Perfectoid pure thresholds of lifts of rational double points

Abstract: We study the perfectoid pure threshold with respect to $p$, an invariant of singularities in mixed characteristic $(0,p)$ arising from perfectoid purity. In this paper, we compute perfectoid pure thresholds for lifts of rational double points. We show that the set of such thresholds is contained in $\mathbb{Q}$ and satisfies the ascending chain condition. In characteristic $2$, all reciprocals of positive integers occur, and $0$ is the unique accumulation point.

• The paper develops multi-heights and Fedder-type criteria to compute perfectoid pure thresholds for mixed-characteristic lifts of rational double points, with pre-periodic multi-heights implying rational values.
• Thresholds for these lifts are rational and satisfy the ascending chain condition; at characteristic 2, the set contains every reciprocal 1/m and has 0 as its only accumulation point.
• The results extend discreteness phenomena beyond regular rings, while leaving the complete classification of D-type thresholds and general multi-height pre-periodicity as open problems.

The perfectoid pure threshold $\ppt(R,p)$ is a mixed characteristic $(0,p)$ invariant of singularities introduced via the theory of perfectoid purity, playing the role of the $F$-pure threshold in positive characteristic and the log canonical threshold in characteristic zero. The paper under review, by Takamatsu and Yoshikawa (2603.24926), computes this invariant for $W(k)$-lifts of rational double points (RDPs) over an algebraically closed field of characteristic $p$, and derives discreteness properties of the resulting set of thresholds. The main theorem establishes that the set $\Sigma$ of all such thresholds is contained in $\mathbb{Q}$ and satisfies the ascending chain condition (ACC); when $p=2$, it contains every reciprocal $1/m$ with $m \in \mathbb{Z}_{\ge 1}$, and its only accumulation point is $(0,p)$0.

Background and context

Mixed characteristic singularity theory has developed rapidly through works such as those establishing the mixed characteristic minimal model program and Skoda-type theorems. Within this framework, the second author previously proved that for regular local rings, the set of perfectoid pure thresholds with respect to $(0,p)$1 coincides exactly with the set of $(0,p)$2-pure thresholds of pairs in characteristic $(0,p)$3:

$(0,p)$4

so discreteness results known for $(0,p)$5-pure thresholds—rationality, ACC, and control of accumulation points—transfer immediately to the regular case. The present paper extends these phenomena beyond regular rings, to lifts of surface canonical singularities, where no such dictionary is available.

Multi-height and Fedder-type criteria

The central technical innovation is the multi-height $(0,p)$6, a refinement of the mixed-characteristic quasi-$(0,p)$7-split height. It is defined inductively: given modules built from pushout diagrams involving Frobenius twists $(0,p)$8 and maps $(0,p)$9, one sets

$F$0

A ring admits a multi-height if and only if it is quasi-$F$1-split, in which case the sequence is unique; moreover $F$2 equals the ordinary quasi-$F$3-split height and $F$4 for all $F$5. For a quasi-$F$6-split complete intersection local ring with multi-height $F$7, the threshold admits an explicit expansion:

$F$8

Thus pre-periodicity of the multi-height is equivalent to rationality of $F$9.

To compute the multi-height effectively, the authors prove a Fedder-type criterion: for $W(k)$0 with $W(k)$1 regular admitting a Frobenius lift, splitting of $W(k)$2 is detected by membership of explicitly constructed ideals $W(k)$3 in $W(k)$4, so that

$W(k)$5

Several practical refinements follow. A variant eliminates the ambient ideal terms, reducing computation to non-membership conditions of the form $W(k)$6, together with a recursive colon-ideal test $W(k)$7. A stability result gives sufficient conditions forcing $W(k)$8 for all sufficiently large $W(k)$9: if certain elements lie outside $p$0, the tail of the multi-height stabilizes at $p$1. Finally, a periodicity detection proposition allows verification of a pre-periodic multi-height by checking only finitely many ideal equalities—a feature exploited in computer-assisted computations using Macaulay2.

The authors also introduce a naive multi-height defined purely from the special fiber $p$2, which dominates the true multi-height lexicographically and depends only on $p$3 up to congruence of the defining equations modulo $p$4. Its first term recovers the positive-characteristic quasi-$p$5-split height. The authors note candidly that an intrinsic definition of multi-height in positive characteristic remains unavailable, and that the naive version may depend on the choice of Frobenius lift.

Thresholds of lifts of RDPs

Recall that an RDP over $p$6 is a complete local ring realizing a non-smooth canonical surface singularity in Artin's list, and a $p$7-lift is a $p$8-flat algebra reducing to it modulo $p$9. Any such lift has the form $\Sigma$0 with $\Sigma$1 lifting an Artin equation. The paper's classification yields:

• Taut RDPs: $\Sigma$2 for every lift, since the special fiber is $\Sigma$3-pure.
• Non-taut RDPs other than types $\Sigma$4, $\Sigma$5 ($\Sigma$6) in characteristic 2: the full set of attainable values is tabulated. Representative values include $\Sigma$7 for type $\Sigma$8 at $\Sigma$9; $\mathbb{Q}$0 for type $\mathbb{Q}$1 at $\mathbb{Q}$2; and $\mathbb{Q}$3 for type $\mathbb{Q}$4 at $\mathbb{Q}$5. Each value is realized by an explicit perturbation $\mathbb{Q}$6, and the proofs bound the possible values by analyzing which coefficients of $\mathbb{Q}$7 can be nonzero modulo $\mathbb{Q}$8.

For instance, in the $\mathbb{Q}$9 case at $p=2$0, the argument splits according to whether $p=2$1 or $p=2$2: monomial-order computations show that unless specific low-degree coefficients of $p=2$3 vanish mod 2, the multi-height stabilizes at $p=2$4 after the first term; otherwise it becomes constant ($p=2$5 or $p=2$6 for all $p=2$7), producing thresholds like $p=2$8 or $p=2$9.

The types $1/m$0 and $1/m$1

The remaining cases, occurring only at $1/m$2, are handled by a detailed monomial analysis. Writing $1/m$3 (or the variant with $1/m$4), the behavior depends on whether the coefficient of $1/m$5 in $1/m$6 is divisible by 2. Two auxiliary sequences $1/m$7 and $1/m$8 are defined recursively from $1/m$9 alone; the key lemma shows that either $m \in \mathbb{Z}_{\ge 1}$0 with an explicit leading-monomial congruence, or the multi-height collapses to $m \in \mathbb{Z}_{\ge 1}$1 from some index onward. Since $m \in \mathbb{Z}_{\ge 1}$2 takes values in the finite range $m \in \mathbb{Z}_{\ge 1}$3, it is eventually periodic, and consequently each multi-height lies in a finite set of possibilities. In particular, the multi-height is always pre-periodic here, so $m \in \mathbb{Z}_{\ge 1}$4 for every lift of every $m \in \mathbb{Z}_{\ge 1}$5-type RDP.

Closed formulas emerge in special cases. For the natural lift ($m \in \mathbb{Z}_{\ge 1}$6), if $m \in \mathbb{Z}_{\ge 1}$7 for all $m \in \mathbb{Z}_{\ge 1}$8, then

$m \in \mathbb{Z}_{\ge 1}$9

for the perturbation $(0,p)$00 with $(0,p)$01 for all $(0,p)$02,

$(0,p)$03

Moreover, for the natural lift the multi-height coincides with the naive multi-height of the special fiber, so the threshold equals the minimum among all lifts of that type. As concrete data, for $(0,p)$04 the natural-lift thresholds run from $(0,p)$05 down through values including $(0,p)$06, $(0,p)$07, $(0,p)$08, $(0,p)$09, $(0,p)$10, and finally $(0,p)$11.

Rationality and the ascending chain condition

Combining the table with the $(0,p)$12-type analysis yields the main theorem. Rationality follows since every multi-height encountered is pre-periodic. The containment $(0,p)$13 at $(0,p)$14 follows from the formula $(0,p)$15 applied with $(0,p)$16, $(0,p)$17. The proof of ACC proceeds by contradiction: any strictly increasing chain must eventually consist of $(0,p)$18-type examples with $(0,p)$19; then either $(0,p)$20 attains its maximal value $(0,p)$21, forcing $(0,p)$22 bounded and hence finitely many values, or it does not, in which case $(0,p)$23 and again only finitely many values occur. Accumulation points are handled similarly: a nonzero accumulation point would force $(0,p)$24 to be bounded while matching a fixed first height $(0,p)$25, again contradicting infinitude.

Two structural consequences deserve emphasis. First, unlike the regular case, where thresholds coincide with $(0,p)$26-pure thresholds and inherit their known discreteness, here the ACC is proved directly for a genuinely singular class of rings, generalizing the earlier phenomena. Second, the appearance of all reciprocals $(0,p)$27 at $(0,p)$28 shows that $(0,p)$29 is infinite with $(0,p)$30 as its unique accumulation point—the threshold set is as large as the ACC permits in this respect.

Limitations and open questions

The paper leaves several points unresolved, stated plainly by the authors themselves. The complete list of threshold values for $(0,p)$31-type RDPs at $(0,p)$32 is not obtained; consequently $(0,p)$33 itself is not fully determined in that case, and the ACC argument relies on structural bounds rather than an explicit enumeration. The naive multi-height lacks an intrinsic characterization in positive characteristic and may depend on the choice of Frobenius lift. Pre-periodicity of the multi-height—which would give rationality of $(0,p)$34 in full generality for quasi-$(0,p)$35-split complete intersections—is asserted as a conjecture rather than proved beyond the classes treated here. Whether analogous ACC statements hold for larger classes of mixed characteristic singularities, or for thresholds with respect to general elements rather than $(0,p)$36, remains open.

Conclusion

This paper develops the multi-height machinery and its Fedder-type computational criteria into an effective tool for mixed characteristic thresholds, applies it to give a nearly complete computation of $(0,p)$37 for lifts of rational double points, and proves that the resulting set of thresholds is rational, satisfies the ACC, and—at $(0,p)$38—contains all reciprocals of positive integers with $(0,p)$39 as the sole accumulation point. The work extends discreteness phenomena previously known only in the regular case to a natural class of singular hypersurfaces, and identifies the pre-periodicity conjecture for multi-heights as the central remaining obstacle to a fully general rationality statement.