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Shadow of a Cubic Surface

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Shadow of a Cubic Surface Faculteit B`etawetenschappen Shadow of a cubic surface Bachelor Thesis Rein Janssen Groesbeek Wiskunde en Natuurkunde Supervisors: Dr. Martijn Kool Departement Wiskunde Dr. Thomas Grimm ITF June 2020 Abstract 3 For a smooth cubic surface S in P we can cast a shadow from a point P 2 S that does not lie on one of the 27 lines of S onto a hyperplane H. The closure of this shadow is a smooth quartic curve. Conversely, from every smooth quartic curve we can reconstruct a smooth cubic surface whose closure of the shadow is this quartic curve. We will also present an algorithm to reconstruct the cubic surface from the bitangents of a quartic curve. The 27 lines of S together with the tangent space TP S at P are in correspondence with the 28 bitangents or hyperflexes of the smooth quartic shadow curve. Then a short discussion on F-theory is given to relate this geometry to physics. Acknowledgements I would like to thank Martijn Kool for suggesting the topic of the shadow of a cubic surface to me and for the discussions on this topic. Also I would like to thank Thomas Grimm for the suggestions on the applications in physics of these cubic surfaces. Finally I would like to thank the developers of Singular, Sagemath and PovRay for making their software available for free. i Contents 1 Introduction 1 2 The shadow of a smooth cubic surface 1 2.1 Projection of the first polar . .1 2.2 Reconstructing a cubic from the shadow . .5 3 The 27 lines and the 28 bitangents 9 3.1 Theorem of the apparent boundary . .9 3.2 The last bitangent . 11 3.3 Injectivity . 13 4 Recovering the cubic from a quartic with computer algebra 14 4.1 Calculating the bitangents . 15 4.2 Calculating the points of contact . 16 4.3 Finding a syzygetic quadruple of bitangents . 17 4.4 Finding a fourth bitangent . 18 4.5 Recovering the cubic surface . 18 5 Algebraic geometry in physics: F-theory 19 5.1 D7-branes . 19 5.2 Elliptic fibration . 19 5.3 Intersecting D7-branes . 20 5.4 Born infeld action . 20 6 Appendix: lemmas and notation 23 6.1 Notation . 23 6.2 Lemmas . 23 1 Introduction Two classical results in algebraic geometry are that a smooth cubic surface in P3 contains 27 lines and a smooth quartic curve in P2 contains 28 contact lines. Figure 2: A smooth quartic curve with 28 Figure 1: A smooth cubic surface containing contact lines. 27 lines. In the following paper we will show how by taking the shadow of the smooth cubic surface, also known as the projection from a point, one can obtain a smooth quartic curve and relate the 27 lines to the 28 contact lines. The paper is structured as follows: • In the second chapter we will show how the (closure of the) shadow of a smooth cubic surface is a smooth quartic curve. To show smoothness, we use ideas from section 5.8.12 of [Bel09]. Then for a smooth quartic curve we will give a construction of a smooth cubic surface that has this curve as its shadow. The idea of this construction is from exercise 5.7.11 of [Bel09] and the worked out details in coordinates is new. • In the third chapter we will show how the 27 lines get mapped to the 28 contact lines. Here we use ideas from section 5.8.9 of [Bel09]. • In the fourth chapter we will describe an algorithm that reconstructs a smooth cubic surface from a smooth quartic curve. This is largely based on the paper and code samples from [PSV11]. • In the fifth chapter we will give a short sketch of how algebraic surfaces are used in F-theory, this is unrelated to the previous chapters so can be read separately. The main reference for this chapter is [Wei18]. 2 The shadow of a smooth cubic surface 2.1 Projection of the first polar We will work in 3 = 3 , projective 3-dimensional space over , but the results also hold over P PC C any algebraically closed field of characteristic zero. 1 For a smooth cubic surface S in P3 we can create its shadow by projecting it from a point P 2 S. Because S is an algebraic variety, we expect that the shadow also is algebraic. In this section we will show that if we choose P to not lie on a line of S, then the shadow (or the closure thereof) is a smooth quartic curve. After that we will also show that any smooth quartic can be considered the projection of some smooth cubic surface. Definition 1. For S a smooth hypersurface in P3 we define the shadow Sh(S; P; H) from a point P 2 S onto a hyperplane H not containing P , to be the locus of points PQ \ H where PQ ⊂ TQS is a line tangent to S at some Q 2 S n fP g. ♦ This definition is close to our intuition of a shadow, but it is cumbersome to work with. We can reformulate it using the notion of a first polar. n Definition 2. Let X be a smooth hypersurface in P given by the polynomial f 2 k[x0; : : : ; xn] of degree r. Let P be a point in X. The first polar X1(P ) of X at P is defined as the hypersurface given by the degree r − 1 polynomial n X @f ∆1 f := P : (1) P i @x i=0 i So this is similar to the equation for the tangent space, except that we now leave the argument of the derivatives as variables, and we fix the point in the tangent space. This can be generalized to higher polars, for 0 ≤ k ≤ r we define the k-th polar Xk(P ) as the hypersurface given by the degree r − k polynomial k k X @ f ∆P f := Pi1 ····· Pik : (2) @xi1 : : : @xik i1;:::;ik=0;:::;n ♦ Now we will give an equivalent definition of a shadow using the first polar S1(P ). Note that S was chosen to be smooth, so this construction indeed works. Let π : P3 n fP g ! H be the projection that maps Q to the intersection PQ \ H. Let S1(P ) be the first polar of S at P . If S is given as the zero set of the cubic polynomial f, 1 P3 @f then S1(P ) is the quadric hypersurface with equation ∆ f := Pi . P i=0 @xi If a point Q 6= P lies in S and in S1(P ), then P lies in the tangent space at Q, which means the line PQ is tangent at Q. For every point R 2 Sh(S; P; H) ⊂ H there exists a point Q 2 S \S1(P ) with Q 6= P such that π(Q) = PQ \ H = R, which means we have Sh(S; P; H) ⊂ π(S \ S1(P ) n fP g). Conversely, for Q 2 S \ S1(P ) n fP g, the line PQ intersects H in π(Q), and this intersection lies in Sh(S; P; H), so π(S \ S1(P ) n fP g) ⊂ Sh(S; P; H). This means we get an alternative definition of Sh(S; P; H) = π(S \ S1(P ) n fP g) as the projection of the intersection of S with its first polar at P . Note that the previous discussion applies to any smooth hypersurface in Pn and not just for smooth cubic surfaces in P3. We have that S \ S1(P ) n fP g is not closed in S because it is missing the point P . Similarly Sh(S; P; H) is not closed in H because we are missing the limiting line obtained by taking the limit of the lines PQ as Q approaches P , thus the tangent lines of S at P . Thus if we take the closure of Sh(S; P; H) in H we add these missing points. In the following theorem we will prove that π induces an isomorphism of varieties between S \ S1(P ) n fP g and Sh(S; P; H) and that the closure Sh(S; P; H) in H is a smooth plane quartic. 2 Theorem 1. Let S be a non-singular cubic hypersurface in P3. Let P 2 S a point lying outside the 27 lines of S, and H a hyperplane not containing P . Then Sh(S; P; H), the closure of its shadow in H, is a smooth quartic curve. Proof. We first choose a change of coordinates g 2 GL(4; k) such that g([0 : 0 : 0 : 1]) = P and g(Z(x3)) = H, for the details on the construction of g see Lemma 6. From now on we will use these new coordinates. This means that S is given as the zero set of the homogeneous polynomial 2 f = x3f1 + x3f2 + f3 (3) with f1; f2; f3 homogeneous polynomials of degree 1,2,3. Also we can identify points in H with 2 points in P by the identification [x0 : x1 : x2 : 0] $ [x0 : x1 : x2], which means we can consider the shadow Sh(S; P; H) to be a subset of P2. 3 2 Let π : P n fP g ! P be the projection from a point defined as π([x0 : x1 : x2 : x3]) = [x0 : x1 : x2]. Let C := S \ S1(P ) be the intersection of S with S1(P ), its first polar at P . The first polar at P has equation 3 X @f @f ∆1 f = P = 1 · = 2x f + f : (4) P i @x @x 3 1 2 i=0 i 3 2 This means C = Z(x3f1 + x3f2 + f3; 2x3f1 + f2).
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