Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic

This paper establishes the pseudo-effectivity of the relative canonical divisor for morphisms with non-uniruled generic fibers in positive characteristic by proving that sufficiently ample cyclic covers of non-uniruled bases remain non-uniruled, a result derived from a new cohomological criterion for uniruledness.

The Gist

Imagine you are an architect trying to understand the stability of a massive, multi-story building. In the world of mathematics, this building is a "variety" (a geometric shape), and the "floors" are connected by a structure called a "fibration."

This paper, written by Zsolt Patakfalvi, tackles a very specific problem about the "weight" and "stability" of these buildings, but with a twist: the building is being constructed in a world with very strange physics (mathematical fields with positive characteristic).

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Big Question: Is the Building Heavy Enough?

In the "normal" world of mathematics (characteristic zero, like the real numbers), mathematicians have known for a long time a rule about these buildings:

• If the top floor (the generic fiber) is not a "party hall" (not "uniruled," meaning you can't fill it with endless straight lines or balloons), then the entire building must have a certain amount of "structural weight" (the relative canonical divisor, $K_{X/T}$, is "pseudo-effective").

Think of "pseudo-effective" as meaning the building has enough concrete and steel to stand on its own. If it didn't, it would collapse.

The Problem:
In the "strange physics" world (positive characteristic, like working with clock arithmetic), this rule was known to break down if the top floor was weird. But mathematicians suspected the rule still held true if the top floor was not a party hall. The author proves this suspicion is correct.

2. The Main Challenge: The "Party Hall" Trap

The hardest part of the proof is a logical trap.

• The Trap: To prove the building is stable, you usually try to find a "party" (a rational curve) inside it. If you find one, the building is unstable. If you can't find one, it's stable.
• The Twist: In this strange physics world, it's very hard to prove a building doesn't have a party. Sometimes, the building looks like it has no parties, but it actually does, hidden in a way we can't see.

The author's strategy is: "Let's assume the building is unstable (it has a party). If we can show this leads to a contradiction, then the building must be stable."

3. The Solution: Building a "Mirror" Copy

To catch the hidden party, the author uses a clever trick: The Mirror Cover.

Imagine you have a complex, twisted building (the base $T$) and you aren't sure if it's stable. You can't just look at it; you need to see it from a different angle.

• The Trick: The author builds a new, smooth, perfect copy of the base building (a finite cover).
• The Goal: This new copy must be stable (not a party hall). If the original base was a party hall, this new copy would be too. But if the new copy is not a party hall, we can use it to prove the original building is stable.

How do you build a stable copy?
This is the paper's biggest technical achievement. The author shows that if you take a building and wrap it in a specific type of "cyclic cover" (like wrapping a gift with a specific pattern of ribbon), and you do it with the right amount of ribbon (a high enough degree), the resulting wrapped building cannot be a party hall.

4. The Secret Weapon: The "Witt" Battery

How does the author know the wrapped building isn't a party hall?
He uses a mathematical tool called Witt Cohomology.

• The Analogy: Imagine every building has a hidden battery inside it.
  • If the building is a "party hall" (uniruled), the battery is dead (zero charge).
  • If the building is stable, the battery is charged.
• The Innovation: The author proves that for these specific "wrapped" buildings, the battery charge (a specific dimension of a cohomology group) keeps growing as you wrap it tighter and tighter.
• The Logic: Since the battery charge is growing and never hits zero, the building cannot be a party hall. It must be stable.

5. The Final Showdown: The "Bend and Break"

Once the author has this stable "Mirror Copy," he runs the final test:

1. He assumes the original building is unstable (it has a "party" or a straight line running through it).
2. He pulls this party back into the Mirror Copy.
3. Because the Mirror Copy is stable (no parties allowed), the party line tries to "bend" and "break" (a standard mathematical technique called bend-and-break).
4. In the strange physics world, this bending usually creates more parties.
5. The Contradiction: The math shows that if the building were unstable, the Mirror Copy would have to be full of parties. But we just proved the Mirror Copy has a charged battery and cannot have parties.
6. Conclusion: Therefore, the original assumption was wrong. The building must be stable. The "weight" (pseudo-effectivity) exists.

Summary of the "Takeaway"

• The Result: If you have a geometric building where the top floor isn't a "party hall," the whole building has enough structural integrity (is pseudo-effective), even in the weird world of positive characteristic math.
• The Method: The author built a special, smooth "mirror" version of the base of the building. He proved this mirror version is so stable it can't possibly be a party hall by showing its internal "battery" (Witt cohomology) is always charged.
• Why it Matters: This confirms a fundamental rule of geometry holds true even in these strange mathematical universes. It helps mathematicians understand how shapes behave, how to classify them, and how to build "moduli spaces" (maps of all possible shapes).

In short: The author proved that if the top floor isn't wild, the whole building is solid, by building a super-stable mirror version of the foundation to prove it.

Technical Summary

Here is a detailed technical summary of the paper "Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic" by Zsolt Patakfalvi.

1. Problem Statement and Context

The Core Problem:
The paper addresses the behavior of the relative canonical divisor $K_{X/T}$ for a surjective morphism $f: X \to T$ between smooth projective varieties over an algebraically closed field $k$ of characteristic $p > 0$. Specifically, it investigates the pseudo-effectivity of $K_{X/T}$.

Background:
In characteristic zero, it is a well-established result (Nakayama, Viehweg) that if the geometric generic fiber $X_\eta$ is not uniruled (or equivalently, if $K_{X_\eta}$ is pseudo-effective), then $K_{X/T}$ is pseudo-effective. This property is fundamental for the subadditivity of Kodaira dimension and the construction of moduli spaces.

The Challenge in Positive Characteristic:
In characteristic $p > 0$, the situation is more complex due to "wild" behavior.

1. The condition that $K_{X_\eta}$ is pseudo-effective is not sufficient to guarantee $K_{X/T}$ is pseudo-effective (counterexamples exist, e.g., [CEKZ21]).
2. However, the condition that $X_\eta$ is not uniruled is stronger and accounts for some of this wild behavior.
3. The open question was whether the implication holds if we assume $X_\eta$ is not uniruled: Does $K_{X/T}$ become pseudo-effective?

Main Question:
Let $f: X \to T$ be a surjective morphism with a geometrically integral and non-uniruled generic fiber. Is $K_{X/T}$ pseudo-effective?

2. Methodology and Technical Approach

The proof relies on a contradiction argument using the "bend-and-break" technique, adapted to positive characteristic. The strategy involves the following steps:

1. Assumption of Negation: Assume $K_{X/T}$ is not pseudo-effective. This implies the existence of a moving family of curves $C$ on $X$ such that $K_{X/T} \cdot C < 0$.
2. Base Change to a Non-Uniruled Cover: To apply bend-and-break, one needs the total space to remain "well-behaved" (integral, local complete intersection) while ensuring the base is not uniruled.
   • The author constructs a finite, flat, smooth, non-uniruled cover $S \to T$.
   • By pulling back the fibration to $S$, the new total space $X_S$ retains the necessary singularity properties (local complete intersection) to apply bend-and-break, while the base $S$ is guaranteed to be non-uniruled.
3. Iterated Frobenius Base Changes: The author performs iterated Frobenius base changes ($T_n \to T$) to amplify the negativity of the intersection number $K_{X/T} \cdot C$ relative to the curves, eventually forcing the existence of rational curves on the total space.
4. The Critical Obstacle: The hardest part of the proof is constructing the smooth non-uniruled cover $S \to T$. In characteristic $p$, resolution of singularities is not available, so the cover must be constructed directly to be smooth.

3. Key Contributions and Intermediate Results

To solve the problem of constructing the non-uniruled cover, the author develops several independent theoretical tools:

A. A Cohomological Criterion for Non-Uniruledness (Theorem 1.3)

The author establishes a new criterion to determine if a variety is not uniruled based on the dimensions of semi-stable parts of Witt cohomology groups.

• Statement: Let $X$ be a smooth projective variety of dimension $n$. If
  $$\dim_k H^{n-1}(X, \mathcal{O}_X)^{ss} < \dim_k H^n(X, \mathcal{O}_X)^{ss}$$
  then $X$ is not uniruled.
• Mechanism: Here, $H^i(X, \mathcal{O}_X)^{ss}$ denotes the Frobenius-semi-stable part (the image of high iterations of the Frobenius map $F^*$). The proof utilizes Witt vector cohomology $H^n(X, W\mathcal{O}_X)$ and shows that the inequality implies the non-vanishing of $H^n(X, W\mathcal{O}_X)_\mathbb{Q}$, which implies non-uniruledness for varieties with $W\mathcal{O}$-rational singularities.

B. Construction of Non-Uniruled Covers via Cyclic Covers (Corollary 1.4)

The author proves that for any smooth projective variety $X$ and an ample line bundle $H$, a general cyclic cover of sufficiently high degree (coprime to $p$) is not uniruled.

• Method: By taking a cyclic cover $Y \to X$ defined by a general section of $H^{sd}$, the dimension of the semi-stable part $H^n(Y, \mathcal{O}_Y)^{ss}$ grows indefinitely as $s$ increases, while $H^{n-1}(Y, \mathcal{O}_Y)^{ss}$ remains bounded. This satisfies the cohomological criterion from Theorem 1.3.
• Technical Detail: This involves analyzing the deformation of Frobenius semi-stable subspaces under general divisors (Proposition 3.20) and showing that the semi-stable dimension grows for general choices of the divisor.

C. Singular Versions

The results are extended to singular varieties. The author allows for complete intersection, $W\mathcal{O}$-rational, and Cohen-Macaulay singularities, which are the weakest singularities for which the bend-and-break arguments and cohomological tools remain valid.

4. Main Results

Theorem 1.1 (Smooth Case):
Let $f: X \to T$ be a surjective morphism between smooth projective varieties over $k$ ($char > 0$) with a geometrically integral and non-uniruled generic fiber. Then, the relative canonical divisor $K_{X/T}$ is pseudo-effective.

Corollary 1.2 (Subadditivity of Kodaira Dimension):
Under the assumptions of Theorem 1.1, if $T$ is of general type and $K_{X_\eta}$ is big, then:
$$\kappa(X) \geq \kappa(K_{X_\eta}) + \kappa(T)$$
This confirms the subadditivity of Kodaira dimension in this specific positive characteristic setting.

Corollary 1.5 (Mixed Characteristic Application):
Assuming the Weak Ordinarity Conjecture, the paper shows that for a smooth projective variety $X$ over a field of characteristic 0 satisfying the cohomological inequality ($\dim H^n > \dim H^{n-1}$), the set of primes $p$ for which the reduction $X_p$ is not uniruled is dense in the spectrum of the base.

5. Significance and Impact

1. Resolution of a Conjecture: The paper confirms that the "non-uniruled" condition is the correct substitute for "pseudo-effective canonical bundle" in positive characteristic to ensure the pseudo-effectivity of the relative canonical divisor. This resolves a gap left by previous counterexamples where the fiber was uniruled.
2. New Tools for Positive Characteristic: The development of the Frobenius-semi-stable cohomological criterion provides a powerful new method to detect non-uniruledness without relying on resolution of singularities. This is crucial in characteristic $p$ where resolution is not generally available.
3. Foundation for Moduli Theory: The pseudo-effectivity of $K_{X/T}$ is a prerequisite for constructing moduli spaces of stable varieties (KSBA) and studying hyperbolicity in positive characteristic. This result removes a major obstruction in these fields.
4. Bend-and-Break in Positive Characteristic: The paper successfully adapts the bend-and-break technique to positive characteristic by carefully managing the singularities of the total space through finite flat covers, demonstrating that the technique remains robust if the base is suitably modified.

In summary, Patakfalvi's work bridges a critical gap in the birational geometry of positive characteristic varieties, establishing a robust link between the geometry of the generic fiber and the positivity of the relative canonical divisor, supported by novel cohomological techniques involving Witt vectors and Frobenius actions.