Toric Fano varieties associated to building sets

Toric Fano varieties associated to building sets

Yusuke Suyama Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585 JAPAN d15san0w03@st.osaka-cu.ac.jp
July 14, 2019
Abstract.

We characterize building sets whose associated nonsingular projective toric varieties are Fano. Furthermore, we show that all such toric Fano varieties are obtained from smooth Fano polytopes associated to finite directed graphs.

Key words and phrases:
toric Fano varieties, building sets, nested sets, directed graphs.
2010 Mathematics Subject Classification:
Primary 14M25; Secondary 14J45, 05C20.

1. Introduction

An -dimensional toric variety is a normal algebraic variety over containing the algebraic torus as an open dense subset, such that the natural action of on itself extends to an action on . The category of toric varieties is equivalent to the category of fans, which are combinatorial objects.

A nonsingular projective algebraic variety is said to be Fano if its anticanonical divisor is ample. The classification of toric Fano varieties is a fundamental problem and has been studied by many researchers. In particular, Øbro [2] gave an algorithm that classifies all toric Fano varieties for any given dimension.

There is a construction of nonsingular projective toric varieties from building sets. The class of such toric varieties includes toric varieties corresponding to graph associahedra of finite simple graphs [4]. On the other hand, Higashitani [1] gave a construction of integral convex polytopes from finite directed graphs. There is a one-to-one correspondence between smooth Fano polytopes and toric Fano varieties. He also gave a necessary and sufficient condition for the polytope to be smooth Fano in terms of the finite directed graph.

In this paper, we give a necessary and sufficient condition for the toric variety associated to a building set to be Fano in terms of the building set (Theorem 2.5). The author [5] characterized finite simple graphs whose associated toric varieties are Fano. Theorem 2.5 generalizes this result (Example 2.6 (2)). Furthermore, we prove that any toric Fano variety associated to a building set is obtained from the smooth Fano polytope associated to a finite directed graph (Theorem 4.1).

The structure of the paper is as follows. In Section 2, we state the characterization of building sets whose associated toric varieties are Fano. In Section 3, we give its proof. In Section 4, we show that all such toric Fano varieties are obtained from finite directed graphs.

Acknowledgment.

This work was supported by Grant-in-Aid for JSPS Fellows 15J01000. The author wishes to thank his supervisor, Professor Mikiya Masuda, for his continuing support. Professor Akihiro Higashitani gave me valuable suggestions and comments.

2. Building sets whose associated toric varieties are Fano

We review the construction of a toric variety from a building set. Let be a nonempty finite set. A building set on is a finite set of nonempty subsets of satisfying the following conditions:

  1. If and , then we have .

  2. For every , we have .

We denote by the set of all maximal (by inclusion) elements of . An element of is called a -component and is said to be connected if . For a nonempty subset of , we call the restriction of to . is a building set on . Note that we have for any building set . In particular, any building set is a disjoint union of connected building sets.

Definition 2.1.

A nested set of is a subset of satisfying the following conditions:

  1. If , then we have either or or .

  2. For any integer and for any pairwise disjoint , we have .

The set of all nested sets of is called the nested complex. is a simplicial complex on .

Proposition 2.2 ([6, Proposition 4.1]).

Let be a building set on . Then all maximal (by inclusion) nested sets of have the same cardinality . In particular, if is connected, then the cardinality of a maximal nested set of is .

First, suppose that is a connected building set on . Let . We denote by the standard basis for and we put . For , we denote . For , we denote by the -dimensional cone , where is the set of non-negative real numbers. We define . Then is a fan in and thus we have an -dimensional toric variety . If is not connected, then we define .

Theorem 2.3 ([6, Corollary 5.2 and Theorem 6.1]).

Let be a building set. Then the associated toric variety is nonsingular and projective.

Example 2.4.

Let and . Then the nested complex is

Hence we have the fan in Figure 1. Therefore the corresponding toric variety is blown-up at one point.

Figure 1. the fan .

Our first main result is the following:

Theorem 2.5.

Let be a building set. Then the following are equivalent:

  1. The associated nonsingular projective toric variety is Fano.

  2. For any -component and for any such that and , we have and .

Example 2.6.
  1. If , then a connected building set on is isomorphic to one of the following six types:

    1. : a point, which is understood to be Fano.

    2. : .

    3. : .

    4. : blown-up at one point.

    5. : blown-up at two points.

    6. : blown-up at three points.

    Thus is Fano in every case. Since the disconnected building set yields , it follows that all toric Fano varieties of dimension are obtained from building sets.

  2. Let be a finite simple graph, that is, a finite graph with no loops and no multiple edges. We denote by and its node set and edge set respectively. For , we define a graph by and . The graphical building set of is defined to be . Theorem 2.5 implies that the toric variety is Fano if and only if each connected component of has at most three nodes, which agrees with [5, Theorem 3.1].

  3. If , then a connected building set on whose associated toric variety is Fano is isomorphic to one of the following nine types:

    1. {{1}, {2}, {3}, {4}, {1, 2, 3, 4}}.

    2. {{1}, {2}, {3}, {4}, {1, 2, 3}, {1, 2, 3, 4}}.

    3. {{1}, {2}, {3}, {4}, {1, 2}, {1, 2, 3, 4}}.

    4. {{1}, {2}, {3}, {4}, {1, 2}, {3, 4}, {1, 2, 3, 4}}.

    5. {{1}, {2}, {3}, {4}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}}.

    6. {{1}, {2}, {3}, {4}, {3, 4}, {1, 2, 3}, {1, 2, 3, 4}}.

    7. {{1}, {2}, {3}, {4}, {1, 2}, {3, 4}, {1, 2, 3}, {1, 2, 3, 4}}.

    8. {{1}, {2}, {3}, {4}, {1, 2}, {1, 2, 3}, {1, 2, 4}, {1, 2, 3, 4}}.

    9. {{1}, {2}, {3}, {4}, {1, 2}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 2, 3, 4}}.

    Among 18 types of toric Fano threefolds, 13 types are indecomposable and five types are products of and toric del Pezzo surfaces (see, for example [3, pp.90–92]). This shows that there are nine types of indecomposable toric Fano threefolds that are obtained from building sets. On the other hand, (1) shows that all toric del Pezzo surfaces are obtained from building sets. Thus there are exactly 14 types of toric Fano threefolds that are obtained from building sets.

3. Proof of Theorem 2.5

We recall a description of the intersection number of the anticanonical divisor with a torus-invariant curve, see [3] for details. For a nonsingular complete fan in and , we denote by the set of -dimensional cones of . We denote by the associated toric variety. For , the intersection number of the anticanonical divisor with the torus-invariant curve corresponding to can be computed as follows:

Proposition 3.1.

Let be an -dimensional nonsingular complete toric variety and , where are primitive vectors in . Let and be the distinct primitive vectors in such that and are in . Then there exist integers such that . The intersection number is equal to .

Proposition 3.2.

Let be an -dimensional nonsingular complete toric variety. Then is Fano if and only if is positive for every .

Let be a building set on . For , we call

the link of in . is a simplicial complex on

For a nonempty proper subset of , we call

the contraction of from . is a building set on .

Proposition 3.3 ([6, Proposition 3.2]).

Let be a building set on and let . Then the correspondence

induces an isomorphism of simplicial complexes.

The symmetric difference of two sets and is defined by . The following is the key lemma.

Lemma 3.4.

Let be a connected building set on and let with and . Then the following hold:

  1. There exist with and , , a maximal nested set of and a maximal nested set of such that

    (3.1)

    are nested sets of for . If , then we can choose so that or .

  2. Furthermore, if , then there exists a nested set of such that

    are maximal nested sets of for ( can be empty).

If , then and are understood to be empty.

Proof.

(1) We use induction on . We have . Suppose . We put and . Clearly and . We choose any maximal nested set of . Then and are nested sets of . If , then .

Suppose . We choose , and maximal nested sets and of and , respectively. If

are nested sets of for , then there is nothing to prove. Without loss of generality, we may assume that

(3.2)

is not a nested set of . We find satisfying and as follows:

Case 1. Suppose that (3.2) does not satisfy the condition (1) in Definition 2.1. and are nested sets. For any and , we have . Hence there exists such that and . Then . We put and . Since , it follows that . Thus .

Case 2. Suppose that (3.2) does not satisfy the condition (2) in Definition 2.1, and there exist

for such that are pairwise disjoint and . Then we have for . We put for . Since , we must have or . If , then it follows that . Thus . Similarly, implies .

Case 3. Suppose that (3.2) does not satisfy the condition (2) in Definition 2.1, and there exist such that are pairwise disjoint and . We put and . Since , it follows that . Thus .

In every case, we have and . Hence . By the hypothesis of induction, there exist with and , , a maximal nested set of and a maximal nested set of such that (3.1) are nested sets of for .

Suppose that . If , then by the hypothesis of induction, we have or . Suppose . We may assume that . We put and . We have and . Since and , it follows that . Hence . We have and , since . By the hypothesis of induction, there exist with and , , a maximal nested set of and a maximal nested set of such that (3.1) are nested sets of for .

Therefore the assertion holds for .

(2) We see that

for . Hence by Proposition 2.2, (3.1) are maximal nested sets of . We choose any maximal nested set of . Then

are maximal nested sets of . By Proposition 3.3,

are in for some . Thus

are maximal nested sets of . ∎

Example 3.5.

The proof of Lemma 3.4 (1) gives a method for obtaining explicit and . Let ,

and . Then . We have

and are maximal nested sets of and , respectively. However,

is not a nested set because of and (Case 1). Thus we put and . But . Thus we put and . Then we have

The only maximal nested set of each is the empty set. However,

is not a nested set because (Case 2). Thus we put and . Then we have

We choose as a maximal nested set of . Then

are nested sets of .

Proposition 3.6 ([6, Proposition 4.5]).

Let be a building set on and let and be two maximal nested sets of with the intersection . Then the following hold:

  1. We have and .

  2. If , then .

  3. There exist such that are pairwise disjoint and ( can be empty).

Proof of Theorem 2.5.

Any building set is a disjoint union of connected building sets. The disjoint union of connected building sets corresponds to the product of toric varieties associated to the connected building sets. The product of nonsingular projective toric varieties is Fano if and only if every factor is Fano. Hence it suffices to show that, for any connected building set on , the following are equivalent:

  • is Fano.

  • .

: Suppose that there exist with such that . We will use the notation of Lemma 3.4.

The case where . By Lemma 3.4, we have maximal nested sets

of for . Let

Clearly

Hence by Proposition 3.1, we have . By Proposition 3.2, is not Fano.

The case where and . By Lemma 3.4 (1), we have nested sets

of for , where or . If , then by Lemma 3.4 (2), we have maximal nested sets