Rigid curves on $\bar M_{0,n}$ and arithmetic breaks

TL;DR

This paper explores the construction of rigid curves on the moduli space _{0,n} and demonstrates, using reduction mod p, that these can be expressed as sums of F-curves, providing insights into the F-conjecture.

Contribution

The paper introduces new methods for constructing rigid curves on _{0,n} and shows their classes decompose into sums of F-curves, advancing understanding of the F-conjecture.

Findings

Rigid curves can be constructed on _{0,n} using new methods.

Classes of these rigid curves decompose into sums of F-curves.

Reduction mod p arguments confirm the decompositions.

Abstract

A result of Keel and McKernan states that a hypothetical counterexample to the F-conjecture must come from rigid curves on $\bar {M}_{0,n}$ that intersect the interior. We exhibit several ways of constructing rigid curves. In all our examples, a reduction mod p argument shows that the classes of the rigid curves that we construct can be decomposed as sums of F-curves.