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)\} .