On invariance of plurigenera for foliations on surfaces

TL;DR

The paper proves that the plurigenera of foliations on surfaces remain invariant in families for large m, but can vary for small m, highlighting the importance of the parameter m.

Contribution

It establishes invariance of plurigenera for foliations on surfaces for large m and provides counterexamples for small m, advancing understanding of foliation invariants.

Findings

Invariance holds for sufficiently large m.

Counterexamples show failure for small m.

Highlights the dependence on the parameter m.

Abstract

We show that if $(X_t,\mathcal{F}_t)_t$ is a family of foliations with reduced singularities on a smooth family of surfaces, then invariance of plurigenera holds for sufficiently large $m$. On the other hand, we provide examples on which the result fails, for small values of $m$.