Abundance for slc surfaces over arbitrary fields

This paper establishes the abundance conjecture for projective semi-log canonical (slc) surfaces over arbitrary fields of positive characteristic by leveraging Tanaka's results on log canonical surfaces and Hacon-Xu's normalization techniques, while also deriving applications for dlt threefold pairs and mixed characteristic families.

The Gist

The Big Picture: The "Minimal Model" Puzzle

Imagine you are an architect trying to simplify a massive, messy, and irregular building (a mathematical "variety"). Your goal is to remodel it into its simplest, most efficient form without tearing down the whole structure. This is the Minimal Model Program (MMP).

In this remodeling process, you want to reach a state where the building's "energy" (mathematically, the Canonical Divisor, or $K$) is stable. There are two ways this stability can happen:

1. The Fano Route: The building is so "bouncy" and energetic that it naturally wants to collapse into a smaller shape (like a ball).
2. The Nef Route: The building is stable and flat. Its energy is "nef" (numerically effective), meaning it doesn't point in any "bad" direction.

The Abundance Conjecture is the final boss of this remodeling project. It asks: If the building is stable (nef), does it also have a "blueprint" that allows us to build a perfect, usable model of it? In math terms, if the energy is stable, is it semi-ample? (Semi-ample means you can use that energy to project the building onto a nice, clean canvas).

The Problem: Cracks and Weird Materials

Most of this work has been done on "smooth" buildings (perfectly flat surfaces) or in "characteristic zero" (which is like our standard real-world physics). But mathematicians also want to understand buildings with cracks (singularities) and buildings made of "exotic materials" (fields of positive characteristic, like the finite fields used in cryptography and computer science).

Specifically, this paper tackles slc surfaces.

• slc (semi-log canonical): Imagine a building made of several rooms glued together. Some walls are normal, but some are glued in a way that creates "knots" or "nodes."
• The Glue: Sometimes the glue is "separable" (like two distinct pieces of tape that can be peeled apart easily). Sometimes, in weird mathematical universes (characteristic 2), the glue is "inseparable" (like two pieces of tape fused so tightly they act as one, but with a twist).

The paper asks: If we have a cracked, glued-together building in a weird mathematical universe, and it's stable, can we still find a perfect blueprint for it?

The Solution: The "Normalization" Trick

Posva's proof is like a clever renovation strategy. Here is the step-by-step analogy:

1. The Un-gluing (Normalization)

First, imagine taking your cracked, glued building and carefully cutting all the seams. You separate the rooms so they are all perfect, smooth, individual houses. In math, this is called normalization.

• The Catch: When you cut the seams, you get a "boundary" (the edges where the glue used to be). The original building is just these houses glued back together according to a specific rule (an involution or a "swap").

2. The Easy Part (The Smooth Houses)

Mathematician H. Tanaka had already proven that for these individual, smooth houses, the Abundance Conjecture is true. If a smooth house is stable, it has a perfect blueprint.

• So, we know the individual rooms are fine.

3. The Hard Part (Re-gluing the Blueprint)

Now, we have blueprints for the individual rooms. But we need a blueprint for the whole glued-up building.

• The Challenge: If you just take the blueprints of the rooms and try to glue them back together, they might not match up perfectly at the seams. The "swap" rule might twist the blueprint.
• The Strategy (Hacon-Xu Technique): Posva uses a technique developed by Hacon and Xu. Imagine the blueprints of the rooms are projected onto a giant screen (a fibration).
  • Because the rooms are glued together, the images on the screen overlap.
  • Posva proves that these overlaps are "finite" (they don't go on forever in a messy way).
  • Because the overlaps are finite and well-behaved, you can quotient (fold) the screen. You take the messy, overlapping screen and fold it down until the overlapping parts match perfectly.
  • The result is a new, clean screen (a new variety) that represents the whole glued building.

4. The Result

By folding the screen, Posva shows that the "energy" (the canonical divisor) of the original cracked building does come from a nice, clean blueprint.

• Conclusion: Yes! Even if the building is cracked and glued together in a weird way, if it's stable, it has a perfect blueprint.

Why Does This Matter? (The Applications)

The paper doesn't just solve a puzzle; it unlocks doors to other problems.

1. Families of Buildings (The Mixed Characteristic Case)
Imagine you have a building that changes slightly as you move from one year to the next (a "family" of surfaces). Sometimes the "physics" of the world changes (mixed characteristic). Posva's result helps prove that if the building is stable in every year, the whole timeline of buildings is stable and has a blueprint. This is crucial for understanding how mathematical structures evolve over time.

2. 3D Buildings (Threefolds)
The paper is a stepping stone to understanding 3D shapes (threefolds).

• Think of a 3D object as a stack of 2D slices.
• To understand the 3D object, you often need to understand the "boundary" (the surface of the object).
• Posva proves that for 3D objects in "weird" mathematical universes (F-finite fields), if the object is stable and "big" enough, it also has a perfect blueprint. This is a massive leap forward for 3D geometry in positive characteristic.

Summary in a Nutshell

• The Goal: Prove that stable, cracked, glued-together 2D shapes in weird mathematical universes have perfect blueprints.
• The Method:
  1. Cut the shape into perfect pieces (Normalization).
  2. Use the fact that the pieces are already known to be good (Tanaka's work).
  3. Show that the "glue" (the involution) is simple enough that you can fold the pieces back together without breaking the blueprint (The Hacon-Xu descent technique).
• The Result: The blueprint exists! This confirms a major conjecture and helps mathematicians build better models of 3D shapes and time-evolving structures in the most exotic mathematical settings.

It's like proving that even if you build a house out of mismatched, cracked bricks and glue them with super-sticky, weird tape, as long as the house stands straight, there is a perfect architectural plan hidden inside that can be revealed.

Technical Summary

1. Problem Statement

The paper addresses the Abundance Conjecture within the context of the Minimal Model Program (MMP) for algebraic varieties in positive characteristic.

• The Conjecture: The MMP predicts that for a variety $X$ with mild singularities, if the canonical divisor $K_X$ (or $K_X + \Delta$ for a pair) is nef (numerically effective), it should also be semi-ample (some multiple defines a morphism to a projective space).
• The Gap: While the abundance conjecture is well-established for log canonical (lc) surfaces over algebraically closed fields (both characteristic 0 and $p > 0$) and for semi-log canonical (slc) surfaces over algebraically closed fields of positive characteristic, the case for slc surfaces over arbitrary fields of positive characteristic (including imperfect fields) remained open.
• Specific Challenge: Standard MMP techniques often fail for slc varieties because they are not normal. The abundance proof requires descending properties from the normalization $\bar{S}$ of the surface $S$ back to $S$. In arbitrary fields, the normalization map can involve inseparable phenomena (inseparable nodes), complicating the descent of semi-ampleness.

2. Methodology

Posva's proof combines recent results on lc surfaces with a geometric descent technique adapted from Hacon and Xu. The methodology proceeds in two main stages: the absolute case (over a field) and the relative case (over a base scheme).

A. Reduction to Separable Nodes

• Inseparable Nodes: In characteristic $p=2$, slc surfaces may have "inseparable nodes." Posva utilizes a factorization theorem (Proposition 2.9) to separate the purely inseparable part of the normalization from the separable part.
• Strategy: He reduces the problem to the case where the slc surface has only separable nodes. This allows the use of Galois-theoretic arguments and finite equivalence relations, which are more tractable than purely inseparable morphisms.

B. Descent via Finite Equivalence Relations

The core of the proof relies on the technique of descending semi-ampleness from the normalization $\bar{S}$ to the slc surface $S$.

1. Normalization and Involution: Let $(\bar{S}, \bar{D} + \bar{\Delta})$ be the normalization of the slc pair $(S, \Delta)$. The conductor divisor $\bar{D}$ carries an involution $\tau$ induced by the gluing data of $S$.
2. Fibration: Since $K_{\bar{S}} + \bar{D} + \bar{\Delta}$ is nef (by Tanaka's result for lc surfaces), it is semi-ample. This defines a fibration $\phi: \bar{S} \to \bar{T}$.
3. Finite Equivalence Relation: The involution $\tau$ on $\bar{D}$ induces an equivalence relation on the base $\bar{T}$. Posva proves that this relation is finite (Claim 3.10). This is the most technical step, requiring:
   • Analysis of the Kodaira dimension of components of the conductor.
   • Finiteness of pluricanonical representations on curves of genus 0 and 1 (Propositions 2.11, 2.16).
   • Control over the "special set" $Z$ where fibers of the fibration behave non-trivially (Claims 3.11–3.14).
4. Quotient Construction: Because the equivalence relation is finite, the geometric quotient $T_0 = \bar{T} / (\bar{D} \Rightarrow \bar{T})$ exists as a scheme.
5. Descent of Divisors: Posva demonstrates that the hyperplane divisor $H$ on $\bar{T}$ descends to an ample divisor $H_0$ on $T_0$. Consequently, the composition $\bar{S} \to \bar{T} \to T_0$ factors through $S$, proving that $K_S + \Delta$ is semi-ample.

C. Relative Case (Families)

To handle the relative setting ($f: S \to B$), Posva adapts the strategy of Tanaka:

• Compactification: He compactifies the base $B$ and the total space $S$ while preserving the slc structure and the gluing data (Claim 3.18).
• MMP over Base: He runs a relative MMP to ensure the boundary behaves well, ensuring the involution $\tau$ extends to the compactification.
• Reduction to Absolute: Once compactified, the relative abundance is reduced to the absolute case over the function field of the base, combined with local-to-global arguments.

3. Key Contributions and Results

Main Theorem (Theorem 1)

Abundance for slc surfaces over arbitrary fields:
Let $(S, \Delta)$ be an slc surface pair and $f: S \to B$ a projective morphism, where $B$ is quasi-projective over a field of positive characteristic. If $K_S + \Delta$ is $f$-nef, then it is $f$-semi-ample.

• Significance: This extends the abundance theorem to imperfect fields and arbitrary base schemes, removing the requirement for the field to be algebraically closed.

Application 1: Mixed Characteristic Families (Theorem 2)

The paper proves abundance for families of slc surfaces over excellent regular one-dimensional schemes (e.g., spectra of DVRs) in mixed characteristic, provided no component is of equicharacteristic zero. This combines the main theorem with recent results by Witaszek on mixed characteristic semi-ampleness.

Application 2: Good Minimal Models for Threefolds (Theorem 5.4)

The paper establishes the existence of good minimal models for lc threefolds of general type over F-finite fields of characteristic $p > 5$.

• Mechanism: The proof relies on the abundance of the restriction of $K_X + \Delta$ to the reduced boundary $\Delta=1$.
• Theorem 5.3: Proves that for a Q-factorial dlt threefold $(X, \Delta)$ with $p > 5$, if $K_X + \Delta$ is nef, then its restriction to the boundary $(K_X + \Delta)|_{\Delta=1}$ is semi-ample. This uses the main theorem (Theorem 1) applied to the slc surface formed by the S2-ification of the boundary.
• Result: If $K_X + \Delta$ is nef and big, it is semi-ample.

4. Significance and Impact

1. Completion of the Surface Theory: This work completes the picture of the abundance conjecture for surfaces in positive characteristic, covering the most general class of singularities (slc) and the most general base fields (arbitrary, including imperfect).
2. Bridge to Higher Dimensions: The results are a critical prerequisite for the Minimal Model Program in dimension 3. The abundance of the boundary in threefolds is a necessary step to prove the abundance of the threefold itself.
3. Handling Imperfect Fields: By working over arbitrary fields (not just algebraically closed ones), the paper enables the application of the MMP to arithmetic geometry and moduli problems where the base field is naturally imperfect.
4. Technical Refinement: The paper provides a concrete, explicit realization of Kollár's abstract theory of "sources and springs" for crepant log structures. While Kollár's theory provides the axiomatic framework, Posva explicitly constructs the finite equivalence relations and descent maps in the specific context of surfaces, making the theory accessible for practical application in positive characteristic.
5. F-finiteness: The application to threefolds clarifies the role of F-finiteness, showing that the main obstruction to extending MMP results to arbitrary fields in dimension 3 is the existence of projective log resolutions, rather than the abundance conjecture itself.

In summary, Posva's paper resolves a long-standing gap in the MMP for positive characteristic by proving abundance for slc surfaces over arbitrary fields, thereby unlocking the existence of good minimal models for lc threefolds in characteristic $p > 5$.