THE MINIMAL PROJECTIVE BUNDLE DIMENSION AND TORIC 2-FANO MANIFOLDS

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