A New Subadditivity Formula for Test Ideals

TL;DR

This paper introduces a sharper subadditivity formula for test ideals on singular varieties using Cartier algebras, improving understanding of singularities and their impact on test ideal behavior.

Contribution

It presents a novel subadditivity formula employing Cartier algebras, offering a sharper containment than previous formulas and connecting to Takagi's adjoint test ideal in singular settings.

Findings

New subadditivity formula for test ideals using Cartier algebras

Sharper containment than previous subadditivity results

Combinatorial construction of Cartier algebra in toric cases

Abstract

We exhibit a new subadditivity formula for test ideals on singular varieties using an argument similar to those of Demailly-Ein-Lazarsfeld and Hara-Yoshida. Any subadditivity formula for singular varieties must have a correction term that measures the singularities of that variety. Whereas earlier subadditivity formulas accomplished this by multiplying by the Jacobian ideal, our approach is to use the formalism of Cartier algebras. We also show that our subadditivity containment is sharper than ones shown previously by Takagi and Eisenstein. The first of these results follows from a Noether normalization technique due to Hochster and Huneke. The second of these results is obtained using ideas of Takagi and Eisenstein to show that the adjoint ideal $\mathscr{J}_X(A, Z) $ reduces mod $p$ to Takagi's adjoint test ideal, even when the ambient space is singular, provided that $A$ is regular at the generic point of $X$. One difficulty of using this new subadditivity formula in practice is the computational complexity of computing its correction term. Thus, we discuss a combinatorial construction of the relevant Cartier algebra in the toric setting.