Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 34 (1993), No. 2, 151-156.

Representation of the Hirzebruch-Kleinschmidt Varieties by Quadrics

Günter Ewald and Alexa Schmeinck

Fakultät für Mathematik, Ruhr-Universität Bochum Universitätsstraße 150, W-4630 Bochum 1

Abstract. Let X_{n}(a_{1},…, a_{k}) be an n-dimensional smooth projective toric variety which Kleinschmidt [5] introduced as a higher-dimensional generalization of Hirzebruch surfaces [3]. We show that for r := n − k + ∑_{i=1}^{k} \binom{a_{i}+1+n−k}{n−k} there is a projective embedding ϕ : X_{n}(a_{1},…, a_{k}) ↪ P^{r} such that the ideal of ϕ(X_{n}(a_{1},…, a_{k})) is generated by quadratic binomials.

1. Introduction and result

If P ⊂ R^{n} is a lattice polytope (vertices in Z^{n}) one associates with P an algebraic variety as follows. Let {m_{0},…, m_{r}} = P ∩ Z^{n} be the set of all lattice points of P (in some order). We define parameter functions

(1)  x_{i} := z^{m_{i}} := z_{1}^{m_{i1}} … z_{n}^{m_{in}}, i = 0,…, r

where z = (z_{1},…, z_{n}) ∈ C^{n} (C can be replaced by a more general field). Let [x_{0} : x_{1} : ⋯: x_{r}] be homogeneous coordinates of the complex projective r-space P^{r}. Any affine relation

(2)  \begin{aligned}α_{i_{0}}m_{i_{0}} + ⋯ + α_{i_{s}}m_{i_{s}} & = α_{i_{s+1}}m_{i_{s+1}} + … + α_{i_{r}}m_{i_{r}} \\ α_{i_{0}} + ⋯ + α_{i_{s}} & = α_{i_{s+1}} + … + α_{i_{r}} \end{aligned}

where α_{i_{j}} ≥ 0 (j = 0,…, r) are integers induces a homogeneous monomial equation

(3)  x_{i_{0}}^{α_{i_{0}}} ⋯ x_{i_{s}}^{α_{is}} = x_{i_{s+1}}^{α_{i_{s+1}}} ⋯ x_{i_{r}}^{α_{ir}} .

The set of common solutions of all equations (3) is an algebraic variety X_{P}. The binomials obtained from (3) by subtracting one side from the other generate the ideal of X_{P}. The number of equations (3) can be assumed to be finite. The algebraic torus T = C^{∗n} is naturally embedded into X_{P} as a dense subset, and it acts on X_{P} by diagonal matrices

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1993 Heldermann Verlag, Berlin

(z_{i} ⟶ c_{i}z_{i}, c_{i} ≠ 0, i = 1,…, n). Each k-dimensional face of P defines a (complex) kdimensional subvariety of X_{P} with a k-dimensional torus acting on it. If a toric varity X is defined in more general terms (compare [2], [4], or [7]) then X is compact (complete) and projective if and only if there exists a polytope P and an equivariant algebraic isomorphism ϕ : X ⟶ X_{P}. The lattice polytope P hereby is only determined up to strict combinatorial isomorphism.

We contribute to the problem of finding a generating system for the equations (3) with monomials whose degree is as low as possible. R.J Koelman [6] has shown that for any 2-dimensional variety X_{P} the equations (3) can all be deduced from a finite set of quadratic monomial equations

(4)  x_{i_{1}}x_{i_{2}} = x_{i_{3}}x_{i_{4}}

provided P possesses at least 4 lattice points on its boundary. Batyrev [1] has proved that for arbitrary n it is possible to choose P such that the degree of the equations (3) can be restricted to be ≤ n + 1.

In the present paper we extend Koelman’s result to a class of higher-dimensional, smooth projective toric varieties. The smoothness of X_{P} is characterized by the following property of P (see, for example, [7], p. 15):

(5) P is simple, that is, from each vertex v_{0} of P there emanate precisely n edges of P, and for the adjacent lattice points v_{1},…, v_{n} of v_{0} on such edges we have

det(v_{1} − v_{0},…, v_{n} − v_{0}) = ±1.

Figure 1  Figure 2  Figure 3

We restrict the Picard number of X_{P} to 2, or, equivalently, P to have n + 2 facets. As Kleinschmidt [5] has shown such varieties can be described explicitely. The smallest polytopes which occur for a representation X_{P} can be characterized as follows (which is readily seen from Kleinschmidt’s analysis):

(6) There exist an integer k, 0 < k < n, and integers a_{i} ≥ 0, i = 1,…, k, such that, up to a unimodular transformation, P is obtained from the simplices Δ := conv{0, e_{1},…, e_{k}}, Δ′ := conv{0, e_{k+1},…, e_{n}} (e_{1},…, e_{n} the canonical basis of R^{n}) as

P = conv[Δ′ ∪ (e_{1} + (a_{1} + 1)Δ′) ∪ ⋯ ∪ (e_{k} + (a_{k} + 1)Δ′)].

For n = 2 we have Δ = [0, e_{1}] (line segment), Δ′ = [0, e_{2}] (Figure 1). For n = 3, we have either k = 1 or k = 2 and obtain polytopes as are illustrated in Figure 2 or Figure 3, respectively.

Each lattice point lies in one of the simplices Δ′, e_{1} + (a_{1} + 1)Δ′,…, e_{k} + (a_{k} + 1)Δ′. So the total number of lattice points of P is

(7)  r + 1 = 1 + n − k + ( \begin{matrix}a_{1} + 1 + n − k \\ n − k \end{matrix} ) + ⋯ + ( \begin{matrix}a_{k} + 1 + n − k \\ n − k \end{matrix} ) .

We shall prove:

Theorem. For each Hirzebruch-Kleinschmidt variety X_{n}(a_{1},…, a_{k}) and the embedding ϕ : X_{n}(a_{1},…, a_{k}) ↪ P^{r} according to (1), (3) and (6) the ideal of ϕ(X_{n}(a_{1},…, a_{k})) is generated by quadratic binomials

(8)  x_{i_{1}}x_{i_{2}} − x_{i_{3}}x_{i_{4}}, 0 ≤ i_{j} ≤ r, j = 1, 2, 3, 4,

and hence ϕ(X_{n}(a_{1},…, a_{k})) is an intersection of quadrics.

Examples. For X_{2}(2) (Figure 1) we obtain r = 5 and the equations x_{4}^{2} = x_{3}x_{5}, x_{3}^{2} = x_{2}x_{4}, x_{2}x_{5} = x_{3}x_{4}, x_{1}x_{3} = x_{0}x_{2}, x_{1}x_{4} = x_{0}x_{3}, x_{1}x_{5} = x_{0}x_{4}. In case X_{3}(2) (Figure 2) we find r = 12 and 35 equations (8). For X_{3}(1, 2) (Figure 3) we obtain r = 8 and 15 equations.

Remark. It is an open question if our theorem can be extended to all smooth projective toric varieties. Koelman’s result for n = 2 takes care of all smooth cases. For n ≥ 3 it is not sufficient to have enough lattice prints on the boundary of P, as is true for n = 2. This can be seen in the following example: Let P := conv{m_{0} := 0, m_{1} := e_{1}, m_{2} := e_{2}, m_{i} := e_{1} + e_{2} + (i − 2)e_{3}, i = 3,…, k; k ≥ 4} ⊂ R^{3}. Then m_{0} + 2m_{3} = m_{1} + m_{2} + m_{4} implies a cubic equation which cannot be reduced to quadratic ones.

2. Proof of the theorem by a ”game”

The set of equations (8) is finite since there are only finitely many lattice points in P. We consider P ∩ Z^{n} as a game board. The proof of our theorem can in the following way be reduced to a game on P ∩ Z^{n}, as follows.

In order to build up a monomial equation (3) by equations deduced from (8) we apply two operations:

(a) Multiply each side of a monomial equation by the same variable x_{i}.

(b) Given two monomial equations set the product of the left sides equal to the product of the right sides.

The establishing of (3) is achieved heuristically by applying the following operations to the given affine relation (2):

(a′) Delete an m_{i} if it occurs on both sides of the equation. (b′) If m_{i} + m_{p} = m_{j} + m_{q} for m_{i}, m_{q}, m_{j}, m_{p} ∈ P ∩ Z^{n} and if m_{i} + m_{p} occurs on one side

of (2), replace it by m_{j} + m_{q}. If 0 can be established on both sides, the inverse operations lead to (2) and hence (3).

(a′), (b′) can now be turned into game rules: If m_{i} occurs with coefficient α_{i} > 0 on the left side of (2) we place α_{i} ”red” figures on m_{i}. If m_{j} occurs on the right side of (2) with coefficient α_{j} > 0, we place α_{j} ”green” figures on m_{j}. By assumption there are, at the beginning of the game, on each m_{i} either only red figures or only green figures or no figures at all. (a′), (b′) are equivalent to the following rules: (a′′) If a red figure moves to a point where there is a green one it ”catches” the green figure

and leaves with it the game board. The same applies, conversely, to any green figure. (b′′) A red figure may move from m_{i} to m_{j} if another red figure moves from m_{p} to m_{q}

where m_{i} +m_{p} = m_{j} +m_{q} (or m_{j} −m_{i} = m_{p} −m_{q}), provided m_{i}, m_{j}, m_{p}, m_{q} ∈ P ∩Z^{n} (compare Figure 4). The same rule applies to green figures.

Figure 4

We assert that the game can always be played until the board is empty. (The player who catches the most figures wins the game; one moves alternately unless a player catches a figure in which case he may continue. In the following proof, however, each player keeps moving as long as he can).

In fact: Let α := α_{i_{0}} + ⋯ + α_{i_{s}} = α_{i_{s+1}} + ⋯ + α_{i_{r}}. Then α^{−1}(α_{i_{0}}m_{i_{1}} + ⋯ + α_{i_{s}}m_{i_{s}}) can be considered to be the centroid of the red figures, α^{−1}(α_{i_{s+1}} + ⋯ + α_{i_{r}}m_{i_{r}}) that of the green figures. Clearly:

(9) The centroid of the red figures equals that of the green figures; equality of the centroids remains unchanged if an operation (a′′) or (b′′) is applied.

We set T_{0} := Δ′, T_{1} := e_{1} + (a_{1} + 1)Δ′,…, T_{k} := e_{k} + (a_{k} + 1)Δ′. Any set T_{0} ∪ ⋯ ∪ T_{i−1} ∪ T_{i+1} ∪ ⋯ ∪ T_{k} is contained in a facet of P, i = 0,…, k. Therefore: If there is a red figure in T_{i}, there is also a green one in T_{i}, since, otherwise, the centroids were different. The same is true for a green figure in T_{i}.

Let u be a red figure and v a green figure in T_{i}.

Let H_{i}^{1},…, H_{i}^{n−k+1} be the hyperplanes relative to affT_{i} which are parallel to the facets F_{i}^{1},…, F_{i}^{n−k+1} of T_{i}, respectively, and which pass through u. If v lies in one of these hyperplanes, we can, up to renumeration, assume v to lie in H_{i}^{1},…, H_{i}^{h}, but not in H_{i}^{h+1},…, H_{i}^{n−k+1}. One of the hyperplanes H_{i}^{h+1},…, H_{i}^{n−k+1}, H_{i}^{h+1} say, separates v from the facet F_{i}^{h+1} or contains F_{i}^{k+1} (compare Figure 5 for n = 4, k = 2, h = 1, i = 2).

We denote the regular (n−k −h)-simplex H_{i}^{1} ∩⋯∩H_{i}^{h} ∩T_{i} by T_{i}^{(h)}, and set T_{i}^{(0)} = T_{i} in case v is not contained in any of the H_{i}^{j}, j = 1,…, n − k − h.

Figure 5

Now two possibilities occur: (A) There is a red figure w ≠ u in one of the T_{j}, j = 0,…, h which does not lie in the

facet of T_{j} parallel to F_{i}. (B) All red figures ≠ u lie in the facets F_{j} parallel to F_{i}.

In case (A) we apply (b′′) to u, u′, w, w+(u−u′) instead of m_{p}, m_{q}, m_{i}, m_{j}, respectively, where u′ is chosen to have a shorter distance from v than u has, however, such that H̃_{i}^{h+1} := H_{i}^{h+1} + u′ − u still separates v from F_{j}^{h+1}. If v lies on H̃_{i}^{h+1} we set T_{i}^{(h+1)} := T_{i}^{(h)} ∩ H̃_{i}^{h+1}. If not we repeat the same procedure until we either arrive at a T_{i}^{(h+1)} or (B) becomes true.

In the latter case we interchange the roles of u and v and proceed analogously. Going on with red and green figures we either arrive at T_{i}^{(n−k)} so that a red and a green figure are on the same place, or (B) becomes valid. In the former case, we apply (a′′) and take both figures from the board.

So suppose (B) is true for red figures and, analogously, for green figures, where the green figures ≠ v lie in facets F̃_{j} ≠ F_{j} of T_{j}, j = 0,…, k. We set P_{u} := conv({u} ∪ F_{0} ∪ F_{1} ∪ ⋯ ∪ F_{k}) and P_{v} := conv({v} ∪ F̃_{0} ∪ F̃_{1} ∪ ⋯ ∪ F̃_{k}). Clearly, P_{u} and P_{v} do not have a relative interior point in common. Therefore, the centroid of the red figures and the centroid of the green figures do not coincide, a contradiction.

Hence we can take off two figures and continue the game until the game board is empty.

References

[1] Batyrev, V.: Letter to B. Sturmfels, 1992.

[2] Ewald, G.: Combinatorial Convexity and Algebraic Geometry. Graduate Texts in Mathematics, Springer 1994.

[3] Hirzebruch, F.: Über eine Klasse von einfach zusammenhängenden komplexen Mannigfaltigkeiten. Math. Ann. 124, (1951), 77-86.

[4] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal embeddings I. Lecture Notes in Math. 339, (1973).

[5] Kleinschmidt, P.: A classification of toric varieties with few generators. Äquationes Math. 35, (1988), 254-266.

[6] Koelman, R.: A criterion for the ideal of a projectively embedded toric surface to be generated by quadrics. Beiträge zur Algebra und Geometrie 34 (1993), 57-62.

[7] Oda, T.: Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties. Ergebnisse der Math., Springer 1988.