Invariance of plurigenera fails in positive and mixed characteristic

TL;DR

This paper constructs examples of elliptic surface families in positive and mixed characteristic where the plurigenera vary, demonstrating that invariance does not follow from the minimal model program and Abundance Conjectures.

Contribution

It provides explicit counterexamples showing the failure of plurigenera invariance in positive and mixed characteristic settings.

Findings

Constructs smooth families with varying plurigenera

Shows invariance of plurigenera does not follow from MMP and Abundance

Demonstrates failure of invariance in positive/mixed characteristic

Abstract

We construct smooth families of elliptic surface pairs with terminal singularities over a DVR of positive or mixed characteristic $(X,B)\to \mathrm{Spec}R$, such that $P_m(X_k,B_k)>P_m(X_K,B_K)$ for all sufficiently divisible $m>0$. In particular, this shows that invariance of all sufficiently divisible plurigenera does not follow from the MMP and Abundance Conjectures.