The minimal projective bundle dimension and toric 2-Fano manifolds

Motivated by the problem of classifying toric 2 2 -Fano manifolds, we introduce a new invariant for smooth projective toric varieties, the minimal projective bundle dimension. This invariant m (X) ∈ 1, …, dim ⁡ (X) m (X) \1, , (X) \ captures the minimal degree of a dominating family of rational curves on X X or, equivalently, the minimal length of a centered primitive relation for the fan of X X. We classify smooth projective toric varieties with m (X) ≥ dim ⁡ (X) − 2 m (X) (X) -2, and show that projective spaces are the only 2 2 -Fano manifolds among smooth projective toric varieties with m (X) ∈ 1, dim ⁡ (X) − 2, dim ⁡ (X) − 1, dim ⁡ (X) m (X) \1, (X) -2, (X) -1, (X) \.