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 $m(X)\in \{1,\dots ,\dim (X)\}$ 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 $m(X)\ge \dim (X)-2$, and show that projective spaces are the only $2$-Fano manifolds among smooth projective toric varieties with $m(X)\in \{1,\dim (X)-2,\dim (X)-1,\dim (X)\}$.