The minimal projective bundle dimension and toric $2$-Fano manifolds

TL;DR

This paper introduces the minimal projective bundle dimension for smooth toric varieties, classifies those with high dimension, and identifies projective spaces as the unique 2-Fano examples within certain bounds.

Contribution

It defines a new invariant for classifying toric varieties and characterizes 2-Fano manifolds among them based on this invariant.

Findings

Classified smooth projective toric varieties with high minimal projective bundle dimension.

Proved that projective spaces are the only 2-Fano toric varieties within certain invariant bounds.

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