Counterexamples To Bertini Theorems for Test Ideals

TL;DR

This paper demonstrates that Bertini theorems, which hold for multiplier ideals in algebraic geometry, do not extend to test ideals in characteristic p > 0, revealing limitations in their generalization.

Contribution

The paper provides counterexamples showing that Bertini theorems fail for test ideals in positive characteristic, highlighting a fundamental difference from multiplier ideals.

Findings

Bertini theorems hold for multiplier ideals in characteristic zero.

Counterexamples show failure of Bertini theorems for test ideals in characteristic p > 0.

The result emphasizes limitations in extending classical geometric theorems to test ideals.

Abstract

In algebraic geometry, Bertini theorems are an extremely important tool. A generalization of the classical theorem to multiplier ideals show that multiplier ideals restrict to a general hyperplane section. In characteristic $p > 0$, the test ideal can be seen to be the characteristic $p > 0$ analog of the multiplier ideal. However, in this paper it is shown that the same type of Bertini type theorem does not hold for test ideals.