Plus-Pure Threshold (PPT) in Mixed Characteristic

• Plus-Pure Threshold (PPT) is a mixed-characteristic singularity invariant that measures the purity of maps through absolute integral closure, paralleling F-pure and log-canonical thresholds.
• Computations for cusp-like hypersurfaces such as p^a+x^b illustrate that PPT often agrees with the equal-characteristic F-pure threshold under explicit congruence and ramification conditions.
• Advanced techniques using ideal-membership reformulations and perfectoid methods enable precise comparisons between PPT, F-pure thresholds, and log-canonical thresholds, revealing novel mixed-characteristic phenomena.

Searching arXiv for the most relevant papers on plus-pure thresholds in mixed characteristic. The plus-pure threshold is a mixed-characteristic singularity invariant defined for a complete Noetherian local domain of residue characteristic $p>0$. It is intended as the mixed-characteristic analogue of the $F$-pure threshold in characteristic $p$ and, more indirectly, of the log-canonical threshold in characteristic $0$. In the regular complete local setting, it is defined by testing purity of the map $R\xrightarrow{1\mapsto f^t}\widehat{R^+}$, or equivalently by membership of $f^t$ in the maximal ideal times the absolute integral closure. Recent work has shown that this invariant is computable in nontrivial families, especially for cusp-like hypersurfaces of the form $p^a+x^b$, and that it exhibits behavior both analogous to and distinct from its equal-characteristic counterparts (Cai et al., 13 Jan 2025, Benozzo et al., 8 Sep 2025).

1. Definition and formal framework

Let $(S,\mathfrak m)$ be a complete Noetherian local domain of residue characteristic $p>0$, and let $0\neq f\in \mathfrak m$. The plus-pure threshold is defined by

$F$0

This is the formulation given for complete local domains in mixed characteristic, with $F$1 denoting the absolute integral closure (Cai et al., 13 Jan 2025). In the setting emphasized in later work, where $F$2 is a complete Noetherian local domain of mixed characteristic $F$3, the same invariant is written

$F$4

using the $F$5-adic completion $F$6 (Benozzo et al., 8 Sep 2025).

For a regular local ring, the definition admits a concrete ideal-membership reformulation. One has

$F$7

This criterion is central in explicit computations, because it reduces the threshold to deciding whether a rational power $F$8 lies in $F$9 (Benozzo et al., 8 Sep 2025). The earlier mixed-characteristic treatment also records equivalent conditions in the regular case: purity or splitting of the map to $p$0 or $p$1, splitting after passage to finite intermediate extensions, membership $p$2, and triviality of the plus-test ideal $p$3 (Cai et al., 13 Jan 2025).

This places the invariant inside the big Cohen–Macaulay and absolute integral closure framework. A plausible implication is that the plus-pure threshold is best viewed not merely as an analogue by formal resemblance, but as the first threshold detected by mixed-characteristic test-ideal methods.

2. Relation to $p$4-pure and log-canonical thresholds

A basic comparison established for cusp-like hypersurfaces is

$p$5

for

$p$6

and

$p$7

Here $p$8 is the expected log-canonical threshold in the corresponding characteristic-zero model, while $p$9 is the equal-characteristic $0$0 $0$1-pure threshold (Cai et al., 13 Jan 2025). More generally, in mixed-characteristic regular local rings one has the structural inequalities

$0$2

where $0$3 is the reduction modulo the uniformizer (Benozzo et al., 8 Sep 2025).

These inequalities clarify the role of the invariant. It is bounded above by the birational threshold and below by the characteristic-$0$4 Frobenius threshold, but it is not determined by either one in general. This is not just a formal possibility: explicit examples show that $0$5 can coincide with $0$6, coincide with neither $0$7 nor $0$8, and approach $0$9 only after ramifying the coefficient DVR (Cai et al., 13 Jan 2025, Benozzo et al., 8 Sep 2025).

The comparison with $R\xrightarrow{1\mapsto f^t}\widehat{R^+}$0-pure thresholds is especially strong for cusp-like singularities. If

$R\xrightarrow{1\mapsto f^t}\widehat{R^+}$1

then automatically

$R\xrightarrow{1\mapsto f^t}\widehat{R^+}$2

in that family (Cai et al., 13 Jan 2025). By contrast, later work emphasizes that unramified mixed characteristic can force genuinely new behavior: $R\xrightarrow{1\mapsto f^t}\widehat{R^+}$3 may lie strictly between $R\xrightarrow{1\mapsto f^t}\widehat{R^+}$4 and $R\xrightarrow{1\mapsto f^t}\widehat{R^+}$5 (Benozzo et al., 8 Sep 2025).

3. Cusp-like hypersurfaces $R\xrightarrow{1\mapsto f^t}\widehat{R^+}$6

The first substantial computational body of work concerns hypersurfaces

$R\xrightarrow{1\mapsto f^t}\widehat{R^+}$7

The main theorem states that, under explicit hypotheses, the mixed-characteristic threshold agrees with the equal-characteristic $R\xrightarrow{1\mapsto f^t}\widehat{R^+}$8-pure threshold of

$R\xrightarrow{1\mapsto f^t}\widehat{R^+}$9

Precisely, assume

$f^t$0

choose $f^t$1 such that

$f^t$2

and suppose

$f^t$3

Then

$f^t$4

A stronger version replaces $f^t$5 by an integer $f^t$6 and uses the function

$f^t$7

to formulate the criterion (Cai et al., 13 Jan 2025).

This yields many explicit congruence-dependent families where equality holds. For $f^t$8, the paper records the following cases.

$f^t$9	Congruence/prime conditions for $p^a+x^b$0
$p^a+x^b$1	all $p^a+x^b$2
$p^a+x^b$3	all $p^a+x^b$4
$p^a+x^b$5	all $p^a+x^b$6
$p^a+x^b$7	all $p^a+x^b$8 with $p^a+x^b$9
$(S,\mathfrak m)$0	all $(S,\mathfrak m)$1 with $(S,\mathfrak m)$2
$(S,\mathfrak m)$3	all $(S,\mathfrak m)$4
$(S,\mathfrak m)$5	all $(S,\mathfrak m)$6 with $(S,\mathfrak m)$7
$(S,\mathfrak m)$8	all $(S,\mathfrak m)$9 with $p>0$0
$p>0$1	all $p>0$2 with $p>0$3
$p>0$4	all $p>0$5 with $p>0$6
$p>0$7	all $p>0$8 with $p>0$9
$0\neq f\in \mathfrak m$0	all $0\neq f\in \mathfrak m$1 with $0\neq f\in \mathfrak m$2
$0\neq f\in \mathfrak m$3	all $0\neq f\in \mathfrak m$4 with $0\neq f\in \mathfrak m$5

The logic behind these formulas is explicit. The paper imports known algorithms for

$0\neq f\in \mathfrak m$6

often expressed via base-$0\neq f\in \mathfrak m$7 expansions, and then checks whether the equal-characteristic threshold lies close enough to $0\neq f\in \mathfrak m$8 to trigger the comparison theorem (Cai et al., 13 Jan 2025).

The case $0\neq f\in \mathfrak m$9 illustrates the outcome particularly well. One has

$F$00

and the equal-characteristic value is

$F$01

This suggests that, at least for many cusp-like binomials, the mixed-characteristic threshold is governed with surprising accuracy by the mod-$F$02 Frobenius geometry (Cai et al., 13 Jan 2025).

4. Ramification and the limit to the $F$03-pure threshold

A major later development is the discovery that plus-pure thresholds converge to $F$04-pure thresholds under increasing ramification of the base DVR. Let $F$05 be a mixed characteristic complete DVR, let

$F$06

and let $F$07 reduce to $F$08 modulo $F$09. Then

$F$10

This is a sharp asymptotic statement: ramifying by adjoining deeper and deeper $F$11-power roots of the uniformizer drives the mixed-characteristic threshold to the equal-characteristic $F$12 threshold (Benozzo et al., 8 Sep 2025).

Moreover, if the $F$13-pure threshold has terminating base-$F$14 expansion,

$F$15

then equality occurs already at that finite ramification level: $F$16 This finite-level stabilization provides exact formulas in many ramified situations (Benozzo et al., 8 Sep 2025).

The phenomenon is especially transparent for ramified diagonal equations. If

$F$17

and if $F$18, then

$F$19

where

$F$20

This suggests a robust heuristic: sufficient ramification tends to erase specifically mixed-characteristic obstructions and recover the characteristic-$F$21 threshold exactly (Benozzo et al., 8 Sep 2025).

The limit theorem does not imply finite stabilization in general. The paper gives

$F$22

and shows

$F$23

for every $F$24, while the limit remains $F$25. This separates convergence from finite stabilization (Benozzo et al., 8 Sep 2025).

5. Unramified behavior and departures from equal-characteristic analogies

While cusp-like calculations often yield $F$26, later work shows that unramified mixed characteristic has threshold phenomena with no exact equal-characteristic analogue. A central theorem states that certain mixed-characteristic analogues of positive-characteristic extremal singularities do not attain the corresponding extremal value. For example, for

$F$27

or

$F$28

in $F$29, the reduction mod $F$30 is an extremal singularity, but

$F$31

This shows that unramified mixed characteristic can force a threshold strictly above the mod-$F$32 extremal value (Benozzo et al., 8 Sep 2025).

A related example involves

$F$33

For odd $F$34, one proves

$F$35

for any big Cohen–Macaulay $F$36-algebra $F$37, hence

$F$38

This is a nonreduced-mod-$F$39 analogue of the same rigidity phenomenon (Benozzo et al., 8 Sep 2025).

Such results suggest that the unramified plus-pure threshold detects arithmetic constraints not visible in either $F$40 or $F$41 alone. A plausible implication is that ramification is not just a technical device for computation; it separates two genuinely different regimes of mixed-characteristic singularity theory.

6. $F$42-th roots modulo $F$43, upper bounds, and perfectoid purity

Another structural theme is the effect of congruence to a $F$44-th power. In an unramified regular local ring of mixed characteristic, if $F$45 admits a $F$46-th root modulo $F$47, meaning

$F$48

then one has the uniform bound

$F$49

This follows from a more general theorem analyzing the extension $F$50 and showing that suitable étaleness in codimension one forces the threshold below $F$51 (Benozzo et al., 8 Sep 2025).

This has a notable consequence for hypersurfaces of the form

$F$52

In a complete unramified regular local ring of mixed characteristic $F$53, such an equation never defines a perfectoid pure singularity, because

$F$54

Since $F$55 is equivalent in that setting to perfectoid purity of the hypersurface $F$56, the result shows that a full $F$57-adic neighborhood of radius $F$58 around a $F$59-th power consists of non-perfectoid-pure forms (Benozzo et al., 8 Sep 2025).

The same paper also proves a stronger upper bound in the presence of a primitive $F$60-th root of unity. If $F$61 admits a $F$62-th root modulo $F$63, then

$F$64

This is the tamely ramified analogue of the unramified $F$65 bound (Benozzo et al., 8 Sep 2025).

These results tie the plus-pure threshold to perfectoid purity in a concrete way. The threshold is not merely an abstract singularity exponent; it controls whether hypersurfaces are perfectoid pure, and it is highly sensitive to congruence patterns of the defining equation.

7. Elliptic and cubic examples

The paper on ramified regular rings also studies hypersurfaces related to elliptic curves. For

$F$66

with $F$67, one proves

$F$68

Combined with the general lower bound $F$69, this yields

$F$70

For $F$71, this sharpens to

$F$72

Since the characteristic-$F$73 reduction satisfies

$F$74

and the log canonical threshold is

$F$75

this gives a concrete example where

$F$76

Thus the plus-pure threshold is neither the corresponding $F$77-pure threshold nor the log-canonical threshold (Benozzo et al., 8 Sep 2025).

A ramified counterpart behaves more simply. If $F$78 with $F$79 containing a $F$80-th root of $F$81, and $F$82 is a homogeneous cubic defining a nonsingular elliptic curve $F$83, then

$F$84

This is another instance of the general principle that sufficient ramification collapses the mixed-characteristic threshold to the familiar characteristic-$F$85 value (Benozzo et al., 8 Sep 2025).

Earlier work also gave the sporadic example

$F$86

for which

$F$87

The exact value was left open. This was among the first clear demonstrations that $F$88 can exceed $F$89 even for simple cusp-like forms (Cai et al., 13 Jan 2025).

8. Methods of computation

The computational methods used across these papers are remarkably concrete. One basic device is the ideal-membership reformulation

$F$90

which turns threshold calculations into containment problems in absolute integral closures or big Cohen–Macaulay algebras (Cai et al., 13 Jan 2025, Benozzo et al., 8 Sep 2025).

Another method is the use of perfectoid-style containment lemmas. If $F$91, then one has statements of the form

$F$92

which serve as mixed-characteristic substitutes for Frobenius manipulations (Cai et al., 13 Jan 2025, Benozzo et al., 8 Sep 2025). These are especially important in passing from equal-characteristic containments to mixed-characteristic ones after adjoining $F$93-power roots of the uniformizer.

Blow-up geometry also enters. For cusp-like equations $F$94, the normalized blowup yields an exceptional divisor $F$95, and one computes

$F$96

with different

$F$97

These formulas produce the upper bound $F$98 and allow adjunction arguments that relate $F$99 to $p$00-regularity on the exceptional divisor (Cai et al., 13 Jan 2025).

A further ingredient is the finite-cover or cyclic-cover principle

$p$01

which links mixed-characteristic geometry to equal-characteristic binomial $p$02-singularity computations (Cai et al., 13 Jan 2025).

Finally, later work uses valuation and divisibility arguments, including Kummer-type valuations of binomial coefficients, to prove upper bounds such as

$p$03

This suggests that plus-pure threshold computations sit at an unusual intersection of perfectoid algebra, valuation theory, and explicit arithmetic combinatorics (Benozzo et al., 8 Sep 2025).

9. Conceptual significance and open directions

The plus-pure threshold has emerged as the mixed-characteristic threshold analogue of the $p$04-pure threshold, but it is not merely a transplanted equal-characteristic invariant. Several facts support this assessment.

First, it behaves functorially with respect to ramification: $p$05 under adjoining deep $p$06-power roots of the uniformizer (Benozzo et al., 8 Sep 2025). Second, in many low-complexity cusp-like families one has exact equality

$p$07

often with explicit congruence conditions on $p$08 (Cai et al., 13 Jan 2025). Third, in unramified settings it can sit strictly between $p$09 and $p$10, and it can fail to realize extremal mod-$p$11 values (Benozzo et al., 8 Sep 2025). This combination of approximation and divergence is distinctive.

Several open problems remain explicit in the literature. One is whether, for binomials $p$12, the only possible values of $p$13 are

$p$14

This is posed directly after the cusp-like computations (Cai et al., 13 Jan 2025). Another is to determine unresolved small cases such as

$p$15

Later work also raises questions about when the ramification limit theorem stabilizes at finite level, and how far characteristic-$p$16 jumping-number phenomena extend to mixed characteristic (Benozzo et al., 8 Sep 2025).

A further distinction from characteristic $p$17 is that $p$18 need not be a jumping number. The $p$19-adic cubic example supplies this directly, in contrast with the characteristic-$p$20 theory of $p$21-jumping numbers (Benozzo et al., 8 Sep 2025). This suggests that the threshold theory attached to $p$22 is structurally richer than a direct Frobenius analogue.

10. Summary

The plus-pure threshold is defined for mixed-characteristic complete local domains by

$p$23

and, in regular local rings, equivalently by

$p$24

It is bounded by

$p$25

but its actual value is often subtler than either endpoint (Cai et al., 13 Jan 2025, Benozzo et al., 8 Sep 2025).

For cusp-like singularities $p$26, extensive computations show that $p$27 frequently agrees with the mod-$p$28 $p$29-pure threshold $p$30, and explicit congruence conditions on $p$31 can force this equality (Cai et al., 13 Jan 2025). Under increasing ramification of the coefficient DVR, one always has

$p$32

and sometimes equality occurs at finite ramification level (Benozzo et al., 8 Sep 2025). By contrast, in unramified mixed characteristic, the invariant exhibits strictly new phenomena: extremal mod-$p$33 values may not be attained, forms congruent to $p$34-th powers modulo $p$35 satisfy upper bounds such as $p$36, and elliptic-type examples show $p$37 can lie strictly between $p$38 and $p$39 (Benozzo et al., 8 Sep 2025).

This suggests that the plus-pure threshold is best understood as a genuinely mixed-characteristic singularity invariant, controlled by absolute integral closure and perfectoid big Cohen–Macaulay methods, related to but not reducible to the familiar thresholds of equal characteristic.