TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS

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1 TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS YUSUKE SUYAMA Abstract. We give a necessary and sufficient condition for the nonsingular projective toric variety associated to a building set to be weak Fano in terms of the building set. 1. Introduction A toric Fano variety is a nonsingular projective toric variety over C whose anticanonical divisor is ample. It is known that there are a finite number of isomorphism classes of toric Fano varieties in any given dimension. The classification problem of toric Fano varieties has been studied by many researchers. In particular, Øbro [] gave an explicit algorithm that classifies all toric Fano varieties for any dimension. A nonsingular projective algebraic variety is said to be weak Fano if its anticanonical divisor is nef and big. Sato [5] classified toric weak Fano 3-folds that are not Fano but are deformed to Fano under a small deformation, which are called toric weakened Fano 3-folds. We can construct a nonsingular projective toric variety from a building set. Since a finite simple graph defines a building set, which is called the graphical building set, we can also associate to the graph a toric variety (see, for example [8]). The author [6, 7] characterized finite simple graphs whose associated toric varieties are Fano or weak Fano, and building sets whose associated toric varieties are Fano. In this paper, we characterize building sets whose associated toric varieties are weak Fano (see Theorem.4). Our theorem is proved combinatorially by using the fact that the intersection number of the anticanonical divisor with a torus-invariant curve can be computed in terms of the building set. A toric weak Fano variety defines a reflexive polytope. Higashitani [1] constructed integral convex polytopes from finite directed graphs and gave a necessary and sufficient condition for the polytope to be terminal and reflexive. We also discuss a difference between the class of reflexive polytopes defined by toric weak Fano varieties associated to building sets, and that of reflexive polytopes associated to finite directed graphs. The structure of the paper is as follows: In Section, we review the construction of a toric variety from a building set and state the characterization of building sets whose associated toric varieties are weak Fano. In Section 3, we consider reflexive polytopes associated to building sets and finite directed graphs. In Section 4, we give a proof of the main theorem. Date: October 13, Mathematics Subject Classification. Primary 14M5; Secondary 14J45, 5B0, 05C0. Key words and phrases. toric weak Fano varieties, building sets, nested sets, reflexive polytopes, directed graphs. 1

2 YUSUKE SUYAMA Acknowledgment. This work was supported by Grant-in-Aid for JSPS Fellows 15J The author wishes to thank his supervisor, Professor Mikiya Masuda, for his valuable advice, and Professor Akihiro Higashitani for his useful comments.. The main result Let S be a nonempty finite set. A building set on S is a finite set B of nonempty subsets of S satisfying the following conditions: (1) I, J B with I J implies I J B. () {i} B for every i S. We denote by B max the set of all maximal (by inclusion) elements of B. An element of B max is called a B-component and B is said to be connected if B max = {S}. For a nonempty subset C of S, we call B C = {I B I C} the restriction of B to C. B C is a building set on C. Note that B C is connected if and only if C B. For any building set B, we have B = C B max B C. In particular, any building set is a disjoint union of connected building sets. Definition.1. Let B be a building set. A nested set of B is a subset N of B \ B max satisfying the following conditions: (1) If I, J N, then we have either I J or J I or I J =. () For any integer k and for any pairwise disjoint I 1,..., I k N, the union I 1 I k is not in B. Note that the empty set is a nested set for any building set. The set N (B) of all nested sets of B is called the nested complex. N (B) is in fact an abstract simplicial complex on B \ B max. Proposition. ([8, Proposition 4.1]). Let B be a building set on S. Then every maximal (by inclusion) nested set of B has the same cardinality S B max. In particular, if B is connected, then the cardinality of every maximal nested set of B is S 1. We are now ready to construct a toric variety from a building set. First, suppose that B is connected and S = {1,..., n + 1}. We denote by e 1,..., e n the standard basis for R n and we put e n+1 = e 1 e n. For a nonempty subset I of S, we denote e I = i I e i. Note that e S = 0. For N N (B) \ { }, we denote by R 0 N the N -dimensional cone I N R 0e I in R n, where R 0 is the set of nonnegative real numbers, and we define R 0 to be {0} R n. Then (B) = {R 0 N N N (B)} forms a fan in R n and thus we have an n-dimensional toric variety X( (B)). If B is not connected, then we define X( (B)) = C B max X( (B C )). Theorem.3 ([8, Corollary 5. and Theorem 6.1]). Let B be a building set. Then the associated toric variety X( (B)) is nonsingular and projective. The following is our main result: Theorem.4. Let B be a building set. Then the following are equivalent: (1) The associated toric variety X( (B)) is weak Fano. () For any B-component C and for any I 1, I B C such that I 1 I, I 1 I and I I 1, we have at least one of the following: (i) I 1 I B C. (ii) I 1 I = C and (B I1 I ) max.

3 TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS 3 Remark.5. In a previous paper [6], we proved that the toric variety associated to a graphical building set is weak Fano if and only if every connected component of the graph does not have a cycle graph of length 4 or a diamond graph as a proper induced subgraph. However, it is unclear whether this result can be obtained from Theorem.4. Example.6. Theorem.4 implies that if S 4, then the toric variety X( (B)) is weak Fano for any connected building set B on S. Any building set is a disjoint union of connected building sets, and the disjoint union corresponds to the product of toric varieties associated to the connected building sets. Since the product of toric weak Fano varieties is also weak Fano, it follows that all toric varieties of dimension 3 associated to building sets are weak Fano. We recall a description of the intersection number of the anticanonical divisor with a torus-invariant curve, see [3] for details. Let be a nonsingular complete fan in R n and let X( ) be the associated toric variety. For 0 r n, we denote by (r) the set of r-dimensional cones in. For τ (n 1), the intersection number of the anticanonical divisor K X( ) with the torus-invariant curve V (τ) corresponding to τ can be computed as follows: Proposition.7. Let be a nonsingular complete fan in R n and τ = R 0 v R 0 v n 1 (n 1), where v 1,..., v n 1 are primitive vectors in Z n. Let v and v be the distinct primitive vectors in Z n such that τ + R 0 v and τ + R 0 v are in (n). Then there exist unique integers a 1,..., a n 1 such that v + v + a 1 v a n 1 v n 1 = 0. Furthermore, the intersection number ( K X( ).V (τ)) is equal to + a a n 1. Proposition.8 ([4, Proposition 6.17]). Let X( ) be an n-dimensional nonsingular projective toric variety. Then X( ) is weak Fano if and only if ( K X( ).V (τ)) is nonnegative for every (n 1)-dimensional cone τ in. Example.9. Let S = {1,, 3, 4, 5} and B = {{1}, {}, {3}, {4}, {5}, {1,, 3, 4}, {, 3, 4, 5}, {1,, 3, 4, 5}}. Then the nested complex N (B) consists of {{1}, {}, {3}, {1,, 3, 4}}, {{1}, {}, {4}, {1,, 3, 4}}, {{1}, {3}, {4}, {1,, 3, 4}}, {{}, {3}, {4}, {1,, 3, 4}}, {{}, {3}, {4}, {, 3, 4, 5}}, {{}, {3}, {5}, {, 3, 4, 5}}, {{}, {4}, {5}, {, 3, 4, 5}}, {{3}, {4}, {5}, {, 3, 4, 5}}, {{1}, {}, {3}, {5}}, {{1}, {}, {4}, {5}}, {{1}, {3}, {4}, {5}} and their subsets. The pair I 1 = {1,, 3, 4} and I = {, 3, 4, 5} does not satisfy the condition () in Theorem.4. Hence the 4-dimensional toric variety X( (B)) is not weak Fano. In fact, there exists a 3-dimensional cone τ in (B) such that ( K X( (B)).V (τ)) 1. Let Then we have N 1 = {{}, {3}, {4}, {1,, 3, 4}}, N = {{}, {3}, {4}, {, 3, 4, 5}}. R 0 N 1 = R 0 e + R 0 e 3 + R 0 e 4 + R 0 (e 1 + e + e 3 + e 4 ), R 0 N = R 0 e + R 0 e 3 + R 0 e 4 + R 0 ( e 1 ).

4 4 YUSUKE SUYAMA Let us consider τ = R 0 N 1 R 0 N = R 0 e +R 0 e 3 +R 0 e 4. Since (e 1 +e +e 3 + e 4 )+( e 1 ) e e 3 e 4 = 0, Proposition.7 gives ( K X( (B)).V (τ)) = 3 = 1. Therefore X( (B)) is not weak Fano by Proposition Reflexive polytopes associated to building sets An n-dimensional integral convex polytope P R n is said to be reflexive if 0 is in the interior of P and the dual P = {u R n u, v 1 for any v P } is also an integral convex polytope, where, denotes the standard inner product in R n. Let be a nonsingular complete fan in R n. If the associated toric variety X( ) is weak Fano, then the convex hull of primitive generators of rays in (1) is a reflexive polytope. For a building set B such that the associated toric variety X( (B)) is weak Fano, we denote by P B the corresponding reflexive polytope. Higashitani [1] gave a construction of integral convex polytopes from finite directed graphs (with no loops and no multiple arrows). We describe his construction briefly. Let G be a finite directed graph whose node set is V (G) = {1,..., n+1} and whose arrow set is A(G) V (G) V (G). For e = (i, j) A(G), we define ρ( e ) R n+1 to be e i e j. We define P G to be the convex hull of {ρ( e ) e A(G)} in R n+1. P G is an integral convex polytope in the hyperplane H = {(x 1,..., x n+1 ) R n+1 x x n+1 = 0}. In a previous paper we proved that if X( (B)) is Fano, then P B can be obtained from a finite directed graph: Theorem 3.1 ([7, Theorem 4.]). Let B be a building set. If the associated toric variety X( (B)) is Fano, then there exists a finite directed graph G such that P B is equivalent to P G, that is, there exists a linear isomorphism f : R n H such that f(z n ) = H Z n+1 and f(p B ) = P G. However, there exist infinitely many reflexive polytopes associated to building sets that cannot be obtained from finite directed graphs. The following proposition provides such examples: Proposition 3.. Let S = {1,..., n + 1} and B = S \ { }. Then X( (B)) is weak Fano by Theorem.4 but the reflexive polytope P B cannot be obtained from any finite directed graph for n 3. Proof. Suppose that there exists a finite directed graph G such that P B is equivalent to P G. Since 0 P G, there exists a nonempty subset A of A(G) and positive real numbers a e for e A such that e A a e ρ( e ) = 0. If (i 1, i ) A, then we must have (i, i 3 ) A for some i 3 V (G). Continuing this process, eventually we obtain a directed cycle of G. In general, if G has a nonhomogeneous cycle (a directed cycle is a nonhomogeneous cycle), then the dimension of P G is V (G) 1 (see [1, Proposition 1.3]). Hence we have V (G) = n + 1. Since G has at most n(n + 1) arrows, P G has at most n(n + 1) vertices. On the other hand, P B has n+1 vertices. Thus we have the inequality n+1 n(n + 1), but this inequality does not hold for n 3. This is a contradiction. Thus we proved the proposition. Example 3.3. There also exists a reflexive polytope associated to a finite directed graph that cannot be obtained from any building set. Let G be the finite directed graph defined by V (G) = {1,, 3, 4}, A(G) = {(1, ), (, 3), (3, 1), (1, 4), (4, 3)}.

5 TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS 5 Then P G cannot be obtained from any building set. P G is a reflexive 3-polytope with six lattice points. However, there are only three types of reflexive 3-polytopes with six lattice points that are obtained from building sets. They are realized by the following building sets: {{1}, {}, {3}, {4}, {1, }, {1,, 3, 4}}, {{1}, {}, {3}, {4}, {1,, 3}, {1,, 3, 4}}, {{1}, {}, {3}, {1,, 3}, {4}, {5}, {4, 5}}. All the building sets yield reflexive polytopes not equivalent to P G. Figure 1. the directed graph G whose reflexive polytope cannot be obtained from any building set. First we introduce some notation. 4. Proof of Theorem.4 Definition 4.1. Let B be a building set on S. (1) We denote by N (B) max the set of all maximal (by inclusion) nested sets of B. N (B) max is a subset of N (B). () For C B \ B max, we call N (B) C = {N (B \ B max ) \ {C} N {C} N (B)} the link of C in N (B). N (B) C is an abstract simplicial complex on {I (B \ B max ) \ {C} {I, C} N (B)}. (3) For a nonempty proper subset C of S, we call C \ B = {I S \ C I ; I B or C I B} the contraction of C from B. C \ B is a building set on S \ C. The symmetric difference of two sets X and Y is defined by X Y = (X Y ) \ (X Y ). Lemma 4.. Let B be a connected building set on S and let I 1, I B with I 1 I, I 1 I, I I 1 and I 1 I 3. Suppose that such that i 1 I 1 \ I, i I \ I 1, N N (B I1 I ) max, N N (B (I1 I )\{i 1,i }) max (4.1) {I k } N (B I1 I ) max N (B (I1 I )\{i 1,i }) max is not a nested set of B for some k = 1,. Then there exist I 1, I B such that I 1 I 1, I I, i 1 I 1 \ I, i I \ I 1, I 1 I I 1 I and I 1 I = I 1 I.

6 6 YUSUKE SUYAMA Proof. The proof is similar to a part of the proof of [7, Lemma 3.4 (1)]. Without loss of generality, we may assume k = 1. Note that {I 1 } N (B I1 I ) max and N (B (I1 I )\{i 1,i }) max are nested sets of B. Thus (4.1) falls into the following three cases. Case 1. Suppose that (4.1) does not satisfy the condition (1) in Definition.1. Then there exist K {I 1 } N (B I1 I ) max, L N (B (I1 I )\{i 1,i }) max such that K L, L K and K L. If K N (B I1 I ) max, then K L =, a contradiction. Thus we must have K = I 1. Then I 1 L B. We put I 1 = I 1 L and I = I. Since L I 1 I, it follows that L \ I 1 (I 1 I ) \ (I 1 I ). Thus I 1 I I 1 I. Case. Suppose that (4.1) does not satisfy the condition () in Definition.1, and there exist K 1,..., K r N (B I1 I ) max, L 1,..., L s N (B (I1 I )\{i 1,i }) max for r, s 1 such that K 1,..., K r, L 1,..., L s are pairwise disjoint and K 1 K r L 1 L s B. Then we have I k L 1 L s B for each k = 1,. We put I k = I k L 1 L s for k = 1,. L 1 L s I 1 I implies I k I k for some k = 1,. Since I k \ I k (I 1 I ) \ (I 1 I ), we have I 1 I I 1 I. Case 3. Suppose that (4.1) does not satisfy the condition () in Definition.1, and there exist L 1,..., L s N (B (I1 I )\{i 1,i }) max such that I 1, L 1,..., L s are pairwise disjoint and I 1 L 1 L s B. We put I 1 = I 1 L 1 L s and I = I. Since L 1 L s I, it follows that L 1 L s (I 1 I ) \ (I 1 I ). Thus I 1 I I 1 I. In every case, we have i 1 I 1 \ I, i I \ I 1 and I 1 I = I 1 I. This completes the proof. Lemmas 4.3 and 4.5 play key roles in the proof of Theorem.4. Lemma 4.3. Let B be a connected building set on S and let I 1, I I 1 I, I 1 I, I I 1 and I 1 I / B. Then there exist J 1, J B, j 1 J 1 \ J, j J \ J 1, B with N N (B J1 J ) max, N N (B (J1 J )\{j 1,j }) max such that J 1 J, J 1 J / B, J 1 J I 1 I and {J k } N (B J1 J ) max N (B (J1 J )\{j 1,j }) max is a nested set of B for each k = 1,. If J 1 J = {j 1, j }, then N and (B (J1 J )\{j 1,j }) max are understood to be empty. Proof. We use induction on I 1 I. We have I 1 I. Suppose I 1 I =. We put J 1 = I 1 and J = I. Then J 1 J, J 1 J / B and J 1 J = I 1 I. We pick N N (B J1 J ) max. Then {J k } N (B J1 J ) max is a nested set of B for each k = 1,. Suppose I 1 I 3. We pick i 1 I 1 \ I, i I \ I 1, N N (B I1 I ) max and N N (B (I1 I )\{i 1,i }) max. If {I k } N (B I1 I ) max N (B (I1 I )\{i 1,i }) max

7 TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS 7 is a nested set of B for each k = 1,, then there is nothing to prove. Otherwise, by Lemma 4., there exist I 1, I B such that I 1 I 1, I I, i 1 I 1 \ I, i I \ I 1, I 1 I I 1 I and I 1 I = I 1 I. Case 1. Suppose I 1 I / B. We have I 1 I = I 1 I I 1 I < I 1 I I 1 I = I 1 I. By the hypothesis of induction, there exist J 1, J B, j 1 J 1 \ J, j J \ J 1, N N (B J1 J ) max, N N (B (J1 J )\{j 1,j }) max such that J 1 J, J 1 J / B, J 1 J I 1 I = I 1 I and {J k } N (B J1 J ) max N (B (J1 J )\{j 1,j }) max is a nested set of B for each k = 1,. Case. Suppose I 1 I B. We may assume that I 1 I 1. Subcase.1. Suppose I 1 I B. We put I 1 = I 1 and I = I 1 I. Then we have I 1 I = I 1 I / B, i 1 I 1 \I and I 1 \I 1 I \I 1. Since i (I 1 I )\(I 1 I ), we have I 1 I I 1 I. Subcase.. Suppose I 1 I / B. We put I 1 = I 1 and I I 1 I = I 1 I / B, i 1 I 1 \ I, i I \ I 1 and I 1 I Since I 1 \ I 1 (I 1 I ) \ (I 1 I ), we have I 1 I I 1 I. In every subcase, we have I 1 I = I 1 I I 1 I I 1 I. By the hypothesis of induction, there exist J 1, J B, j 1 J 1 \ J, j J \ J 1, N N (B J1 J ) max, N N (B (J1 J )\{j 1,j }) max such that J 1 J, J 1 J / B, J 1 J I 1 I I 1 I and {J k } N (B J1 J ) max N (B (J1 J )\{j 1,j }) max is a nested set of B for each k = 1,. Therefore the assertion holds for I 1 I. Example 4.4. Let S = {1,, 3, 4, 5, 6} and = I. Then we have = I 1 I = I 1 I. < I 1 I I 1 I = B = {{1}, {}, {3}, {4}, {5}, {6}, {1,, 3, 4}, {, 3, 4, 5}, {3, 4, 5, 6}, {1,, 3, 4, 5}, {, 3, 4, 5, 6}, {1,, 3, 4, 5, 6}}. Let us consider I 1 = {1,, 3, 4} and I = {3, 4, 5, 6}. We pick i 1 = 1 and i = 6. Then B I1 I = {{3}, {4}}, B (I1 I )\{i 1,i } = {{}, {5}}. The only maximal nested set of each is the empty set. However, {I 1 } (B I1 I ) max (B (I1 I )\{i 1,i }) max = {{1,, 3, 4}, {3}, {4}, {}, {5}} is not a nested set because {3} {4} {} {5} = {, 3, 4, 5} B (Lemma 4., Case ). Thus we put I (1) 1 = I 1 {, 3, 4, 5} = {1,, 3, 4, 5}, I (1) = I {, 3, 4, 5} = {, 3, 4, 5, 6}. We have I (1) 1 I (1) = {, 3, 4, 5} B (Lemma 4.3, Case ) and I 1 I (1) 1. Since I (1) 1 I = {3, 4, 5} / B (Subcase.), we put I () 1 = I (1) 1 = {1,, 3, 4, 5}, I () = I = {3, 4, 5, 6}.

8 8 YUSUKE SUYAMA We pick i () 1 = 1 and i () = 6. Then B () I = {{3}, {4}, {5}}, B 1 I() () (I 1 I() )\{i() 1,i() } = {{}}. The only maximal nested set of each is the empty set. {I () 1 } (B I () 1 I() = {{1,, 3, 4, 5}, {3}, {4}, {5}, {}} ) max (B () (I }) 1 I() )\{i() 1,i() max is not a nested set because {3} {4} {5} {} = {, 3, 4, 5} B (Lemma 4., Case ). Thus we put I (3) 1 = I () 1 {, 3, 4, 5} = {1,, 3, 4, 5}, I (3) = I () {, 3, 4, 5} = {, 3, 4, 5, 6}. We have I (3) 1 I (3) = {, 3, 4, 5} B (Lemma 4.3, Case ) and I () I (3). Since I () 1 I (3) = {, 3, 4, 5} B (Subcase.1), we put I (4) 1 = I () 1 I (3) = {, 3, 4, 5}, I (4) = I () = {3, 4, 5, 6}. Then B (4) I = {{3}, {4}, {5}}, I (4) 1 I(4) 1 I(4) =. The only maximal nested set of B (4) I is the empty set and 1 I(4) are nested sets of B. {I (4) 1 } (B I (4) {I (4) } (B I (4) 1 I(4) 1 I(4) ) max = {{, 3, 4, 5}, {3}, {4}, {5}}, ) max = {{3, 4, 5, 6}, {3}, {4}, {5}} Lemma 4.5. Let B be a connected building set on S and let I 1, I B with I 1 I, I 1 I, I I 1 and (B I1 I ) max 3. Then there exist J 1, J B, j 1 J 1 \ J, j J \ J 1, N N (B J1 J ) max, N N (B (J1 J )\{j 1,j }) max such that J 1 J, J 1 J / B and {J k } N (B J1 J ) max N (B (J1 J )\{j 1,j }) max is a nested set of B for each k = 1,. Furthermore, we have J 1 J I 1 I or (B J1 J ) max 3. If J 1 J = {j 1, j }, then N and (B (J1 J )\{j 1,j }) max are understood to be empty. Proof. We use induction on I 1 I. We have I 1 I. Suppose I 1 I =. We put J 1 = I 1 and J = I. Then J 1 J and (B J1 J ) max 3. We pick N N (B J1 J ) max. Then {J k } N (B J1 J ) max is a nested set of B for each k = 1,. Suppose I 1 I 3. We pick i 1 I 1 \ I, i I \ I 1, N N (B I1 I ) max and N N (B (I1 I )\{i 1,i }) max. If {I k } N (B I1 I ) max N (B (I1 I )\{i 1,i }) max is a nested set of B for each k = 1,, then there is nothing to prove. Otherwise, by Lemma 4., there exist I 1, I B such that I 1 I 1, I I, i 1 I 1 \ I, i I \ I 1, I 1 I I 1 I and I 1 I = I 1 I.

9 TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS 9 Case 1. Suppose (B I 1 I ) max 3. We have I 1 I = I 1 I I 1 I < I 1 I I 1 I = I 1 I. By the hypothesis of induction, there exist J 1, J B, j 1 J 1 \ J, j J \ J 1, N N (B J1 J ) max, N N (B (J1 J )\{j 1,j }) max such that J 1 J, J 1 J / B and {J k } N (B J1 J ) max N (B (J1 J )\{j 1,j }) max is a nested set of B for each k = 1,. Furthermore, we have J 1 J I 1 I = I 1 I or (B J1 J ) max 3. Case. Suppose (B I 1 I ) max. For any K (B I1 I ) max, there exists unique L K (B I 1 I ) max such that K L K. Hence there exists L (B I 1 I ) max that contains more than one element of (B I1 I ) max. Let K 1,..., K r be all elements of (B I1 I ) max contained in L. Note that I 1 I L is the disjoint union of K 1,..., K r. If L I 1 I, then B L = I 1 I L = K 1 K r / B, a contradiction. Thus L I 1 I. We may assume L I 1. Subcase.1. Suppose I 1 L B. If L I, then B I 1 L = I 1 I L = K 1 K r / B, a contradiction. Thus L I. We put I 1 = I 1 L and I = I. Then we have I 1 I = I 1 I L / B, L \ I I 1 \ I and i I \ I 1. Since i 1 (I 1 I ) \ (I 1 I ), we have I 1 I I 1 I. Subcase.. Suppose I 1 L / B. We put I 1 = I 1 and I = L. Then we have I 1 I = I 1 L / B, i 1 I 1 \I and I \I 1 = L\I 1. Since i (I 1 I )\(I 1 I ), we have I 1 I I 1 I. In every subcase, by Lemma 4.3, there exist J 1, J B, j 1 J 1 \ J, j J \ J 1, N N (B J1 J ) max, N N (B (J1 J )\{j 1,j }) max such that J 1 J, J 1 J / B, J 1 J I 1 I I 1 I and {J k } N (B J1 J ) max N (B (J1 J )\{j 1,j }) max is a nested set of B for each k = 1,. Therefore the assertion holds for I 1 I. Example 4.6. Let S = {1,, 3, 4, 5, 6, 7} and B = {{1}, {}, {3}, {4}, {5}, {6}, {7}, {, 4, 6}, {, 3, 4, 5}, {1,, 3, 4, 5}, {, 3, 4, 5, 6}, {3, 4, 5, 6, 7}, {1,, 3, 4, 5, 6}, {, 3, 4, 5, 6, 7}, {1,, 3, 4, 5, 6, 7}}. Let us consider I 1 = {1,, 3, 4, 5} and I = {3, 4, 5, 6, 7}. We pick i 1 = 1 and i = 7. Then B I1 I = {{3}, {4}, {5}}, B (I1 I )\{i 1,i } = {{}, {6}}. The only maximal nested set of each is the empty set. However, {I 1 } (B I1 I ) max (B (I1 I )\{i 1,i }) max = {{1,, 3, 4, 5}, {3}, {4}, {5}, {}, {6}} is not a nested set because {4} {} {6} = {, 4, 6} B (Lemma 4., Case ). Thus we put I (1) 1 = I 1 {, 4, 6} = {1,, 3, 4, 5, 6}, I (1) = I {, 4, 6} = {, 3, 4, 5, 6, 7}.

10 10 YUSUKE SUYAMA We have I (1) 1 I (1) = {, 3, 4, 5, 6} B (Lemma 4.5, Case ) and L = {, 3, 4, 5, 6}. Since L I 1 and I 1 L = {, 3, 4, 5} B (Subcase.1), we put I () 1 = I 1 L = {, 3, 4, 5}, I () = I = {3, 4, 5, 6, 7}. Now we apply Lemma 4.3 to I () 1 and I (). We pick i() 1 = and i () = 7. Then B () I = {{3}, {4}, {5}}, B 1 I() () (I 1 I() )\{i() 1,i() } = {{6}}. The only maximal nested set of each is the empty set. {I () 1 } (B I () 1 I() = {{, 3, 4, 5}, {3}, {4}, {5}, {6}} ) max (B () (I }) 1 I() )\{i() 1,i() max is not a nested set because {, 3, 4, 5} {6} = {, 3, 4, 5, 6} B (Lemma 4., Case 3). Thus we put I (3) 1 = {, 3, 4, 5, 6}, I (3) = I () = {3, 4, 5, 6, 7}. We have I (3) 1 I (3) = {3, 4, 5, 6} / B (Lemma 4.3, Case 1) and B (3) I = {{3}, {4}, {5}, {6}}, I (3) 1 I(3) 1 I(3) =. The only maximal nested set of B (3) I is the empty set and 1 I(3) {I (3) 1 } (B I (3) {I (3) } (B I (3) 1 I(3) 1 I(3) ) max = {{, 3, 4, 5, 6}, {3}, {4}, {5}, {6}}, ) max = {{3, 4, 5, 6, 7}, {3}, {4}, {5}, {6}} are nested sets of B. Furthermore, we have I (3) 1 I (3) I 1 I. Example 4.7. Let S = {1,, 3, 4, 5, 6, 7} and B = {{1}, {}, {3}, {4}, {5}, {6}, {7}, {, 6}, {4, 5, 6}, {, 4, 5, 6}, {1,, 3, 4, 5}, {, 3, 4, 5, 6}, {3, 4, 5, 6, 7}, {1,, 3, 4, 5, 6}, {, 3, 4, 5, 6, 7}, {1,, 3, 4, 5, 6, 7}}. Let us consider I 1 = {1,, 3, 4, 5} and I = {3, 4, 5, 6, 7}. We pick i 1 = 1 and i = 7. Then B I1 I = {{3}, {4}, {5}}, B (I1 I )\{i 1,i } = {{}, {6}, {, 6}}. The only maximal nested set of B I1 I is the empty set. We choose N = {{}} N (B (I1 I )\{i 1,i }) max. {I 1 } (B I1 I ) max N (B (I1 I )\{i 1,i }) max = {{1,, 3, 4, 5}, {3}, {4}, {5}, {}, {, 6}} is not a nested set because {1,, 3, 4, 5} {, 6} = {} = (Lemma 4., Case 1). Thus we put I (1) 1 = I 1 {, 6} = {1,, 3, 4, 5, 6}, I (1) = I = {3, 4, 5, 6, 7}. We have I (1) 1 I (1) = {3, 4, 5, 6} = {3} {4, 5, 6} (Lemma 4.5, Case ) and L = {4, 5, 6}. Since L I 1 and I 1 L = {4, 5} / B (Subcase.), we put I () 1 = I 1 = {1,, 3, 4, 5}, I () = L = {4, 5, 6}. Now we apply Lemma 4.3 to I () 1 and I (). We pick i() 1 = 1 and i () = 6. Then B () I = {{4}, {5}}, B 1 I() () (I = {{}, {3}}. 1 I() )\{i() 1,i() }

11 TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS 11 The only maximal nested set of each is the empty set. {I () } (B I () 1 I() = {{4, 5, 6}, {4}, {5}, {}, {3}} ) max (B () (I }) 1 I() )\{i() 1,i() max is not a nested set because {4, 5, 6} {} {3} = {, 3, 4, 5, 6} B (Lemma 4., Case 3). Thus we put I (3) 1 = I () 1 = {1,, 3, 4, 5}, I (3) = {, 3, 4, 5, 6}. We have I (3) 1 I (3) = {, 3, 4, 5} / B (Lemma 4.3, Case 1) and B (3) I = {{}, {3}, {4}, {5}}, I (3) 1 I(3) 1 I(3) =. The only maximal nested set of B (3) I is the empty set and 1 I(3) {I (3) 1 } (B I (3) {I (3) } (B I (3) 1 I(3) 1 I(3) ) max = {{1,, 3, 4, 5}, {}, {3}, {4}, {5}}, ) max = {{, 3, 4, 5, 6}, {}, {3}, {4}, {5}} are nested sets of B. Furthermore, we have I (3) 1 I (3) I 1 I. Proposition 4.8 ([8, Proposition 3.]). Let B be a building set on S and let C B \ B max. Then the correspondence { I \ C (C I), I I (C I) induces an isomorphism N (B) C N (B C (C \ B)) of simplicial complexes. Lemma 4.9. Let B be a connected building set on S and J 1, J B with J 1 J, J 1 J, J J 1 and J 1 J S. Let N N (B J1 J ) such that {J k } N N (B J1 J ) max for each k = 1,. Then there exists M N (B) such that {J k, J 1 J } N M N (B) max for each k = 1,. Proof. We pick M N ((J 1 J ) \ B) max. Then {J k } N M N (B J1 J ((J 1 J ) \ B)) max for each k = 1,. Hence by Proposition 4.8, there exists M N (B) such that {J k } N M are maximal simplices of N (B) J1 J. Hence {J k, J 1 J } N M N (B) max for each k = 1,. Proposition 4.10 ([8, Proposition 4.5]). Let B be a building set on S and let I 1, I B with I 1 I and N N (B) such that N {I 1 }, N {I } N (B) max. Then the following hold: (1) We have I 1 I and I I 1. () If I 1 I, then (B I1 I ) max N. (3) There exists {I 3,..., I k } N such that I 1 I, I 3,..., I k are pairwise disjoint and I 1 I k N B max ({I 3,..., I k } can be empty). We are now ready to prove Theorem.4. Proof of Theorem.4. The disjoint union of connected building sets yields the product of toric varieties associated to the connected building sets. Since the product of nonsingular projective toric varieties is weak Fano if and only if every factor is weak Fano, it suffices to show that, for any connected building set B on S = {1,..., n + 1}, the following are equivalent:

12 1 YUSUKE SUYAMA (1 ) The associated toric variety X( (B)) is weak Fano. ( ) For any I 1, I B such that I 1 I, I 1 I and I I 1, we have at least one of the following: (i ) I 1 I B. (ii ) I 1 I = S and (B I1 I ) max. (1 ) ( ): Let I 1, I B such that I 1 I, I 1 I, I I 1 and I 1 I / B. We show that if I 1 I S or (B I1 I ) max 3, then the toric variety X( (B)) is not weak Fano. Case 1. Suppose I 1 I S. By Lemma 4.3, there exist J 1, J B, j 1 J 1 \ J, j J \ J 1, N N (B J1 J ) max, N N (B (J1 J )\{j 1,j }) max such that J 1 J, J 1 J / B, J 1 J I 1 I S and (4.) {J k } N (B J1 J ) max N (B (J1 J )\{j 1,j }) max is a nested set of B for each k = 1,. Since the cardinality of (4.) is J 1 J 1, (4.) is a maximal nested set of B J1 J. By Lemma 4.9, there exists M N (B) such that {J k, J 1 J } N (B J1 J ) max N (B (J1 J )\{j 1,j }) max M N (B) max for k = 1,. Let Clearly τ = R 0 ({J 1 J } N (B J1 J ) max N (B (J1 J )\{j 1,j }) max M). e J1 + e J Since (B J1 J ) max, Proposition.7 gives C (B J1 J ) max e C e J1 J = 0. ( K X( (B)).V (τ)) = (B J1 J ) max 1 1 = 1. Therefore X( (B)) is not weak Fano by Proposition.8. Case. Suppose that I 1 I = S and (B I1 I ) max 3. By Lemma 4.5, there exist J 1, J B, j 1 J 1 \ J, j J \ J 1, N N (B J1 J ) max, N N (B (J1 J )\{j 1,j }) max such that J 1 J, J 1 J / B and (4.3) {J k } N (B J1 J ) max N (B (J1 J )\{j 1,j }) max is a nested set of B for each k = 1,. Furthermore, we have J 1 J I 1 I = S or (B J1 J ) max 3. If J 1 J S, then a similar argument shows that X( (B)) is not weak Fano. Suppose that J 1 J = S and (B J1 J ) max 3. Then (4.3) is a maximal nested set of B. Let τ = R 0 (N (B J1 J ) max N (B (J1 J )\{j 1,j }) max ). Since e J1 J = e S = 0, it follows that e J1 + e J C (B J1 J ) max e C = 0.

13 TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS 13 Proposition.7 gives ( K X( (B)).V (τ)) = (B J1 J ) max 3 = 1. Therefore X( (B)) is not weak Fano by Proposition.8. ( ) (1 ): Let I 1, I B with I 1 I and N N (B) such that N {I 1 }, N {I } N (B) max. We need to show that ( K X( (B)).V (R 0 N)) 0. Case 1. Suppose I 1 I =. By Proposition 4.10 (3), there exists {I 3,..., I k } N such that I 1 I, I 3,..., I k are pairwise disjoint and I 1 I k N B max = N {S}. Since e I1 + e I + e I3 + + e Ik e I1 I k = 0, Proposition.7 gives { k 1 (I1 I ( K X( (B)).V (R 0 N)) = k N), k (I 1 I k = S). Hence ( K X( (B)).V (R 0 N)) 1. Case. Suppose I 1 I. By Proposition 4.10 (1), we have I 1 I and I I 1. (i ) Suppose I 1 I B. By Proposition 4.10 (), we have {I 1 I } = (B I1 I ) max N. By Proposition 4.10 (3), there exists {I 3,..., I k } N such that I 1 I, I 3,..., I k are pairwise disjoint and I 1 I k N B max = N {S}. Since e I1 + e I e I1 I + e I3 + + e Ik e I1 I k = 0, Proposition.7 gives { k (I1 I ( K X( (B)).V (R 0 N)) = k N), k 1 (I 1 I k = S). Hence ( K X( (B)).V (R 0 N)) 0. (ii ) Suppose that I 1 I = S and (B I1 I ) max. By Proposition 4.10 (), we have (B I1 I ) max N. Since e I1 I = e S = 0, it follows that e I1 + e I e C = 0. C (B I1 I ) max Proposition.7 gives ( K X( (B)).V (R 0 N)) = (B I1 I ) max = 0. Therefore X( (B)) is weak Fano by Proposition.8. This completes the proof of Theorem.4. References [1] A. Higashitani, Smooth Fano polytopes arising from finite directed graphs, Kyoto J. Math. 55 (015), [] M. Øbro, An algorithm for the classification of smooth Fano polytopes, arxiv: [3] T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties, Ergeb. Math. Grenzgeb. (3) 15, Springer-Verlag, Berlin, [4] H. Sato, Toward the classification of higher-dimensional toric Fano varieties, Tohoku Math. J. 5 (000), no. 3, [5] H. Sato, The classification of smooth toric weakened Fano 3-folds, Manuscripta Math. 109 (00), no. 1, [6] Y. Suyama, Toric Fano varieties associated to finite simple graphs, Tohoku Math. J., to appear; arxiv:

14 14 YUSUKE SUYAMA [7] Y. Suyama, Toric Fano varieties associated to building sets, Kyoto J. Math., to appear; arxiv: [8] A. Zelevinsky, Nested complexes and their polyhedral realizations, Pure Appl. Math. Q. (006), no. 3, Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka JAPAN address: