Restriction of test ideals to hypersurfaces

In positive characteristic algebraic geometry and commutative algebra, one of the most fundamental invariants of a variety $X$, or more generally a pair of a variety $X$ with a divisor $\Delta$, is the test ideal $\tau(X,\Delta)$. Larger test ideals imply that the singularities of $X$ and $\Delta$ are mild, while smaller test ideals imply more severe singularities. In characteristic zero, the notion of the multiplier ideal $\mathcal{J}(X,\Delta)$ serves a similar purpose. In addition, the restriction theorem implies that the singularities of a hypersurface $H$ inside $X$ are worse than that of $X$. However, for most choices of such a restriction, the severity of the singularities are unchanged. In this work, it is demonstrated that in positive characteristic, the corresponding statement is false. In particular, there is a large class of examples for which almost every hypersurface has distinctly worse singularities than the ambient variety.