Article Lorentzian Lattices and E-Polytopes Adrian Clingher 1 and Jae-Hyouk Lee 2,* 1 Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, MO 63121, USA; [email protected] 2 Department of Mathematics, Ewha Womans University 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea * Correspondence: [email protected]; Tel.: +82-2-3277-3346 Received: 25 July 2018; Accepted: 17 September 2018; Published: 28 September 2018 Abstract: We consider certain En-type root lattices embedded within the standard Lorentzian lattice + Zn 1 (3 ≤ n ≤ 8) and study their discrete geometry from the point of view of del Pezzo surface + geometry. The lattice Zn 1 decomposes as a disjoint union of affine hyperplanes which satisfy a certain periodicity. We introduce the notions of line vectors, rational conic vectors, and rational cubics vectors and their relations to E-polytopes. We also discuss the relation between these special vectors and the combinatorics of the Gosset polytopes of type (n − 4)21. Keywords: lorentzian lattice; weyl group; root lattice; dual lattice; lines; gosset polytope; E-polytope MSC: 03G10; 05B25; 52B20; 14J26 1. Introduction Lattices and their related discrete geometry appear naturally in the study of algebraic surfaces. In this context, the Picard group Pic(S) of a del Pezzo surface S is a typical example of Lorentzian lattice, determining, in turn, a root lattice with Weyl group W of E-type [1–7]. Special divisor classes in Pic S, such as lines, rulings and exceptional systems are of interest [5–7]. The convex hull of the set of lines is a Gosset polytope and some special divisor classes correspond to the facets of these Gosset polytopes [3,5]. Such divisor classes were studied in [6]. The present article builds on this work. + Consider the Lorentzian lattice Zn 1 (3 ≤ n ≤ 8) with signature (1, n), identified geometrically with H2(S, Z) where S is a general del Pezzo surface of degree 9 − n. This lattice carries a canonical ? element Kn with length (Kn, Kn) = 9 − n. The orthogonal sub-lattice Kn turns out to be a root lattice a of En-type, in the terminology of [3]. We study the affine lattice hyperplanes Ln consisting of lattice a elements D with (D, Kn) = a. We prove an identification between Ln and elements of the discriminant ?_ ? a group Kn /Kn . This fact is used to explain a periodicity appearing in the lattice structure on Ln. Motivated by considerations in [6], we introduce the notions of lines, rational conic vectors and + rational cubics as elements in Zn 1 ⊗ Q. These special elements form orbits under the action of the Weyl group Wn and their lattice structures are and naturally related to the ones of certain En-polytopes, such as the Gosset polytopes (n − 4)21, 2(n−4)1, 1(n−4)2. We compute the total numbers of these subsets via theta series associated with root lattices and their duals. We also consider lattice elements that can be written as sum of lines and study the configurations of lines analog to the discrete geometry of the Gosset polytopes. As an application, we show that each ? root d in Kn can be written as a difference of two distinct perpendicular lines. In the next article, we will consider certain type of K3 surfaces related to del Pezzo surfaces via the involutive automorphism. The lattice structures of the K3 surfaces are also Lorentzian ones with Weyl action, and play key roles to understand the geometry of K3 surfaces. We expect the study of E-polytopes of del Pezzo surfaces can be extended to the discrete geometry of K3 surfaces. Symmetry 2018, 10, 443; doi:10.3390/sym10100443 www.mdpi.com/journal/symmetry Symmetry 2018, 10, 443 2 of 13 2. Root Lattices and Hyperplanes n+1 Let Z be the Lorentzian lattice with rank n + 1 and standard basis f ei j 0 ≤ i ≤ ng satisfying (e0, e0) = 1, (ei, ei) = −1 for i = 1, ..., n, (ei, ej) = 0 for i 6= j . We denote by (, ) the lattice inner product. We shall refer to (v, v) as the length of the lattice point v. Motivated by the geometry of del Pezzo surfaces in algebraic geometry, we consider the following specific integral vector Kn := −3e0 + e1 + ... + en which we shall refer to as the canonical vector. We also restrict the range of the parameter n to 3 ≤ n ≤ 8, so that Kn has positive length (Kn, Kn) = 9 − n. ? n+1 Let Kn ⊂ Z be the orthogonal complement sub-lattice n o ? n+1 Kn := D 2 Z (D, Kn) = 0 ? ? It follows that the restricted inner product ( . ) on Kn is negative definite and Kn is in fact a root lattice (see [3,4]) with a root system given by: n n+1 o Rn := D 2 Z j (D, D) = −2, (D, Kn) = 0 . The overall number of roots is as given below: n 3 4 5 6 7 8 jRnj 12 20 40 72 126 240 Total numbers of the roots of Rn A set of simple roots in Rn is constructed as: d0 = e0 − e1 − e2 − e3, di = ei − ei+1, 1 ≤ i ≤ n − 1, with an associated Dynkin diagram of En-type (see [3,5,6]): d0 u ud1 ud2 ud3 ud4 r r r udn−1 Dynkin diagram of En n≥3 We shall therefore refer to the list below as the extended list of En’s: n 3 4 5 6 7 8 En A1×A2 A4 D5 E6 E7 E8 Again borrowing terminology from algebraic geometry, we shall refer to the product deg(D) = (D, −Kn) + as the degree of a vector D. The length and degree of a vector D in Zn 1 satisfy the inequality 2 2 (D, D)(Kn, Kn) ≤ (D, −Kn) = deg(D) Symmetry 2018, 10, 443 3 of 13 This follows from the Lemma below (the backwards Cauchy-Schwartz in Lorentzian setting). + + Lemma 1. Let v in Zn 1 with positive length. Then any D in Zn 1 satisfies, (D, D)(v, v) ≤ (D, v)2 The equality holds if and only if D and v are scalar multiple of each other. ? + Proof. Since v has positive length, every point of its orthogonal complement fvg ⊂ Zn 1 ⊗ R has non-positive length. Hence: (D, v) (D, v) D − v, D − v ≤ 0 . (v, v) (v, v) The needed inequality follows. r n+1 Next, we introduce the fixed-degree hyperplanes Ln ⊂ Z defined as n o r n+1 Ln := D 2 Z (D, −Kn) = r . 0 ? In particular, Ln is the root lattice Kn . r n+1 Consider furthermore the subsets Mn of Z ⊗ Q defined by r r r Mn := Kn + Ln . (Kn, Kn) r ?_ r Lemma 2. The following statements hold: (1) Mn is a subset of the dual lattice Kn , (2) For any v 2 Mn, one has r ? ? Mn = v + Kn = f v + g j g 2 Kn g . r Proof. (1) Note that, for each u in Ln, we have r Kn + u, Kn = 0 . (Kn, Kn) r 0 0 Hence, Mn is a subset in Ln ⊗ Q. Furthermore, for each w 2 Ln, we have r Kn + u, w = (v, w) 2 Z (Kn, Kn) r ?_ One concludes that Mn is a subset of the dual lattice Kn . r ? r (2) Note that, for any two v1 and v2 in Mn, v1 − v2 2 Kn . Thus, for any choice v 2 Mn, we get r ? Mn ⊂ v + Kn . ? Conversely, for any g in Kn , one verifies: r v + g − Kn, −Kn = − (v, Kn) + 0 + r = r . (Kn, Kn) Moreover: r r n+1 v + g − Kn = v − Kn + g 2 Z . (Kn, Kn) (Kn, Kn) ? r r ? Hence, one has v + Kn ⊂ Mn. One concludes Mn = v + Kn . We note that Lemma2 above provides a canonical group morphism: ? j : Z ! D(Kn ) Symmetry 2018, 10, 443 4 of 13 r ? which associates to any fixed-degree subset Ln a coset v + Kn . Here, _ ? ? ? D(Kn ) = Kn /Kn ? denotes the discriminant group of the negative-definite root lattice Kn , which is known to be a cyclic group of order 9 − n. One can easily see that the morphism j is surjective. r Let us also note that the choice of v in Mn representing the coset may be selected in a canonical ? way. Consider the basis of simple roots B = f di 2 Rn j i = 1, ..., ng of Kn , as introduced earlier. Then: _ ? ? [ Kn : Kn ] = det(Gram) = (Kn, Kn) = 9 − n where Gram is the n × n symmetric matrix whose (i, j) entry is given by the pairing (di, dj) of the corresponding simple roots. Consider, in addition, the following concept, as introduced, for instance, in [8], Section4: Definition 1. The fundamental parallelepiped associated with B is, by definition: ( ) n _ ? ? P(B) := ∑ aidi ai 2 [0, 1] for each i ⊂ Kn ⊗ R ⊂ Kn ⊗ R i=1 ?_ It follows then (see Lemma 4.2 of [8]) that for each w 2 Kn one has a unique decomposition w = uo + vo ?_ ? where uo 2 P(B) \ Kn and vo 2 Kn . We then have the following: ?_ r ? Theorem 1. For each r 2 Z, there exists a unique uo 2 P(B) \ Kn such that Mn = uo + Kn . r Proof. We know that each element w in Mn has a unique decomposition w = uo + vo ?_ ? with uo 2 P(B) \ Kn and vo 2 Kn . In fact, the element uo is independent of the choice of w.