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Inversion of adjunction for quotient singularities III: semi-invariant case

This paper establishes the precise inversion of adjunction formula and the semi-continuity of minimal log discrepancies for finite linear group quotients of complete intersection varieties defined by semi-invariant equations, thereby extending previous results on varieties defined by invariant equations.

Original authors: Yusuke Nakamura, Kohsuke Shibata

Published 2026-05-01

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CC BY 4.0

Original authors: Yusuke Nakamura, Kohsuke Shibata

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to understand the "roughness" or "smoothness" of a landscape. In the world of mathematics, specifically geometry, these landscapes are called varieties, and the rough spots are called singularities.

This paper is about a specific tool mathematicians use to measure how rough a spot is, called the Minimal Log Discrepancy (mld). Think of the mld as a "smoothness score." A higher score means the spot is smoother; a lower score means it's very jagged.

The authors, Yusuke Nakamura and Kohsuke Shibata, are trying to solve a long-standing puzzle: Does this smoothness score behave predictably when you move around the landscape? Specifically, they want to prove that if you have a spot with a certain smoothness score, all the spots nearby will have scores that are at least that good (or better). This is called the Lower Semi-Continuity (LSC) Conjecture.

The Setting: The "Group Party" and the "Folded Map"

To understand their breakthrough, you need to understand the setting:

The Original Space: Imagine a perfectly smooth, infinite grid (like a giant sheet of graph paper).
The Group: A group of friends (a finite group $G$ ) decides to play a game. They take this grid and fold it over itself in specific ways.
The Quotient Space: When you unfold the paper, you get a new shape. Because of the folding, some points on this new shape might have "kinks" or "crinkles" where the folds met. This new shape is called a quotient variety.

The paper focuses on shapes defined by equations (rules that tell you which points belong to the shape).

Previous Work: In their earlier paper, the authors only looked at shapes defined by invariant equations. Imagine a rule that says, "Stay the same no matter how the group folds the paper."
This Paper: They expand the rules to semi-invariant equations. These are trickier. Imagine a rule that says, "If the group folds the paper, you might change your color or size, but you must change in a very specific, predictable way."

The Problem: The "Semi-Invariant" Twist

The authors' previous method for measuring smoothness worked perfectly for the "invariant" rules. But when they tried to apply it to the "semi-invariant" rules, the math broke.

The Analogy:
Imagine you are trying to measure the depth of a hole in a folded piece of paper.

Invariant Case: The paper folds symmetrically. You can use a standard ruler.
Semi-Invariant Case: The paper folds, but the ink on the paper also stretches or shrinks as it folds. If you use the standard ruler, your measurement is wrong because the ink itself has changed scale.

The authors realized that for semi-invariant equations, the "ruler" (the mathematical tool they use to measure smoothness) needed to be recalibrated. They had to invent a new way to account for how the equations "stretch" under the group's folding.

The Solution: The "Arc Space" and the "Time Machine"

To fix the broken ruler, the authors used a concept called Arc Spaces.

The Metaphor: Imagine a time-lapse camera. Instead of just looking at a single point on the landscape, you look at a "path" (an arc) that moves through the point over time.
By studying these paths, mathematicians can see the hidden structure of the rough spots.

The authors built a new "time-lapse camera" specifically for the semi-invariant case. They had to adjust the lens to account for the "stretching" of the equations (the $w_\gamma(f)$ term in the paper). Once they adjusted the lens, they could finally measure the smoothness scores correctly.

The Big Results

With their new, recalibrated measuring tool, they proved two major things:

The PIA Conjecture (Inversion of Adjunction):
- The Idea: Imagine you have a 3D mountain range (the variety) and you want to know the smoothness of a specific cliff face (a divisor) on that mountain.
- The Result: They proved that the smoothness of the cliff face is exactly the same as the smoothness of the mountain if you treat the cliff face as a separate, smaller mountain.
- Why it matters: It means you don't have to study the whole complex mountain to understand the cliff; you can just study the cliff itself. This works even with the tricky "semi-invariant" folding rules.
The LSC Conjecture (Lower Semi-Continuity):
- The Idea: If you find a spot on the mountain that is "smooth enough" (has a good score), then all the spots right next to it are also "smooth enough." The smoothness score doesn't suddenly drop as you walk a few steps.
- The Result: They proved this is true for their new class of shapes (defined by semi-invariant equations).

Why Should You Care? (According to the Paper)

The authors mention that this isn't just abstract math. Their new ability to handle "semi-invariant equations" allows them to study 3-dimensional terminal singularities.

The Metaphor: Think of 3D shapes that are the "building blocks" of the universe in string theory or high-level physics. Some of these blocks are very complex.
The Connection: A famous classification of these 3D blocks shows that they often look like "folded paper" with "semi-invariant" rules. By proving their formulas work for these specific rules, the authors have successfully described the smoothness of these fundamental building blocks.

Summary

In simple terms, Nakamura and Shibata took a complex mathematical tool used to measure the "roughness" of geometric shapes. They realized the tool was broken for a specific, tricky type of shape (those defined by semi-invariant equations). They fixed the tool by adding a new "calibration factor" to account for how the shapes stretch. With the fixed tool, they proved that the smoothness of these shapes behaves predictably, solving a major conjecture and opening the door to understanding the geometry of 3-dimensional "terminal" singularities.

1. Problem Statement and Context

The paper addresses two central conjectures in birational geometry regarding minimal log discrepancies (mld):

The LSC Conjecture (Lower Semi-Continuity): Proposed by Ambro, this conjecture states that the function mapping a closed point $x$ to its minimal log discrepancy $\text{mld}_x(X, \mathfrak{a})$ is lower semi-continuous.
The PIA Conjecture (Inversion of Adjunction): This conjecture relates the mld of a pair $(X, D, \mathfrak{a})$ to the mld of the divisor $D$ itself, specifically asserting that $\text{mld}_x(X, D, \mathfrak{a}) = \text{mld}_x(D, \mathfrak{a}|_D)$ under certain conditions.

Previous Work:
The authors previously proved these conjectures for finite linear group quotients of complete intersection varieties defined by invariant equations (where $g(f) = f$ for all $g \in G$ ) in their paper [NS22].

The Gap:
Many important singularities, particularly 3-dimensional terminal singularities, arise from quotients of varieties defined by semi-invariant equations (where $g(f) = \chi(g)f$ for a character $\chi$ ). The previous invariant-only framework was insufficient to handle these cases. The goal of this paper is to extend the results from invariant equations to semi-invariant equations.

2. Methodology

The authors employ the theory of arc spaces (specifically the Denef-Loeser theory) to compute minimal log discrepancies. The core methodology involves the following steps:

A. Setup and Definitions

Varieties: Let $A = \mathbb{A}^N_k$ and $G \subset \text{GL}_N(k)$ be a finite group. Let $Y \subset A$ be a complete intersection defined by $c$ semi-invariant polynomials $f_1, \dots, f_c$ .
Quotient: The variety of interest is $Y/G$ .
Semi-invariance: A polynomial $f$ is $G$ -semi-invariant if for every $\gamma \in G$ , $\gamma(f) = \xi^{a} f$ for some root of unity $\xi$ and integer $a$ . The authors define a weight $w_\gamma(f) = a/d$ (where $d = |G|$ ).

B. Modification of the Arc Space Construction

In the invariant case ([NS22]), the authors constructed a scheme $B^{(\gamma)}$ over $k[t]$ to study the arc space of the quotient. For semi-invariant equations, the standard construction fails because the transformed equations $\lambda^*_\gamma(f_i)$ are not necessarily in $R[t]$ .

Key Technical Innovation:

Weighted Transformation: The authors introduce a modified homomorphism $\lambda^*_\gamma$ and define the scheme $B^{(\gamma)}$ using the ideal generated by weighted terms:
$\lambda^*_\gamma(f_i) t^{-w_\gamma(f_i)}$
This ensures the generators lie in $R[t]$ , allowing the construction of a well-defined $k[t]$ -scheme.
Extension to $k[t^{1/d}]$ : To handle the sheaf of differentials and Jacobian ideals correctly, the authors extend the base from $k[t]$ to $k[t^{1/d}]$ . They define a scheme $B^{(\gamma)}_{1/d}$ and a specific invertible sheaf $\mathcal{L}_{B^{(\gamma)}_{1/d}}$ that plays the role of the relative canonical sheaf.
Comparison of Jacobians: They establish precise relations between the Jacobian ideals of the morphisms involved (specifically $\mu_\gamma$ and $p$ ) and the Nash ideals of the original variety $B$ . This involves calculating the order of the Jacobian of the morphism between arc spaces.

C. The Main Formula

The authors derive a formula for the minimal log discrepancy at the origin $x$ of the quotient variety $Y/G$ :
$\text{mld}_x(Y/G, \mathfrak{a}^\tau) = \inf_{\gamma \in G, b_1, b_2 \in \mathbb{Z}_{\ge 0}} \left\{ \text{codim}(C_{\gamma, b_1, b_2}) + \text{age}(\gamma) - w_\gamma(f) - b_2 - \tau b_1 \right\}$
Here, $w_\gamma(f) = \sum w_\gamma(f_i)$ is the new term introduced by the semi-invariant nature of the equations. The term $\text{age}(\gamma)$ is the standard age of the group element, and $C_{\gamma, b_1, b_2}$ are specific contact loci in the arc space of the modified scheme $B^{(\gamma)}$ .

3. Key Contributions and Results

A. Theorem 1.5 (Inversion of Adjunction for Quotients)

This is the central technical result. It establishes a precise equality between the mld of the quotient variety $Y/G$ and the mld of the ambient space $A/G$ with a specific ideal sheaf:
$\text{mld}_y(Y, \mathfrak{a}\mathcal{O}_Y) = \text{mld}_y\left(X, (f_1^d \cdots f_c^d)^{1/d} \mathfrak{a}\right)$
This formula effectively reduces the problem on the singular quotient $Y$ to a problem on the ambient quotient $X$ , adjusted by the product of the defining equations raised to the power of the group order.

B. Theorem 1.4 (PIA Conjecture for Semi-Invariant Cases)

Using the result above, the authors prove the PIA conjecture for finite linear group quotients of complete intersections defined by semi-invariant equations. Specifically, if $D$ is a klt Cartier divisor on $Y$ , then:
$\text{mld}_y(Y, D, \mathfrak{a}) = \text{mld}_y(D, \mathfrak{a}|_D)$

C. Theorem 1.2 (LSC Conjecture for Semi-Invariant Cases)

The paper proves that the minimal log discrepancy function is lower semi-continuous for varieties $Y/G$ defined by semi-invariant equations.

Significance: This extends the known validity of the LSC conjecture to a broader class of singularities, including 3-dimensional terminal singularities.

4. Significance and Impact

Resolution of 3D Terminal Singularities: The classification of 3-dimensional terminal singularities (by Reid and Mori) shows they are quotients of smooth points or cDV points. These often involve semi-invariant equations. By proving the LSC and PIA conjectures for this case, the authors provide a complete theoretical foundation for the behavior of mld in dimension 3.
Generalization of Arc Space Theory: The paper successfully adapts the Denef-Loeser theory of arc spaces to handle semi-invariant equations. The introduction of the weight $w_\gamma(f)$ and the extension to $k[t^{1/d}]$ schemes are novel technical tools that may be applicable to other problems involving group actions with non-trivial characters.
Foundation for Termination of Flips: Since the LSC and ACC conjectures imply the termination of flips (a crucial step in the Minimal Model Program), this result strengthens the theoretical underpinnings of the MMP in the context of quotient singularities.

5. Conclusion

Nakamura and Shibata successfully generalize their previous work on invariant equations to the more complex semi-invariant case. By carefully modifying the construction of arc spaces and introducing a weight correction term $w_\gamma(f)$ , they prove the Inversion of Adjunction and the Lower Semi-Continuity of minimal log discrepancies for a wide class of quotient singularities. This work is a critical step toward a full understanding of singularities in higher-dimensional birational geometry, particularly those arising from finite group actions.