Quasi-$F$-singularities and singularities in birational geometry

Abstract: We give an overview of the theory of quasi-$F$-singularities, focusing on their connection with singularities in birational geometry.

• The paper introduces quasi-F-singularities as a refined extension of classical F-singularity notions, effectively bridging Frobenius methods with birational singularity types.
• It develops explicit criteria and analyzes deformation properties using Witt vectors and Fedder-type methods, which accurately assess klt, lc, and pseudo-rational behaviors.
• The study establishes sharp numerical bounds and equivalences for surface and threefold singularities, offering actionable insights for the minimal model program in positive characteristic.

Quasi-$F$-Singularities and Their Connections to Birational Geometry

Overview and Motivation

The theory of $F$-singularities, formulated via the Frobenius endomorphism in positive characteristic, has established deep connections with the classification of singularities in birational geometry, including klt (Kawamata log terminal), lc (log canonical), rational, and Du Bois singularities. However, classical $F$-singularities such as $F$-purity and $F$-regularity are often too rigid to detect subtler birational phenomena, especially in low dimensions or for singularities failing to be $F$-split.

Recent developments have introduced the notion of quasi-$F$-singularities—generalizations of $F$-singularities leveraging the structure of Witt vectors and new pushout constructions—to broaden the scope of Frobenius-based methods in birational geometry. This paper provides a comprehensive overview of quasi-$F$-singularities, their deformation properties, their precise relationship to singularities in the minimal model program, and their effective criteria in the hypersurface setting.

Background: $F$-Singularities and Birational Geometry

$F$0-singularities are defined using the behavior of the Frobenius endomorphism on local or module structures. The four fundamental classes—$F$1-regular, $F$2-pure, $F$3-rational, and $F$4-injective—are connected, via deep theorems ([Smith97], [hw02], [Schwede09_2]), to corresponding classes in characteristic zero:

• $F$5-regular $F$6 klt
• $F$7-pure $F$8 lc
• $F$9-rational $F$0 pseudo-rational
• Cohen-Macaulay $F$1-injective $F$2 pseudo-Du Bois

However, these implications are strict; the converse fail even for surface singularities. For example, cones over supersingular elliptic curves are lc but not $F$3-pure ([Fedder83]).

Quasi-$F$4-Singularities: Definition and Fundamental Properties

Motivations and Construction

The original impetus for quasi-$F$5-splitting arose in Yobuko's work on liftability of Calabi-Yau varieties and extended by Nakkajima–Yobuko to Kodaira-type vanishing theorems ([yobuko19], [NY21]). The formalism centers on modules and endomorphisms over truncated Witt vector rings $F$6, incorporating a pushout $F$7 of the Frobenius and restriction maps. A local ring $F$8 is defined to be $F$9-quasi-$F$0-split if $F$1 splits as a (Witt module) homomorphism.

The systematic study ([KTTWYY1]–[KTTWYY3], [TWY]) has introduced variants of quasi-$F$2-singularities: quasi-$F$3-split, quasi-$F$4-regular, quasi-$F$5-rational, and quasi-$F$6-injective, paralleling but extending the classical classes. Notably, for Gorenstein and surface singularities, quasi-$F$7-splitting coincides with quasi-$F$8-splitting ([KTTWYY3]).

Connections with Birational Singularity Classes

Quasi-$F$9-singularities align much more closely with standard birational singularities than classical $F$0-singularities. The main results include:

• Quasi-$F$1-regular implies klt, and the converse holds in dimension two. For $F$2-Gorenstein surface singularities, being klt is equivalent to being quasi-$F$3-regular ([KTTWYY3]).
• Quasi-$F$4-split implies lc, with equivalence for surfaces of residue characteristic $F$5 ([STY]).
• Quasi-$F$6-rational implies pseudo-rational, and in dimension two (again, under mild conditions), the converse also holds ([KTTWYY3]).
• Quasi-$F$7-injective implies pseudo-Du Bois; if the residue field is perfect, the converse holds in dimension two.

These results are precise: for instance, while $F$8-purity is not equivalent to lc even in surfaces, quasi-$F$9-splitting bridges this gap.

Deformation and Inversion of Adjunction

Deformation properties are central in the minimal model program. Classical $F$0-regularity and $F$1-purity do not generally deform, but $F$2-rationality and $F$3-injectivity do under Cohen-Macaulayness ([Fedder83], [HH94]). For quasi-$F$4-singularities, the deformation picture is subtler:

• Quasi-$F$5-rationality deforms under ambient quasi-$F$6-injectivity: If $F$7 forms a regular sequence, $F$8 quasi-$F$9-rational and $F$0 quasi-$F$1-injective $F$2 $F$3 is quasi-$F$4-rational.
• Quasi-$F$5-regularity, quasi-$F$6-splitting, and quasi-$F$7-injectivity do not deform in general, even for Gorenstein rings.

Higher Dimensional Perspectives and Sharpness

In dimension three, for $F$8-factorial klt singularities over a perfect field of characteristic $F$9, the singularity is quasi-$F$0-regular ([KTTWYY2]), and this lower bound is sharp. However, for lc threefold singularities, one can construct lc but non-quasi-$F$1-split examples in all characteristics using cones over supersingular K3 or abelian surfaces. In dimension four and higher, the divergence widens further ([TY26]).

Explicit Hypersurface Criteria: Fedder-Type and Quasi-$F$2-Splitting Heights

The practical diagnosis of Frobenius properties often depends on explicit homological criteria, historically via Fedder's criterion for hypersurfaces. The paper extends these to quasi-$F$3-splitting via sequences of ideals $F$4, governed by Witt-theoretic analogues of the Frobenius trace and differential operators. The quasi-$F$5-split height is determined by containment of $F$6 in powers of the maximal ideal:

$F$7

Applications:

• Determination of log canonical singularities in characteristic zero via reduction and quasi-$F$8-splitting in positive characteristic.
• Computation of Artin–Mazur heights for Calabi-Yau complete intersections ([KTY22], [BGP]).
• Construction of varieties with arbitrary Artin–Mazur heights ([KTY25]).

Quasi-$F$9-Pure Thresholds

Quasi-$F$0-pure thresholds $F$1 interpolate between $F$2-pure thresholds and log canonical thresholds, refining the connection in dimension two:

$F$3

The paper provides explicit computation methods for these thresholds, extending Mustaţă–Takagi–Watanabe's approach to the quasi-$F$4 setting.

Theoretical and Practical Implications

Theoretically, quasi-$F$5-singularities yield a framework that more closely tracks the expected correspondence between Frobenius-based singularity theory and birational geometry than $F$6-singularities, particularly for low-dimensional, (quasi-)Gorenstein or $F$7-factorial cases. Their stability under quotient and completion, their compatibility with local cohomology, and the precise links with rational and Du Bois singularities, make quasi-$F$8-singularities a robust toolkit for furthering the minimal model program in positive characteristic.

Practically, the explicit criteria—Fedder-type, direct computations in the Witt vector setting—enable the construction and analysis of singularities otherwise inaccessible via classical $F$9-singularities, including the calibration of invariants such as singularity heights and thresholds relevant for lifting, vanishing, and degeneration problems.

Numerical Results and Contrasting Claims:

• Sharp bounds on $F$0 for threefold quasi-$F$1-regularity ($F$2).
• Counterexamples to deformation for quasi-$F$3-splitting in explicit Gorenstein hypersurfaces.
• Existence of Calabi-Yau varieties of arbitrary Artin–Mazur height over $F$4.

Open Problems and Future Directions

Key open questions include:

• Do normal quasi-$F$5-split singularities imply quasi-$F$6-injectivity for all $F$7?
• Are all quasi-$F$8-regular singularities Cohen-Macaulay beyond dimension three or without the $F$9-Gorenstein hypothesis?
• For regular local rings and $F$00, does $F$01 imply quasi-$F$02-splitness of $F$03?

These problems touch on subtle invariants of singularities and their relationships to the birational and arithmetic structure of algebraic varieties. Progress in this direction will clarify the landscape between Frobenius methods and geometric classification, with possible implications for liftability, deformation theory, and the behavior of cycles and cohomology in mixed and positive characteristic.

Conclusion

Quasi-$F$04-singularities represent a critical extension of the Frobenius-based approach to singularity theory in algebraic geometry, resolving known limitations of classical $F$05-singularities and furnishing refined correspondences with birational classification. Their robust properties, explicit criteria, and deep connections with both the structure of singularities and arithmetic invariants mark them as central objects of study for further advances in positive characteristic geometry and its applications to vanishing, extension, and lifting phenomena.