The minimal projective bundle dimension and toric 2-Fano manifolds

Abstract

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 centrally symmetric 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)}.