A Q-factorial complete toric variety with Picard number 2 is projective

Embedding non-projective Mori dream space

“…Since every Mori chamber, in the sense of Hu-Keel, is a filling cell, such a condition is automatically assumed when considering a MDS. As a byproduct, directly following from an analogous result for Q-factorial complete toric varieties, jointly obtained with Lea Terracini [30,Thm. 3.2] and here recalled by Theorem 2.43, one gets that the Hu-Keel result (1) holds for a complete wMDS with Picard number r ≤ 2.…”

“…It is clearly a GKZ-cone contained in the boundary ∂γ 1 . In particular γ is the g-cell corresponding to both the two different fans Σ, Σ ∈ SF(V ) giving rise to complete and non-projective Q-factorial toric varieties, as described in [30,Rem. 3.1].…”

“…Projective embedding of lower rank complete wMDS. In the recent paper [30], jointly written with L. Terracini, we proved 1 the following Let us conclude the present section by recalling that every smooth and complete toric variety of Picard number r ≤ 3 is projective, by a well known result of Kleinschmidt and Sturmfels [19]. This clearly gives the following Notice that the Oda's example recalled in Remark 2.38 is then a sharp counterexample with respect to an extension of Corollary 2.47 for higher values of the Picard number r: in fact it is a smoothly fillable wMDS with r = 4 and non-projective.…”

“…Theorem 2.43 (Thm. 3.2 in[30]). Every Q-factorial complete toric variety of Picard number r ≤ 2 is projective.First of all let us observe that this result allows us to prove a converse of Corollary 2.37 for a complete fillable wMDS of Picard number r ≤ 2.…”

“…In[30] we assumed C as ground field. Actually all techniques and arguments therein employed are completely extendable to a general field K = K with char K = 0.…”