Planar, rational curves over ${\mathbb F}_2$ whose only singularity is a double point

This paper demonstrates the existence of planar rational curves over the finite field $\mathbb{F}_2$ with arbitrarily large degrees that possess a unique singularity of multiplicity 2, a phenomenon that is restricted to degrees of at most 6 in characteristic 0.

The Gist

Imagine you are an architect trying to build a bridge (a mathematical curve) across a river. In the world of "standard" mathematics (what the paper calls Characteristic 0, which is like our everyday reality), there are strict building codes. You can only build bridges of certain lengths (degrees) if you want them to have exactly one weak spot (a singularity) that is just a simple kink (a double point).

According to the old rules, if your bridge gets too long (degree higher than 6), it's impossible to build it with just one weak spot. It would either need to be perfectly smooth (no weak spots) or have a messy pile-up of many weak spots.

The Twist: A Different World of Rules
This paper, written by mathematician János Kollár, explores a strange, parallel universe called Characteristic 2 (specifically over the field $\mathbb{F}_2$). Think of this universe as a place where the laws of physics are slightly different—like a video game where gravity works in reverse or where you can walk through walls.

In this weird universe, Kollár discovers that you can build these super-long bridges with exactly one weak spot, even when they are huge.

The Core Discovery: The "Magic" Curve

Kollár constructs a specific type of curve (a bridge) that:

1. Is Rational: It can be drawn with a single, continuous line without lifting your pen.
2. Is Planar: It lives on a flat sheet of paper.
3. Has One Singularity: It has only one "kink" or "crunch" in its entire length.
4. Is Huge: It can be as long as you want (degree $2n+2$).

In our normal world, a curve this long with only one kink is impossible. But in the "Characteristic 2" world, Kollár found a secret recipe (using something called Artin-Schreier polynomials) to build them.

The "Lifting" Problem: Why This Matters

Here is the most exciting part of the story. Mathematicians often try to "lift" objects from this weird, restricted universe (Characteristic 2) back to our normal universe (Characteristic 0). It's like taking a blueprint from a video game and trying to build the actual structure in real life.

Usually, if you have a structure in the video game, you can find a matching structure in real life that looks very similar. The "weak spots" (singularities) usually translate over perfectly.

Kollár's Surprise:
He found that for these specific, huge curves, you cannot lift them.
If you try to take his "Characteristic 2" bridge and build it in "Characteristic 0," the laws of physics (mathematics) break. The bridge either collapses, or the single weak spot turns into a messy pile of many weak spots.

The Analogy:
Imagine you have a sculpture made of ice (Characteristic 2) that has a single, perfect crack. You try to melt it down and recast it in gold (Characteristic 0). You expect the gold sculpture to have the same single crack.
But Kollár found that for these specific shapes, the gold version cannot have just one crack. It must shatter into many pieces. The "single crack" is a feature that only exists in the ice world; it simply cannot exist in the gold world.

Why Should We Care?

1. Breaking the Rules: This proves that the "building codes" of mathematics are not universal. What is impossible in our world is possible in others.
2. The "Resolution" Puzzle: Mathematicians use a process called "blowing up" to fix these cracks (like smoothing out a crumpled piece of paper). Kollár shows that the way you smooth out the crack in the weird world is fundamentally different from how you would do it in our world. You can't just "translate" the instructions.
3. A Warning to Theorists: There was a hope that all these weird mathematical objects could be "lifted" to our world to help us understand them better. Kollár's paper says, "Not so fast! Some things are unique to their own universe and cannot be brought home."

The "K3" Side Story

The paper also briefly touches on "K3 surfaces" (which are like 3D versions of these curves). He mentions that while some weird shapes in the "Characteristic 2" world can be lifted to our world, others (like certain supersingular surfaces) cannot. It's like finding a few specific LEGO sets that, when you try to rebuild them with real bricks, just don't fit together the same way.

Summary

János Kollár has found a way to build mathematical "monsters" in a strange, binary universe that have a single, unique flaw. He then showed that these monsters are so alien that they cannot be recreated in our normal mathematical universe. This challenges our understanding of how different mathematical worlds relate to each other and proves that some mathematical structures are truly unique to their own environment.

Technical Summary

Here is a detailed technical summary of the paper "Planar, Rational Curves over $\mathbb{F}_2$, Whose Only Singularity is a Double Point" by János Kollár.

1. Problem Statement

The paper addresses the construction and classification of planar rational curves with a unique singular point in positive characteristic, specifically over the field $\mathbb{F}_2$.

• The Core Question: Can one construct rational plane curves of arbitrarily large degree $d$ that possess exactly one singular point, which is a double point (multiplicity 2)?
• The Characteristic 0 Constraint: In characteristic 0, such curves are highly restricted. It is a known result (Lemma 4) that rational curves with a single singularity of multiplicity 2 exist only for degrees $d \leq 6$.
• The Liftability Issue: A central theme is whether these positive characteristic examples can be "lifted" to characteristic 0 while preserving their resolution of singularities (specifically, the sequence of blow-ups and the associated discrepancies). The paper investigates if the existence of these curves in characteristic 2 contradicts or refines expectations regarding deformation theory and log canonical thresholds.

2. Methodology

Kollár employs a combination of explicit algebraic construction, singularity theory, and deformation analysis.

• Explicit Parametrization: The author constructs curves using Artin-Schreier-type maps. Instead of relying solely on implicit equations, the curves are defined as the image of a map $\phi_{q,r}: \mathbb{A}^1 \to \mathbb{A}^2$ given by:
  $$t \mapsto (t^q, t^{q+r} + t)$$
  where $q$ is a power of the characteristic $p$ and $r$ is a multiple of $p$.
• Closure in $\mathbb{P}^2$: The affine curves are closed in the projective plane to analyze their behavior at infinity and their global degree.
• Singularity Analysis: The paper analyzes the type of singularity (specifically $A_m$ singularities) using normal forms and embedded resolution graphs. It compares the multiplicity of the singularity ($m$) against the degree of the curve ($d$).
• Discrepancy Preservation: The author tests the "liftability" of these curves by examining whether the sequence of blow-ups required to resolve the singularity in characteristic 2 can be deformed into a valid resolution sequence in characteristic 0 without changing the discrepancies of the exceptional divisors.

3. Key Contributions and Results

A. Construction of High-Degree Counterexamples (Example 3)

The primary contribution is the explicit construction of a family of curves $C_{2n,2} \subset \mathbb{P}^2_{\mathbb{F}_2}$ defined by the equation:
$$z^2 y^{2n} = x^{2n+2} + x z^{2n+1}$$

• Properties:
  • Degree: $d = 2n + 2$.
  • Rationality: The curve is rational.
  • Singularity: It has a unique singular point at infinity $(0:1:0)$.
  • Multiplicity: The singularity is a cusp of multiplicity 2.
  • Type: The singularity is of type $A_m$ where $m = 2n(2n+1)$.
• Significance: For any $n \geq 1$, this yields a curve of degree $d \geq 4$ with a single double-point singularity. As $n$ increases, the degree can be arbitrarily large, directly contrasting with the characteristic 0 bound of $d \leq 6$.

B. Comparison with Characteristic 0 Bounds (Lemma 4 & Remark 6)

• Lemma 4: Citing Gusein-Zade and Nekhoroshev, the paper notes that for a curve of degree $2d$in$\mathbb{C}\mathbb{P}^2$with an$A_m$singularity,$m \leq 3d(d-1) + 1$.
• The Discrepancy: The constructed curves $C_{2n,2}$ have $m \approx \frac{1}{2}(\deg C)^2$. While this fits within the general upper bound for any curve, the specific constraint of being rational and having only one singularity makes these examples impossible in characteristic 0 for $d > 6$.
• Example: The curve $C_{4,2}$ has degree 6 and an $A_{20}$ singularity. In characteristic 0, the maximal singularity for a degree 6 rational curve is $A_{19}$ (the curve $Y_6$). The paper suggests $Y_6$ might specialize to $C_{4,2}$, but the singularity type jumps from $A_{19}$ to $A_{20}$.

C. Non-Liftability of Resolution Sequences (Remark 7)

The paper demonstrates that the standard resolution of singularities for these curves cannot be lifted to characteristic 0.

• The Mechanism: To resolve the singularity of $C_{2n,2}$, one must perform a specific sequence of blow-ups. The discrepancies of the exceptional divisors are determined by the multiplicities of the points being blown up.
• The Counterexample: The curve $C_{8,2}$ (degree 10) has an $A_{72}$ singularity.
  • In characteristic 0, Lemma 4 implies a reduced decic curve ($d=10$) can have at most an $A_{60}$ singularity.
  • Therefore, there is no curve in characteristic 0 with degree 10 and an $A_{72}$ (or even $A_{71}$) singularity.
  • Consequently, the resolution sequence of $C_{8,2}$ (which preserves discrepancies) cannot be lifted to characteristic 0. This refutes the idea that discrepancy-preserving deformations always exist from characteristic $p$ to 0.

D. Log Canonical Thresholds (Remark 8)

• The log canonical threshold (LCT) of $C_{2n,2}$ is calculated as $\frac{1}{2} + \frac{1}{m+1}$.
• For $n \geq 2$, no curve in characteristic 0 with the same degree shares this specific LCT.
• However, the paper notes that the LCT of the associated surface $X(g) \subset \mathbb{A}^3$ is $3/(2n+2)$, which is achievable in characteristic 0. This suggests that while the specific planar curve geometry is rigid, the associated surface singularities might behave differently.

E. Context on K3 Surfaces (Example 9)

The author briefly discusses K3 surfaces to show that non-liftable singularities are not unique to planar curves.

• Supersingular K3 surfaces in characteristic $p$ can have Picard numbers (22) higher than the maximum in characteristic 0 (20).
• Specific examples (e.g., a degree 4 K3 with a $D_{21}$ singularity in char 2) are shown to be non-liftable as quartic surfaces, though they may exist as surfaces in weighted projective spaces.

4. Significance and Implications

1. Refutation of Universal Liftability: The paper provides concrete counterexamples to the hypothesis that singularities in positive characteristic can always be lifted to characteristic 0 while preserving the "standard" resolution sequence and discrepancies.
2. Clarification of [Ish25]: The results are presented as a check on the program outlined in Ishii's work ([Ish25]). While [Ish25] suggests that discrepancy-preserving liftings are often possible, Kollár shows that for planar rational curves with a single double point, this fails for degrees $d \geq 10$.
3. Distinction in Singularity Theory: The work highlights a fundamental divergence between characteristic 0 and characteristic $p$ regarding the moduli of rational curves. In characteristic 0, the geometry of rational curves with few singularities is rigid; in characteristic 2, Artin-Schreier phenomena allow for "surprising" high-degree examples with minimal singularities.
4. Log Canonical Thresholds: While the curves themselves do not immediately resolve the question of whether the set of log canonical thresholds depends on the characteristic (Remark 8), they provide a testing ground for understanding how singularities behave under deformation across characteristics.

In summary, Kollár uses explicit Artin-Schreier constructions to demonstrate that the geometric constraints on rational plane curves in characteristic 2 are significantly looser than in characteristic 0, leading to high-degree curves with unique double-point singularities that cannot be lifted to characteristic 0 without altering their resolution data.