F-pure Threshold (FPT) Overview

• F-pure Threshold (FPT) is a rational invariant that quantifies how an ideal or hypersurface fails to split under the Frobenius endomorphism in prime characteristic.
• Explicit formulas for FPT exist for monomials, binomials, and homogeneous polynomials, utilizing base-p expansions and syzygy gap fractals for efficient computation.
• FPT exhibits key properties like semicontinuity, discreteness, and the ascending chain condition, linking arithmetic singularity measurements with log-canonical thresholds.

The F-pure threshold (FPT) is a fundamental invariant in the study of singularities in prime characteristic commutative algebra and algebraic geometry. It measures how tightly a given ideal or hypersurface fails to split under Frobenius endomorphism, providing deep connections between arithmetic, singularity theory, and birational geometry. The F-pure threshold is the characteristic $p$ analogue of the log-canonical threshold (LCT) in characteristic zero, and its precise computation illuminates subtle arithmetic and geometric properties of singularities.

1. Rigorous Definition and Fundamental Properties

Let $R$ be an $F$-finite Noetherian ring of characteristic $p > 0$ and let $I \subset R$ be an ideal containing a nonzerodivisor. For a real number $t > 0$, the pair $(R, I^t)$ is sharply $F$-split if, for infinitely many $e \gg 0$, the natural map

$$I^{\, t(p^e - 1)} \cdot \mathrm{Hom}_R(F_*^e R, R) \longrightarrow R$$

is surjective. The F-pure threshold is defined as

$R$0

In the case where $R$1 is a regular local or polynomial ring, this can be equivalently described as

$R$2

where $R$3 (Baily, 23 Jun 2025, Miller et al., 2012).

Key properties include:

• Rationality: F-pure thresholds are rational numbers.
• Semicontinuity: The FPT behaves upper semi-continuously in families (Stefani et al., 23 Jan 2025).
• Comparison with LCT: If $R$4, then for all large $R$5,

$R$6

• Regularity Criterion: $R$7 if and only if $R$8 is regular (Stefani et al., 23 Jan 2025).
• Discreteness and ACC: The set of F-pure thresholds is discrete and satisfies the ascending chain condition (ACC) (Hernández et al., 2014, Stefani et al., 23 Jan 2025).

2. Relationship to Characteristic Zero Invariants

The F-pure threshold is closely related to the log-canonical threshold, a birational invariant of singularities in characteristic zero. For smooth $R$9-dimensional varieties, Bhatt–Hernández–Miller–Mustaţă establish that the set of all limit points of sequences of F-pure thresholds as $F$0 coincides with the set of LCTs of ideals in dimension $F$1 (Bhatt et al., 2011): $F$2 This provides a dictionary between $F$3-singularities in positive characteristic and log-canonical thresholds in characteristic zero, mediated via the reduction mod $F$4 process. The F-pure threshold also controls the behavior of test ideals, which are characteristic $F$5 analogues of multiplier ideals in characteristic zero.

3. Explicit Computations and Structural Formulas

a) Monomials and Binomials

For monomial ideals, $F$6 (Bhatt et al., 2011). For binomial hypersurfaces, explicit formulas for the FPT are given in terms of the splitting polytope defined by the exponent vectors and their base-$F$7 expansions, with the value determined by a "no-carry" criterion on the base-$F$8 digits and the position of critical points in the polytope (Hernández, 2011).

b) Homogeneous Polynomials in Two Variables: Truncation Formulas

For homogeneous degree $F$9 polynomials in two variables, significant progress has been made in precisely enumerating possible F-pure thresholds. Smith and Vraciu establish that the maximum FPT among all degree $p > 0$0 forms is given by a truncation of the base $p > 0$1 expansion of $p > 0$2, and that all truncations occurring at places determined by the order of $p > 0$3 modulo $p > 0$4 can arise as thresholds of suitable reduced forms: $p > 0$5 where $p > 0$6 and $p > 0$7 is the $p > 0$8-th truncation of its (non-terminating) base-$p > 0$9 expansion. Existence theorems for reduced forms realize every truncation at a suitable place as an FPT, refining earlier necessary conditions (Smith et al., 2022).

A counterexample demonstrates that the necessary list of congruence conditions from Hernández–Núñez-Betancourt–Witt–Zhang is not sufficient for the realization of a given truncation as an FPT, and Smith–Vraciu conjecture that the true constraint is the order of $I \subset R$0 in the multiplicative group $I \subset R$1 (Smith et al., 2022).

c) Generic and Extremal Bounds

For homogeneous polynomials of degree $I \subset R$2 in $I \subset R$3 variables, the generic FPT is the largest truncation $I \subset R$4 satisfying two arithmetic properties depending on $I \subset R$5; in most cases, the generic value is strictly below the characteristic zero log canonical threshold $I \subset R$6 (Smith et al., 2022).

A general sharp lower bound for reduced forms of degree $I \subset R$7 is given by the first nonzero digit in the base-$I \subset R$8 expansion of $I \subset R$9 in two variables or $t > 0$0 generally: $t > 0$1 with sharpness established in a wide range of degrees and characteristics (Smith et al., 2022).

d) Structure of Denominators

For homogeneous two-variable forms, it is shown that if $t > 0$2, then the denominator of $t > 0$3 is always divisible by $t > 0$4 (Hernández et al., 2014). This resolves a question of Schwede and underscores the intricate tightness between the arithmetic of $t > 0$5 and the possible thresholds.

4. Syzygy Gap Fractals and Algorithmic Computation in Two Variables

A crucial combinatorial tool for understanding F-pure thresholds and all $t > 0$6-thresholds in the two-variable homogeneous case is the "syzygy gap fractal" function, defined for polynomials written as products of linear forms. This function encodes the F-threshold function for any collection of exponents, and its critical points correspond to sharp thresholds where the denominator in lowest terms is always a multiple of $t > 0$7 (Hernández et al., 2014).

Main results include:

• When $t > 0$8 (the "expected" value), its denominator is necessarily divisible by $t > 0$9.
• An explicit and efficient algorithm exists for computing F-pure thresholds in two variables, based on iterated colon ideals and base-$(R, I^t)$0 digital expansions (Hernández et al., 2014).

5. Refined Existence and Realization Results

The landscape of realizable F-pure thresholds is highly arithmetic and sensitive to both $(R, I^t)$1 and $(R, I^t)$2. Smith–Vraciu establish:

• For every integer $(R, I^t)$3 not divisible by $(R, I^t)$4, there exists a reduced homogeneous polynomial whose FPT is a truncation of $(R, I^t)$5 at the place $(R, I^t)$6 (Smith et al., 2022).
• Low-degree cases ($(R, I^t)$7) admit complete characterization: all FPTs are precisely those predicted by the truncation and order criterion.

The aforementioned counterexample for $(R, I^t)$8 (with $(R, I^t)$9) demonstrates that additional congruence conditions, as previously believed, are insufficient (Smith et al., 2022).

6. Open Problems and Future Directions

Major open questions concern:

• The full stratification of the parameter space of degree-$F$0 forms by F-pure threshold: for $F$1, existence of reduced forms on every stratum remains unknown.
• A necessary and sufficient combinatorial description of all possible truncations that yield F-pure thresholds for reduced forms, beyond order and congruence constraints.
• Further exploration of connections between F-pure thresholds, Hilbert–Kunz multiplicity, and other Frobenius-splitting invariants.

The refined truncation approach of Smith–Vraciu, algorithmic advances via syzygy gap fractals, and the structural denominators result of Hernández–Núñez-Betancourt–Witt–Zhang collectively provide a comprehensive modern arithmetic theory of the F-pure threshold for homogeneous and binomial hypersurfaces in prime characteristic (Smith et al., 2022, Hernández et al., 2014, Hernández et al., 2014).

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References:

(Smith et al., 2022) Smith and Vraciu: "Values of the F-pure threshold for homogeneous polynomials" (Hernández et al., 2014) Hernández, Teixeira, Witt, Zhang: "F-threshold functions: syzygy gap fractals and the two-variable homogeneous case" (Hernández et al., 2014) Hernández, Núñez-Betancourt, Witt, Zhang: "F-pure thresholds of homogeneous polynomials" (Hernández, 2011) Hernández: "F-pure thresholds of binomial hypersurfaces" (Baily, 23 Jun 2025) Baily: "On lower bounds for the F-pure threshold of equigenerated ideals" (Stefani et al., 23 Jan 2025) Blickle, Schwede, Tucker: "The defect of the F-pure threshold" (Bhatt et al., 2011) Bhatt, Hernández, Miller, Mustaţă: "Log canonical thresholds, F-pure thresholds, and non-standard extensions"