Complete intersections and movable curves on the moduli space of six-pointed rational curves

TL;DR

This paper investigates the relationship between the cone of movable curves and the complete intersection cone on certain moduli spaces of rational curves, revealing they coincide in some cases but differ in others, with implications for understanding curve families.

Contribution

It characterizes when the movable and complete intersection cones coincide on moduli spaces, introducing an algorithm applicable to broader classes of varieties.

Findings

The cones coincide for toric moduli spaces.

The cones differ for the non-toric space M_{0,6}.

An example of a toric threefold where the cones differ.

Abstract

A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by studying the complete intersection cone on a family of blow-ups of complex projective space, including the moduli space of stable six-pointed rational curves, $\bar{M}_{0,6}$, and the permutohedral or Losev-Manin moduli space of four-pointed rational curves. Our main result is that the movable and complete intersection cones coincide for the toric members of this family, but differ for the non-toric member, $\bar{M}_{0,6}$. The proof is via an algorithm that applies in greater generality. We also give an example of a projective toric threefold for which these two cones differ.