On linear $\alpha_p$-quotients

This paper investigates linear $\alpha_p$-quotients on affine spaces by employing explicit stacky resolutions to classify their singularity types and compute stringy motivic invariants, thereby reducing a conjecture by the authors and Fabio Tonini—which posits that these invariants match those of linear $\mathbb{Z}/p$-quotients—to a computationally verified equality of multi-sets.

The Gist

Here is an explanation of the paper "On Linear $\alpha_p$-Quotients" using simple language, analogies, and metaphors.

The Big Picture: Two Different Ways to Fold Space

Imagine you have a giant, infinite sheet of paper (mathematical space). You want to fold it up to make a smaller, compact shape. In mathematics, this is called taking a quotient.

Usually, we fold space using a "clock" mechanism. Imagine a clock with $p$ hours (where $p$ is a prime number like 2, 3, 5, etc.). You rotate the paper by 1 hour, then 2 hours, and so on, until you get back to the start. This is a $\mathbb{Z}/p$-action. It's like a discrete, step-by-step dance.

But in this paper, the authors are studying a different kind of folding called an $\alpha_p$-action. Instead of a clock with distinct steps, imagine a "slippery slide." You can slide the paper a tiny, infinitesimal amount, and because of the slippery nature of the math (specifically, working in a world where the number $p$ acts like zero), the slide behaves differently than the clock.

The Mystery:
For a long time, mathematicians knew that these two ways of folding (the Clock and the Slide) created shapes that looked very different. However, the authors and their colleague Fabio Tonini had a hunch: Deep down, the "folds" and "crumples" created by the Slide are actually identical to those created by the Clock.

They call these crumples singularities. The paper asks: Are the "bad spots" on the Slide-folding exactly the same as the "bad spots" on the Clock-folding?

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Part 1: The "Crumple" Check (Log Canonical, Canonical, Terminal)

When you fold paper, sometimes you get a nice, smooth crease. Other times, you get a sharp, jagged tear. Mathematicians have a grading system for how "bad" a tear is:

1. Log Canonical: A mild tear. The paper is still usable.
2. Canonical: A sharper tear, but the structure is still stable.
3. Terminal: A very sharp, dangerous tear. This is the "gold standard" of badness in modern geometry.

The Discovery:
The authors built a special tool (a "weighted blow-up," think of it as a high-powered magnifying glass that zooms in on the fold) to examine the Slide-folding ($\alpha_p$).

They found a simple rule to predict how bad the tear is. It depends on a number they call $D_d$ (which is just a sum of the sizes of the folds) and the prime number $p$.

• If $D_d$ is big enough, the tear is mild.
• If $D_d$ is huge, the tear is sharp.

The Surprise:
The rule they found for the Slide ($\alpha_p$) is exactly the same as the rule for the Clock ($\mathbb{Z}/p$).

• Analogy: It's like discovering that a car crash at 50mph and a train derailment at 50mph leave the exact same amount of damage, even though the vehicles are totally different.

This is huge because it proves that in this specific mathematical world, the "slippery slide" behaves just like the "stepping clock" when it comes to the severity of the damage.

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Part 2: The "Fingerprint" (Stringy Motivic Invariants)

Mathematicians don't just want to know how bad a tear is; they want a unique fingerprint for the shape. This fingerprint is called the Stringy Motivic Invariant.

Think of it like a DNA test for a geometric shape. If two shapes have the same DNA, they are fundamentally the same, even if they look different on the surface.

The Conjecture:
The authors' team guessed that the DNA of the Slide-shape and the Clock-shape are identical.
$$\text{DNA}(\text{Slide}) = \text{DNA}(\text{Clock})$$

The Problem:
Calculating this DNA is incredibly hard.

• The Clock's DNA formula looks like a complex recipe with many ingredients.
• The Slide's DNA formula looks like a completely different recipe.
• Comparing them is like trying to prove that a cake made of flour and sugar is the same as a cake made of rice and beans, just by looking at the ingredient lists. They look nothing alike!

The Solution:
The authors didn't just guess; they turned the problem into a puzzle.
They realized that if the two DNA formulas are equal, it means two giant lists of numbers (called "multi-sets") must be identical.

• List A: Numbers generated by the Clock.
• List B: Numbers generated by the Slide.

They used a computer (Mathematica) to check thousands of these lists.

• They checked cases where the prime number $p$ was up to 173.
• They checked cases where the dimension of the space was up to 32.
• Result: In every single case they checked, List A and List B were identical.

It's like checking a million different pairs of socks and finding that every left sock perfectly matches a right sock, even though they were made in different factories.

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Part 3: The "Magic Bridge" (Degeneration)

How did they connect the Clock and the Slide?
They built a bridge (mathematically, a "degeneration").

Imagine a movie where a Clock is slowly melting into a Slide.

• At the start of the movie (Time $t=1$), you have a Clock.
• At the end of the movie (Time $t=0$), the Clock has melted into a Slide.
• In the middle, it's a weird hybrid.

The authors showed that this movie exists. Because the Clock and the Slide are connected by this continuous movie, it makes sense that their "DNA" (invariants) should be the same. If you slowly melt a clock into a slide, the "bad spots" shouldn't suddenly change their fundamental nature.

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Summary: What Does This Mean?

1. The Analogy: We have two different ways to fold space (Clock vs. Slide).
2. The Finding: The "bad spots" (singularities) created by both methods are identical in severity.
3. The Proof: They proved the "DNA" (stringy invariants) is likely identical by converting the problem into a number puzzle and checking it with a computer for thousands of cases.
4. The Implication: This suggests that the "slippery slide" ($\alpha_p$) and the "stepping clock" ($\mathbb{Z}/p$) are two sides of the same coin. Even though they look different, they share the same deep mathematical soul.

The "So What?":
This helps mathematicians understand the rules of geometry in "positive characteristic" (a world where numbers wrap around like a clock). It shows that even in this weird, slippery world, the rules are surprisingly consistent with the "normal" world we are used to. It's a step toward unifying different branches of mathematics.

Technical Summary

Here is a detailed technical summary of the paper "On Linear $\alpha_p$-Quotients" by Quentin Posva and Takehiko Yasuda.

1. Problem Statement and Context

The paper investigates the geometry and singularities of linear $\alpha_p$-quotients in positive characteristic $p > 0$.

• Context: In characteristic 0, quotient singularities by finite groups are well-understood (e.g., via the McKay correspondence). In positive characteristic, quotients by infinitesimal group schemes like $\alpha_p$ exhibit "surprising" behavior that depends sensitively on the ratio between the dimension of the space and the characteristic $p$.
• Specific Object: The authors study quotients of the affine space $\mathbb{A}^d$ by a linear action of the group scheme $\alpha_p$. These actions are defined by a specific derivation $\partial_d$ constructed from a sequence of integers $d = \{d_\lambda\}$, where $0 \le d_\lambda \le p-1$. The quotient is denoted$A/(\alpha_p, d)$.
• Motivation: There is a known correspondence between linear $\alpha_p$-actions and linear $\mathbb{Z}/p$-actions (where $\mathbb{Z}/p$ is the constant group scheme). Specifically, $\alpha_p$-actions can be viewed as degenerations of $\mathbb{Z}/p$-actions. The authors aim to compare the singularity types (log canonical, canonical, terminal) and motivic invariants of these two types of quotients.
• Conjecture: Building on previous work by Yasuda and Tonini, the paper addresses the conjecture that the stringy motivic invariants of $A/(\alpha_p, d)$ and $A/(\mathbb{Z}/p, d)$ coincide.

2. Methodology

The authors employ a combination of stacky geometry, weighted blow-ups, and motivic integration.

A. Stacky Partial Resolutions

Instead of working directly with the singular quotient scheme, the authors construct an explicit partial resolution using Deligne–Mumford (DM) stacks:

1. Weighted Blow-up: They perform a weighted blow-up $b: Y \to \mathbb{A}^d$ along the fixed-point locus of the $\alpha_p$-action. The center of the blow-up is defined by the ideal generated by specific coordinate functions with weights corresponding to the indices of the derivation.
2. Regularity of the Foliation: A key technical achievement is proving that the pullback of the 1-foliation (induced by the $\alpha_p$-action) to the stack $Y$ becomes regular.
3. Quotient Stack: Since the foliation on $Y$ is regular, the quotient stack $W = Y / \mathcal{F}_Y$ is a regular tame DM stack. The coarse moduli space $\underline{W}$ provides a schematic partial resolution of the original quotient $A/(\alpha_p, d)$.
4. Stratification: The authors stratify the exceptional divisor of this resolution into loci with tame cyclic quotient singularities. This allows the application of combinatorial formulas for invariants.

B. Discrepancy Calculations

Using the theory of 1-foliations on stacks (extending results from Posva), they compute the discrepancies of the resolution.

• They relate the discrepancies of the stack $W$ to those of the scheme $Y$ and the foliation $\mathcal{F}$.
• This allows them to determine the singularity class (lc, canonical, terminal) of the original quotient based on the combinatorial data of the sequence $d$.

C. Stringy Motivic Invariants

The authors compute the stringy motivic invariant $M_{st}(A/(\alpha_p, d))$ using:

• Batyrev's Formula: Applied to the strata of the partial resolution, which are cyclic quotient singularities.
• Combinatorial Summation: They sum the contributions of all strata, utilizing the structure of the Farey sequence and a specific function $\theta(y)$ derived from the floor function.

3. Key Contributions and Results

A. Classification of Singularities (Theorem 1.0.1 / 4.3.3)

The authors provide a complete characterization of when the linear $\alpha_p$-quotient $A/(\alpha_p, d)$ is log canonical (lc), canonical, or terminal.
Let $D_d = \sum_\lambda \frac{d_\lambda(d_\lambda+1)}{2}$.

• Log Canonical: $A/(\alpha_p, d)$ is lc $\iff D_d \ge p - 1$.
• Canonical: $A/(\alpha_p, d)$ is canonical $\iff D_d \ge p$.
• Terminal: $A/(\alpha_p, d)$ is terminal $\iff D_d \ge p + 1$.

Significance: This result is striking because, unlike in characteristic 0, these singularities are not Cohen–Macaulay in general (specifically when $D_d \ge 3$). This provides new examples of canonical and terminal singularities in positive characteristic that fail the Cohen–Macaulay property.

B. Computation of Stringy Motivic Invariants (Theorem 1.0.2 / 5.2.3)

The authors derive an explicit closed formula for the stringy motivic invariant $M_{st}(A/(\alpha_p, d))$ assuming $D_d > p-1$:
$$M_{st}(A/(\alpha_p, d)) = L^d - L^l + \frac{L^{l-1}}{1 - L^{-1-D_d+p}} \sum_{s/r \in \mathcal{F}} (L^{N_r} - 1) L^{\theta(s/r)}$$
Where:

• $L$ is the class of the affine line.
• $\mathcal{F}$ is the Farey sequence of order $\max(d)$.
• $\theta(y)$ is a specific function involving floor functions.
• $N_r$ counts indices divisible by $r$.

C. Comparison with $\mathbb{Z}/p$-Quotients (Section 6)

The paper addresses the Tonini-Yasuda Conjecture: $M_{st}(A/(\alpha_p, d)) = M_{st}(A/(\mathbb{Z}/p, d))$.

• Reduction: The authors reformulate the conjecture as an equality between two explicit multi-sets of integers (Conjecture 6.0.1). One set involves the function $sht(j)$ from previous $\mathbb{Z}/p$ literature, and the other involves the newly defined $\theta(y)$.
• Verification: Using Mathematica, they verified this combinatorial equality for:
  • All cases where $p \le 173$ and the action is indecomposable ($l=1$).
  • All cases where the total dimension is $\le 32$ and $p \le 131$.
• Stringy Euler Numbers: They prove unconditionally that the stringy Euler numbers coincide:
  $$e_{st}(A/(\alpha_p, d)) = e_{st}(A/(\mathbb{Z}/p, d)) = \frac{D_d}{D_d - p + 1}$$
  This provides strong evidence for the full motivic equality.

4. Significance

1. New Examples in Positive Characteristic: The paper expands the known landscape of singularities in positive characteristic, specifically highlighting the existence of canonical/terminal singularities that are not Cohen–Macaulay.
2. Methodological Advancement: It successfully adapts the machinery of motivic integration and stacky resolutions to the setting of 1-foliations and infinitesimal group schemes ($\alpha_p$), bridging a gap between the study of finite group actions ($\mathbb{Z}/p$) and infinitesimal actions.
3. Deformation Theory: The work reinforces the idea that $\alpha_p$-quotients are degenerations of $\mathbb{Z}/p$-quotients (via a Tate-Oort group scheme family). The coincidence of invariants suggests a deep "equisingularity" between these two seemingly different types of quotients, despite the drastic difference in their algebraic structures (infinitesimal vs. étale).
4. Open Questions: The paper concludes by posing the question of whether the family connecting these quotients admits a simultaneous resolution, a problem that remains open and is central to understanding the geometry of these degenerations.

In summary, the paper provides a rigorous geometric framework for analyzing linear $\alpha_p$-quotients, solves the classification of their singularities, computes their motivic invariants, and offers compelling evidence for a deep connection between $\alpha_p$ and $\mathbb{Z}/p$ quotient singularities in positive characteristic.