the coordinates x, y, z. Then an easy computation shows that the equation of {Cλ = {0 = fλ (x, y, z) = 1−(x +y +z)}}λ∈R>0 can be written for λ = 1, 9 as f9 = 9 x 2 y + xy 2 + x + y − 10xy − x 2 + y 2 , f1 = x 2 y + xy 2 + x + y − 2xy − x 2 + y 2 . (2.17) For the next lemma, recall that P = (1/3, 1/3, 1/3) is a singular point for C9 and Q = Sing(C1 ). Lemma 2.3. The equation for the tangent cone of C9 (resp., of C1 ) passing through P (resp., Q of C1 ) is (x −1/3)2 +(x −1/3)(y −1/3)+(y −1/3)2 (resp., 4(y −1)(x +y)). In particular, P (resp., Q) is a nonordinary double point of C9 at P (resp., Q of C1 ). 2 Proof. The second partial derivatives at P are given as ∂x2 f9 = −2 + 18y, ∂xy f9 = 2 2 2 2 −10+18x +18y, ∂y f9 = −2+18x. Summarizing, ∂x f9 (P ) = 4, ∂y f9 (P ) = 4, ∂xy f9 (P ) = 2. The equation for the tangent cone at P is then 1 1 2 1 1 2 y− +4 y − +4 x − . 4 x− 3 3 3 3 (2.18) The calculation for C1 can be done analogously. Remark 2.4. A direct computation shows that f9 is irreducible over R. Under the linear change of coordinates u = x − 1/3, v = y − 1/3, the equation for f9 = 0 is transformed into f9 = 9uv(u + v) + 2uv + 2(u2 + v 2 ). Under this linear change of coordinates, the cubic curve C9 is transformed into a real cubic with isolated singularity at the origin which is to be expected from the classiﬁcation of irreducible cubic curves over the real numbers ﬁeld. Useful notation. Let e = (λ − 9)(λ − 1), R = (λ − 3 − e)/2λ, S = (λ − 3 + e)/2λ. By Proposition 2.1, R < S, for x, u ∈ (R, S), we will adapt the convention of writing these numbers as x = (R(n − 1) + S)/n, u = (R(m − 1) + S)/m for some n, m > 1. 2336 ISIDRO NIETO We need the following lemma. Lemma 2.5. (1) If n, m ∈ R are chosen so that 1 < n < M = me/(3m + e(m − 1)), 2 < m, then M < 2 < m < e/3 and for such n, m, the numbers x, u satisfy 1 − u < x. In √ particular, e > 6, which is equivalent to λ > 5(1 + 2). (2) If (u, v, w), (x, y, z) are solutions to (2.9) and are chosen so that x > 1 − u and x ∈ {(λ + 3 ± e)/4λ}, then x ≠ v, x ≠ w, x ≠ y, x ≠ z. Indeed, x ∈ {(λ + 3 ± e)/4λ} only if n = (3(λ−1)+e)/2λ or n = (λ−1+e)/2. In particular, if 1 < n < 2 only, the value for λ = (3n − 2)/(n − 1)(2 − n) is possible. Proof of part (1). 1 − u < x if and only if (m − (R(m − 1) + S))/m < (R(n − 1) + S)/n, hence n < (S − R)/(1 − (R(2m − 1) + S)/m). Substitute the values for S, R. The value for the numerator S −R = e/λ and the denominator is equal to (m(3+e)−e)/mλ, hence, by substituting in the original expression, we obtain n < M. The last inequality follows from 1 < M if and only if m < e/3. To prove the inequality, m > M if and only if m > 2e/(e + 3) and note that the last number is always less than 2. Thus, if m > 2, it is suﬃcient. We ﬁnally note that M < 2 if and only if m > 2e/(e + 6), but if m > 2, then m > 2e/(e + 3) > 2e/(e + 6). Proof of part (2). Note that Ω = t(λt + 3) − (λt − 1), Ω t(λt + 3) = − 1 > t − 1. λt − 1 λt − 1 (2.19) Fix u as one solution to (2.11); namely, v = (1/2)(1 − u) + Z/2, where Z = −(1 − u)Ω/(λu − 1). Assume to the contrary that x = v, therefore 2x − (1 − u) = Z. By (2.19), we obtain −Z 2 = (1 − u)(u(λu + 3)/(λu − 1) − 1) > (1 − u)(u − 1), hence (1 − u)2 > Z 2 . For u > 1/λ, the quantity on the right-hand side is positive, hence (1 − u)2 − Z 2 = (1 − u + Z)(1 − u − Z) > 0, which is equal to 2x(2(1 − u) − 2x) > 0. Since x > 0, x < 1 − u, which is a contradiction. Assuming x = w = 1 − (u + v), substituting in v, we obtain 2x − (1 − u) = −Z ≤ 0. If x > 1 − u, then 0 ≥ 2x − (1 − u) > 2(1 − u) − (1 − u) = 1 − u > 0, again a contradiction. To see that none of u, v, w (resp., x, y, z) is equal to the other using the equation v 2 (λu − 1) + v(1 − u)(1 − λu) + u(1 − u) = 0, (2.20) we consider the following cases. (I) If v = u such that u(λu − 1) + (1 − u)(1 − λu) + (1 − u) = 0, hence 2λu2 − u(λ + 3) + 2 = 0. It follows that u = (λ + 3 ± e)/4λ. (II) If u = w = 1 − u − v, v = 1 − 2u, thus v 2 = 1 − 4u(1 − u) and, substituting in (2.20), we obtain again the same quadratic equation in u as in case (I). A completely analogous reasoning applies to the variables x, y, z. The veriﬁcation of (2.20) is an easy veriﬁcation using (2.9). REAL QUARTIC SURFACES CONTAINING 16 SKEW LINES 2337 √ √ To conclude the proof, if we write x = ((σ − ρ)(n − 1) + σ + ρ)/n, then n is to be eliminated from √ √ √ σ± ρ 3 + , σ − ρ (n − 1) + σ + ρ = n 2 2λ (2.21) √ √ √ where the right-hand side is {(λ + 3 ± e)/4λ}. Hence, n(σ − 2 ρ ± ρ − 3/λ) = −4 ρ. We are to solve two cases corresponding to the sign in the last expression. √ √ (I) n(σ − 3 ρ − 3/λ) = −4 ρ; then substituting the deﬁnitions for σ , ρ in the last expression gives n = −4e/(λ − 9 − 3e) or, by multiplying the denominator by λ − 9 + 3e, we obtain n = e(λ − 9 + 3e)/2λ(λ − 9) and, after simpliﬁcation, we obtain the claimed value. To obtain the value for n, one solves the quadratic equation in terms of λ by the value found for n and obtains λ = −(n − 1)(n + 2)/(2 − n) < 0. √ √ (II) n(σ − ρ − 3/λ) = −4 ρ; then as in case (I), we obtain n = −4e/(λ − 9 − e) by multiplying again by λ − 9 + e, and, simplifying the expression, we obtain the claimed value. The value for n is obtained in the same way as in case (I). An application of Lemma 2.5 is the following. √ Corollary 2.6. 7 < e/3 if and only if e > 21 if and only if λ > 5 + 407 and the last number is greater than 25. For example, if λ = 30.0000, then e = 24.6779. Consider values for m < e/3 = 8.2259, say, for example, m = 7; then M = 7e/(21 + 6e) = 1.0217, so choosing n = 1.01 will suﬃce. √ √ √ √ √ √ Proposition 2.7. Let l = ( q0 : η q1 : q2 : η q3 : q4 : η q5 ) be such that qi ∈ R>0 for all i = 0, . . . , 5 with q0 > 1 − q1 such that n = 1.01, m = 7 for λ ≥ 30.00, λ ≠ 104.04, and the triples (q0 , q2 , q4 ), (q1 , q3 , q5 ) are solutions to (2.9); then l ∈ M s . Proof. The point l is not in 5 (1) ᏽ trivially since Πi=0 qi ≠ 0; √ √ (2) Σ since l ∈ Σ, then q0 = q1 , hence q0 = q1 , contrary to the assumption; (3) ᏼ; indeed, we need to check that q0 ∈ {q2 , q3 , q4 , q5 } since already q0 ≠ q1 . But this is the statement of Lemma 2.5; we only need to verify that n, λ are not the exceptional case of the statement of Lemma 2.5(2). For that, note that only λ = 104.04 is possible. The chosen value for m = 7 is in accordance with Corollary 2.6. By hypothesis, l ∈ M and (1), (2), (3) above show that l ∈ M s . 5 Remark 2.8. It is interesting to note that by relaxing the condition, Πi=0 qi ≠ 0. That is, if one qi = 0, then another qj = 0 with j ≠ i (see the commentary after (2.3)), then the quartic surfaces obtained are singular along two skew lines. They have already been studied in [1, Proposition 7.2(a)] and in [6, Proposition 5.1]. Remark 2.9. Fix λ > 9. We will follow the convention previous to Lemma 2.5 in writing the elements of (R, S) in the sequel. Let ᐀ = {x ∈ (R, S) | 1 < n < M, n ≠ (λ − 1 + e)/2} and ᐁ = {u ∈ (R, S) | 2 < m < e/3}, for z ∈ R, let Rz = {x ∈ R | x > z} 2338 ISIDRO NIETO and Ᏽ = (r , s) ∈ R3>0 × R3>0 | r = (x, y, z), s = (u, v, w),v, y as in (2.11) , (2.22) and consider π : Ᏽ → ᐀ × ᐁ such that π (r , s) = (x, u). The construction of Ᏽ implies that π maps bijectively onto ᐂ = ᐀ × ᐁ, hence dim(Ᏽ) = dim(π (Ᏽ)) by [2, Theorem 2.2.8]. Since each of ᐀, ᐁ is one-dimensional, dim(π (Ᏽ)) = dim(᐀ × ᐁ), hence Ᏽ is two-dimensional. In particular, for λ ≥ 30.0000 and λ ≠ 104.04, deﬁne a map ϕ : ᐂ → CP5 such that ϕ(x, u) = (x : ηy : z : ηu : v : ηw) with ((x, y, z), (u, v, w)) ∈ Ᏽ. By Proposition 2.7, ϕ(x, y) belongs to M s and is injective, thus ϕ(ᐂ) is a subset of M s bijective to a two-dimensional variety. 3. The 32 lines on the quartic surface. Fix the three-dimensional real projective space RP3 with coordinates z0 , z1 , z2 , z3 , the quartic surface Xf = {(z0 : z1 : z2 : z3 ) ∈ RP3 , f (z0 , z1 , z2 , z3 ) = 0} given by the homogeneous polynomial f of degree 4 in the variables z0 , z1 , z2 , z3 over the real numbers ﬁeld R, and a line in RP3 which is generated by a two-plane in R4 represented by a 2 × 4 matrix 1 0 ∗ ∗ 0 1 ∗ . ∗ (3.1) 2 4 R and another One coordinate-free approach characterizes a line as a 2-form ω ∈ one is to say that a line is given by a two-dimensional subspace V ⊂ R4 which yields 2 4 R , one obtains the a well-deﬁned point in RP5 . Choosing the canonical basis of Plücker coordinates {pij } (or P-coordinates) of the line as follows. Let z0 Λ= z0 z1 z1 z2 z2 z3 z3 (3.2) be a 2 × 4 matrix and the minors of Λ given by pi,j = zi zj − zj zi , i, j ∈ {0, 1, 2, 3}, (3.3) where i ≠ j. If a matrix Λ is a two-plane, this means that pi,j ≠ 0 for some i, j, and the P-coordinates for this line satisfy the equation p01 p23 − p02 p13 + p03 p12 = 0. (3.4) Conversely, if a point with P-coordinates {pi,j } satisfying the above equation is given, then the point representing {pi,j } is a two-plane (see, e.g., [3, Chapter 1, Section 5]). Let REAL QUARTIC SURFACES CONTAINING 16 SKEW LINES 2339 H t be the subgroup of SL(4, R) spanned by the transformations 0 0 σ1 = 1 0 1 0 0 1 τ1 = 0 0 0 0 0 0 0 1 0 0 −1 0 1 0 0 0 0 1 , 0 0 0 0 , 0 −1 0 1 σ2 = 0 0 1 0 τ2 = 0 0 1 0 0 0 0 0 , 1 0 0 0 0 1 0 −1 0 0 0 0 1 0 0 0 , 0 −1 (3.5) which satisfy the relations σi2 = τi2 = id, σi τi = −τi σi , (3.6) for i = 1, 2. One obtains a central exact sequence of groups: 1 → {±1} → H t → G → 0, (3.7) where G Z42 . The group H t is the Heisenberg group of level two. The explicit action of H t on f for a polynomial, as above, on the variables z0 , z1 , z2 , z3 is given by the usual linear action on the polynomials of degree 4; in particular, σ z0 z1 z2 z3 = z0 z1 z2 z3 ∀σ ∈ H t . (3.8) Apply the following coordinate transformation in C6 to the P-coordinates: x0 = p01 − p23 , x2 = p02 + p13 , x1 = η p01 + p23 , x3 = η p02 − p13 , x5 = η p03 + p12 . x4 = p03 − p12 , (3.9) These are the Klein coordinates (or K-coordinates as a notational convenience) that satisfy 0 = x02 + x12 + x22 + x32 + x42 + x52 = −2 p01 p23 − p02 p13 + p03 p12 (3.10) which is the equation for Gr, the Grassmannian variety, which parametrizes the set of complex lines in CP3 . The K-coordinates are eigenfunctions for the action of H t on them. They are also very useful in studying properties of hypersurfaces in Gr, known classically as the line complex (c.f. [4, Chapter VIII, Section 130 and Chapter XII, Section 221]). An easy consequence of inverting the transformation given by (3.9) is the following corollary. Corollary 3.1. l = (x0 : x1 : x2 : x3 : x4 : x5 ) with P-coordinates {pi,j } deﬁnes a real line (i.e., for all i, j, pi,j ∈ R) if and only if x0 , x2 , x4 are all positive-real and x1 , x3 , x5 are purely imaginary. 2340 ISIDRO NIETO In view of the previous corollary, it is quite natural to introduce the notation of (2.4). Using the deﬁnition of (2.5), the P-coordinates can be expressed in terms of the {qi }coordinates as √ √ q + q p 01 √ 0 √ 1 p03 = q4 + q5 , √ √ p13 q2 − q3 √ √ q2 + q3 p02 p23 = − √q0 − √q1 . √ √ p12 − q4 − q5 (3.11) We consider a line l with coordinates {pi,j } such that p01 ≠ 0. For example, the line with coordinates 1 pa = pb 0 0 1 x u y v (3.12) in RP3 is expressed using (3.11) in the {qi }-coordinates as u = p02 = q2 + q3 , y = −p13 = − q2 − q3 , v = p03 = q4 + q5 , x = −p12 = q4 − q5 . (3.13) Let W be the complex vector space of quartic forms in the variables z0 , z1 , z2 , z3 invariant under H t ; then by, for example, [5, Proposition 4.1.1(ii)], it is of dimension ﬁve; and let Gr be as before. Let Ᏽ be the incidence variety given by Ᏽ = (l, ν) ∈ Gr ×P(W )fν l = 0 . (3.14) If g0 , g1 , g2 , g3 , g4 is a basis of W , then, for every ν ∈ W with ν = (ν0 : · · · : ν4 ), let fν = Σ4i=0 νi gi be the associated quartic polynomial. Let π be the projection of Ᏽ into Gr. Let l ∈ Gr and let Ᏽl be the ﬁbre under π . For diﬀerent points of l ∈ M, Ᏽl has already been calculated in [1, Proposition 7.1] and in [5, Corollary 3.4.3]. What is needed in this situation is an easy extension of [5, Lemma 3.3.1] to the real case and it can be stated as follows. Lemma 3.2. Let l ∈ M s deﬁne a line with real pi,j coordinates. Then Ᏽl = (l, ν) for a unique ν ∈ RP4 . Proof. The point ν is a solution to a system of nonhomogeneous equations with entries over the real pi,j coordinates in [5, Lemma 3.3.1], hence the claim. The proof of the following corollary is a direct consequence of Proposition 2.7 and the calculations are left as an easy veriﬁcation. REAL QUARTIC SURFACES CONTAINING 16 SKEW LINES 2341 √ √ √ √ √ √ Corollary 3.3. Let l = ( q0 : η q1 : q2 : η q3 : q4 : η q5 ) satisfy the hypothesis of Proposition 2.1. By Lemma 3.2, the point ν deﬁnes a real smooth quartic surface Xf = {fν = 0} which contains the G orbit of l and of a line l which will be deﬁned as follows. In the K-coordinates {xi }, the involution 1 1 1 1 1 1 : x0 : x1 : x2 : x3 : x4 : x5 → − : : : : : , x0 x1 x2 x3 x4 x5 (3.15) which is well deﬁned away from the fourfolds {xi = 0} applied to l, gives a line l . Writing √ √ } (for this, let q = −1/ q0 − 1/ q1 ), the P-coordinates associated to this line as {pi,j 1 1 p 1 01 −√ √ q · p = q4 , q 03 5 1 1 q · p13 √ +√ q2 q3 1 1 √ −√ q3 q · p q2 02 1 1 q · p = √ − √ . 23 q q1 0 q · p12 1 1 −√ − √ q4 q5 (3.16) Remark 3.4. The two orbits of lines that are in Xf can be grouped as the “even” lines, that is, as those having an even number of minus signs in their K-coordinates, and the “odd” lines as those having an odd number of minus signs in their K-coordinates (in fact, using these K-coordinates, we have studied in [1, Proposition 4.2] group-theoretical properties of the conﬁguration of these lines). It is clear that if Xf contains the double 16, it contains its even and odd lines. It is now clear from Corollary 3.3 that if Xf is a real H t -invariant quartic surface deﬁned by l, Xf contains the double 16 of Corollary 3.3. The quartic surface above can contain more than the double 16 of lines. If the quartic surface is the image of a polarized abelian surface of type (1, 3) as an irreducible polarized abelian surface, then the image surface contains only the double 16 (c.f. [1, Propositions 6.4 and 6.7]). 4. Description of the program. The program written in Maple 7 (a copy of the program is available upon request) deﬁnes a global variable d (this is the value of λ in Proposition 2.1) which has to be given as initial input in the program. Using the subroutines named Var, Vas, the values for R, S which are polynomial expressions in terms of λ are calculated. Using the intervals for the solutions given in Proposition 2.1 for q0 (resp., for q1 ), these are calculated by two other routines named Np, Jb. In order to evaluate the global variable q2 , one needs to introduce the local variables M, N in terms of d, q0 and ﬁnally evaluate Sq1 . A subroutine then evaluates the positive root of q2 in terms of the local variables N, Sq1 , and dq0 − 1. q4 is evaluated introducing 2342 ISIDRO NIETO the routine rw which uses the equality q4 = 1−(q0 +q2 ). Using the value for q1 , the program uses the routine rz to evaluate q3 , and, applying the routine rw again, it evaluates q5 in exactly the same way. In order to draw the lines, one ﬁrst evaluates the parametric equation for the lines. We introduce the local variables r r , ss in terms of q4 , q5 , q2 , q3 . The last variables are used to give the parametric equation of the line l as given by (3.11). The orbit of l under H t is evaluated using the procedure Graf. The parametric equation of the transversal to l is evaluated using the variables m, n. The procedure Mpoly substitutes the variables x, y, u, v for the obtained values r r , ss, zz, ww together with (3.16), which are used to evaluate the values for m and n. The coeﬃcients for the quartic surface are evaluated by means of the nonhomogeneous system of equations described in the proof of Lemma 3.2. Introducing the routine Mpoly, the matrix solution is saved as a 4 × 1 matrix named K. It then deﬁnes the H t quartic invariant polynomial saved as the variable quar. The polynomial coeﬃcients of quar are deﬁned as the entries of K. This deﬁnes a new quartic polynomial in the variables x, y, u, v named quars. By substituting v = −x − y − u in quars, one obtains a new quartic polynomial quart in the variables x, y, u, which gives us the intersection of the quartic surface deﬁned by quars and the special hyperplane H = {(x, y, u, v) ∈ R4 | v + u + x + y = 0}. The next routine expands quart and this quartic polynomial is saved as quarn. The program then has to graph {quarn = 0} implicitly in terms of a given variable, which we chose as x. For this, we use the display3d command of the library plots of Maple. The body of the command consists of the range for x, y, u and the plotting options consist of the grid values, which we chose (unless otherwise speciﬁed in Table 4.1) as the default value of 25 for the three variables speciﬁed in the range; the style command which speciﬁes how the surface is to be plotted was ﬁxed as patchnogrid; the user-deﬁned lighting in Maple is speciﬁed by the red, green, and blue components of the ambientlight command which was ﬁxed as [0.6, 0.6, 0.6] (note that the default values are all values set equal to 1), the orientation command has to be speciﬁed as a pair (u, v), where u is the horizontal angle, v is the vertical angle, which in the group of test examples we chose to be the values given in the table. In order to visualize the lines, one has to specify the parametric equation of the lines using the variables already calculated, r r , ss, m, n; this is performed by the spacecurve command. As optional commands within the last command, the thickness of the lines which was ﬁxed as 1 is given, the color of the lines and the direction from which each line is to be viewed are given again by the orientation command. Also, as part of the display3d optional command, one has to specify the values for two directional light sources given by the light command, which consists of a quintuple of values: the ﬁrst two values are (v, u), where v is the vertical angle and u is the horizontal angle, and the other three values specify the intensities of the red, green, and blue colors. In the test samples given in the table, these were ﬁxed as [90, −80, 0.7, 0.6, 0.1], [90, 80, 0.7, 0.6, 0.1]. In the image produced by the procedure Graf of the Maple program in all the test examples, the line l was chosen in red. The remaining colors for the disjoint lines were chosen as follows: blue, yellow, sienna, cyan, khaki, pink, turquoise, aquamarine, magenta, plum, violet, brown, green, navy, and gold. The ten transversals to l have been drawn in color black. REAL QUARTIC SURFACES CONTAINING 16 SKEW LINES 2343 Table 4.1 Description of the conﬁguration of double 16 Value for d Angle (u, v) Compilation time Number of transversal lines, other visual properties 30.00 40 , 1 0.01 , Six transversals 1 0.78 , Six transversals 1 10.18 , Six transversals 1 22.04 , 53.74 Six transversals (−52, 48) Six transversals (−10, 82) Five transversals (−34, 60) Six transversals (−22, 52) Five transversals (−52, 44), (−52, 40), (−56, 62) Six transversals (−56, 66), (−54, 52), (−54, 68) Six transversals (−54, 52), (−54, 68), (−64, 60) Six transversals (−62, 30), (−70, 12) Six transversals (42, ) full rotation Six transversals 35.00 on v angle 37.00 40.00 (16, 33), (−12, 47) Six transversals (−48, 89) Six transversals (−56, 47) Seven transversals 2 33.97 , 1 42.19 (6, 25), (−46, 73) Six transversals 1 58.14 , 1 16.57 50.00 (−48, 96) Four transversals (−60, 96) Five transversals (−66, 96) Five transversals (−32, 64) Six transversals 51.04 , 49.89 60.00 40.54 (−171, −54) Four transversals (155, −56) Five transversals 1 42.92 , 1 26.87 70.00 1 27.74 (−28, 52) Four transversals (−12, 42) Five transversals (128, −88) Five transversals (154, −170) Five transversals 2344 ISIDRO NIETO Table 4.1 Continued. Description of the conﬁguration of double 16 Value for d Angle (u, v) Compilation time Number of transversal lines, other visual properties 58.30 , 124.30 80.00 122.70 (−20, 44) Five transversals (−36, 44) Five transversals 43.45 , 40.72 90.00 (166, −30) Five transversals (174, −38),(134, −64) Five transversals 100.00 1 07.28 (−24, 26) Five transversals (144, −60) Five transversals (42, 22) Five transversals (−24, 48) Five transversals 5. Application of the program. We treat the problem of visualizing examples of real quartic surfaces obtained by intersecting a real Heisenberg invariant quartic surface containing the double 16 conﬁguration of lines with the hyperplane H. To show as clearly as possible the double 16 conﬁguration, we only drew, in each case in the table, the red line on the surface and its ten transversals drawn in black including 15 other transversals with the colors mentioned at the end of Section 4. Another remark concerning the values for the orientation of the image given in the table is that the orientation was found by rotating the image and stopped at the angles (u, v) showing the visual properties of the surface and lines as described in the table. The program was run on a Pentium II. In the test examples given in the table, the following properties were tabulated: the value for d, the orientation of the surface speciﬁed by the angle (u, v), the approximate compilation time of the program to calculate the image, and the number of lines of the double 16 visible on the screen. These parameters are tested for each of the examples as given in the table. Note that the ﬁrst row of the table for each value of d (when recorded) gives the compilation time, and, for this value of d, all other entries do not consider this parameter. As one can see from the table, only at most six of the transversals were clearly visible from the given values of d. It is of interest (although not recorded in the table) to note that at the angle (42, 136) without drawing the surface for d = 37.00, one can see the red line intersecting the ten transversals drawn in black. See also Figure 5.1. It would be of interest to check if the surfaces considered in the table are actually singular and irreducible. REAL QUARTIC SURFACES CONTAINING 16 SKEW LINES 2345 Figure 5.1. The quartic surface for d = 30.00 at (−54, 68). Acknowledgments. The author was supported by the Coordinacion de Investigación Cientiﬁca de la UMSNH project 4.8 NBI “Algoritmos para superﬁcies algebraicas reales” during the years 2001 and 2002 and CONACyT project 38678-E “Superﬁcies compactas complejas.” We would like to thank W. Barth and S. Gutierrez for helpful conversations during the initial stages of the project. We would also like to thank one of the referees for giving some indications to improve the ﬁrst draft of the manuscript. References [1] [2] [3] [4] [5] [6] W. Barth and I. Nieto, Abelian surfaces of type (1, 3) and quartic surfaces with 16 skew lines, J. Algebraic Geom. 3 (1994), no. 2, 173–222. J. Bochnak, M. Coste, and M.-F. Roy, Géométrie Algébrique Réelle [Real Algebraic Geometry], Ergebnisse der Mathematik und Ihrer Grenzgebiete (3), vol. 12, Springer-Verlag, Berlin, 1987. P. Griﬃths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. C. M. Jessop, A Treatise on the Line Complex, Chelsea Publishing, New York, 1969. I. Nieto, Invariante Quartiken unter der Heisenberg Gruppe T [Quartics invariant under the Heisenberg group T], Ph.D. thesis, University of Erlangen, Germany, 1989. , Examples of abelian surfaces with polarization type (1, 3), Algebraic Geometry and Singularities (La Rábida, 1991) (C. Lopez and N. Macarro, eds.), Progr. Math., vol. 134, Birkhäuser, Basel, 1996, pp. 319–337. Isidro Nieto: Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoaćan 58260, Mexico E-mail address: inieto@zeus.ccu.umich.mx