A survey of test ideals

Abstract

Test ideals were first introduced by Mel Hochster and Craig Huneke in their celebrated theory of tight closure, and since their invention have been closely tied to the theory of Frobenius splittings. Subsequently, test ideals have also found application far beyond their original scope to questions arising in complex analytic geometry. In this paper we give a contemporary survey of test ideals and their wide-ranging applications.

44 pages, minor corrections and some added/updated references. Comments very welcome (please let us know if we left out something obvious)

References in corpus (5)

• F-thresholds and Bernstein-Sato polynomials
• F-signature exists
• Local syzygies of multiplier ideals
• Transcendence of some Hilbert-Kunz multiplicities (modulo a conjecture)
• Adjoint ideals and a correspondence between log canonicity and F-purity

Cited by in corpus (13)

• Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems
• The non-nef locus in positive characteristic
• Valuations and Frobenius
• IMPANGA lecture notes on log canonical thresholds
• The gamma construction and asymptotic invariants of line bundles over arbitrary fields
• F-signature of pairs and the asymptotic behavior of Frobenius splittings
• Seshadri Constants and Fujita's Conjecture via Positive Characteristic Methods
• Semi-positivity in positive characteristics
• Cartier modules on toric varieties
• Inversion of adjunction for $F$-signature
• F-singularities via alterations
• $F$-singularities under generic linkage
• Test elements in equal characteristic semianalytic algebras