Coxeter theory for curves on blowups of $\mathbb{P}^r$

This paper investigates smooth irreducible rational curves on blowups of $\mathbb{P}^r$ with specific normal bundle splittings, utilizing Coxeter group theory and a bilinear form on the Chow space to establish numerical criteria for identifying when such curves are generated by Weyl group orbits, with particularly sharp results derived for the case of $r=3$.

The Gist

Imagine you are an architect working in a magical, multi-dimensional city called Projective Space ($P^r$). In this city, the most basic buildings are straight lines. But sometimes, you need to modify the city. You decide to "blow up" specific points in the city.

Think of "blowing up" a point like taking a single dot on a map and inflating it into a whole new neighborhood (a sphere of possibilities). If you do this at $s$ different locations, you create a new, complex city called $Y^r_s$.

Now, your job is to study the roads (curves) that can exist in this new city. Specifically, you are looking for roads that are smooth, don't loop back on themselves (rational), and have a very specific "stability" property. The authors call these special roads $(i)$-curves.

Here is the simple breakdown of what this paper does, using everyday analogies:

1. The Three Types of Special Roads

The authors are interested in roads that behave in one of three ways regarding their "flexibility":

• The Rigid Road ($i = -1$): These are like a steel beam. They are stuck in place. You can't wiggle them or move them without breaking them. In math terms, they are "rigid."
• The Flexible Road ($i = 0$): These are like a garden hose. You can move them around a bit, but they have some constraints.
• The Super-Flexible Road ($i = 1$): These are like a piece of string. You can move them almost anywhere.

The paper focuses heavily on the Rigid Roads (the $-1$ curves) because they are the most interesting and difficult to find.

2. The "Magic Mirror" (Cremona Transformations)

The city has a strange property: it has a "Magic Mirror" (called a Cremona transformation). If you stand in front of this mirror, it flips the city inside out.

• A straight line might look like a twisted curve in the reflection.
• A twisted curve might look like a straight line.

The authors discovered that if you start with the simplest possible road (a straight line passing through 1 or 2 points) and look at it through this Magic Mirror over and over again, you generate a whole family of complex roads. They call these $(i)$-Weyl lines.

The Big Question: If you see a complex road in the city, how do you know if it's just a "Magic Mirror reflection" of a simple straight line, or if it's a weird, unique road that doesn't belong to that family?

3. The "Coxeter" Compass

To answer this question, the authors use a mathematical tool called Coxeter Theory.

• The Analogy: Imagine you have a compass that doesn't point North, but points toward "Symmetry." This compass is built on a specific geometric shape (a graph called $T_{2pq}$).
• How it works: This compass gives you a special "score" (a bilinear form) for every road.
  • If a road is a "Weyl line" (a reflection of a simple line), it must have a specific score.
  • However, the authors found that having the right score isn't enough. Some fake roads can trick the compass.

4. The "Shadow" Test (Projections)

To catch the fake roads, the authors invented a new test called the Projection Inequality.

• The Analogy: Imagine shining a light on your 3D road to cast a shadow on a 2D wall.
• The Rule: If your road is a true "Weyl line," its shadow must obey a strict rule: the "length" of the road minus the "weight" of the points it passes through must be positive.
• The Discovery: If a road fails this shadow test, it's definitely not a Weyl line. If it passes, it might be one.

5. The "Noether" Inequality (The Final Filter)

For the specific case of a 3D city ($P^3$), the authors proved a "Noether-type inequality."

• The Analogy: Think of this as a "Bouncer" at a club.
• The Rule: The Bouncer checks the road's "degree" (how complex it is) and its "multiplicities" (how many times it hits the blown-up points).
• The Result: The authors proved that if a road passes the Bouncer's check (satisfies the inequality), you can apply the Magic Mirror to it, and it will become simpler. You can keep doing this until the road shrinks all the way down to a simple straight line.
• Conclusion: If you can shrink a road down to a simple line using these mirrors, it was a Weyl line all along. If you get stuck and can't shrink it, it's a fake.

Summary of the Paper's Achievement

The authors built a checklist to identify these special roads:

1. Check the Score: Does it have the right linear and quadratic numbers? (The Compass)
2. Check the Shadow: Does it pass the Projection Inequality? (The Shadow Test)
3. Check the Bouncer: Does it satisfy the Noether Inequality? (The Final Filter)

If a road passes all three, the authors proved that it is definitely a Weyl line—meaning it is just a complex reflection of a simple straight line.

Why does this matter?
In the world of algebraic geometry, knowing which roads are "movable" and which are "rigid" helps mathematicians understand the shape and structure of the universe (the space $Y^r_s$). This paper provides a reliable, numerical recipe to sort the real roads from the fakes, turning a confusing geometric puzzle into a solvable math problem.

Technical Summary

Here is a detailed technical summary of the paper "Coxeter theory for curves on blowups of $\mathbb{P}^r$" by Olivia Dumitrescu and Rick Miranda.

1. Problem Statement

The paper investigates the geometry of smooth, irreducible, rational curves in $Y^r_s$, the blowup of the projective space $\mathbb{P}^r$ at $s$ general points. Specifically, the authors focus on $(i)$-curves, defined as curves whose normal bundle splits as a direct sum of line bundles $\mathcal{O}_{\mathbb{P}^1}(i)$ for $i \in \{-1, 0, 1\}$.

The central problem is to determine numerical criteria that characterize when a given curve class in the Chow ring $A_{r-1}(Y^r_s)$ corresponds to an $(i)$-Weyl line. An $(i)$-Weyl line is defined as the image of a standard line passing through $1-i$ points under the action of the Weyl group generated by standard Cremona transformations.

While previous work (specifically [DM2]) established that for Mori Dream Spaces (where the Weyl group is finite), $(i)$-Weyl curves could be identified numerically using linear and quadratic invariants, this characterization fails for spaces that are not Mori Dream Spaces (infinite Weyl group cases). The paper aims to provide a complete numerical characterization for these curves, particularly in the case of $r=3$ ($\mathbb{P}^3$), by incorporating conditions related to the irreducibility of the curve and its projections.

2. Methodology

The authors employ a synthesis of algebraic geometry and Coxeter group theory.

• Chow Ring and Intersection Theory: The study is conducted within the Chow ring $A(Y^r_s)$. The authors utilize the standard basis of divisor classes ($H, E_i$) and curve classes ($h, e_i$). They define a specific bilinear form $\langle -, - \rangle$ on the space of curve classes, derived from a quadratic form, which is invariant under the Cremona action. This form is equivalent to the Dolgachev-Mukai pairing.
• Cremona Transformations: The standard Cremona transformation $\phi_I$ (centered at $r+1$ points) acts on the Chow ring. The authors analyze how these transformations affect the degree and multiplicities of curve classes. They define "Cremona reduced" classes as those where the degree cannot be lowered by any Cremona transformation.
• Coxeter Group Theory: The authors identify the action of the Weyl group (generated by permutations and Cremona transformations) on the subspace of curve classes orthogonal to the anticanonical class. They prove this action is isomorphic to the standard geometric representation of a Coxeter group of type $T_{2, r+1, s-r-1}$.
• Tits Cone and Reduction Algorithms: Using the theory of Tits cones and fundamental chambers, the authors develop algorithms to reduce arbitrary curve classes to a "non-negative chamber" (Cremona reduced form). They analyze whether a class lies within the Tits cone to determine if it can be reduced to a standard line class.
• Projection Inequalities: A key tool is the "Projection Inequality," which relates the degree and multiplicities of a curve to the existence of its projection. The authors strengthen this to a "Strong Projection Inequality" to ensure irreducibility is preserved under reduction.

3. Key Contributions and Results

A. Structural Isomorphism (Proposition 4.5)

The authors prove that the action of the Weyl group on the subspace of curve classes orthogonal to the canonical class is isomorphic to the standard geometric representation of the Coxeter group associated with the graph $T_{2, r+1, s-r-1}$. This establishes a rigorous link between the geometry of curves on blowups and abstract Coxeter theory.

B. Classification of $(i)$-Weyl Lines

The paper provides a systematic classification of when a numerical class corresponds to an $(i)$-Weyl line:

• Finite Weyl Group Cases: When the Weyl group is finite (e.g., $r=2$ or specific bounds on $s$), a class is an $(i)$-Weyl line if and only if it reduces to the standard line class ($h-e_1-e_2$, $h-e_1$, or $h$) via the Cremona reduction algorithm.
• Infinite Weyl Group Cases: For cases where the Weyl group is infinite, the authors prove that no Cremona-reduced class with non-negative degree and multiplicities can be an $(i)$-Weyl line (except for the trivial degree 1 cases). This implies that any $(i)$-Weyl line in these cases must be "non-reduced" and can be strictly reduced in degree by applying Cremona transformations.

C. The Projection Inequality

The authors establish that for any $(i)$-Weyl line with degree $d > 1$, the class must satisfy the Projection Inequality:
$$d + m_1 \leq \sum_{j=2}^s m_j + i$$
They further introduce the Strong Projection Inequality, requiring that $\langle c, w(h-e_j) \rangle \geq 1$ for all Weyl group elements $w$ that do not reduce the degree to 1. This condition is necessary to ensure the curve remains irreducible and non-degenerate throughout the reduction process.

D. Noether-Type Inequality and Main Theorem for $\mathbb{P}^3$ (Theorem 6.13)

The most significant result is a sharp numerical criterion for curves in $\mathbb{P}^3$ ($r=3$). The authors prove a Noether-type inequality for curves: if a class satisfies specific linear and quadratic invariants and the strong projection inequality, then the sum of the four largest multiplicities must exceed the degree ($m_1+m_2+m_3+m_4 > d$).

Theorem 6.13 states that for an irreducible curve $C$ in $Y^3_s$ with degree $d > 1$, the following are equivalent:

1. The class $c$ satisfies:
   • The linear invariant: $\langle c, F \rangle = 2 + i(r-1)$.
   • The quadratic invariant: $\langle c, c \rangle = 1 + (i-1)(r-1)$.
   • The Strong Projection Inequality: $\langle c, w(h-e_k) \rangle \geq 1$ for all relevant $w$.
2. $c$ is the class of an $(i)$-Weyl line (i.e., it is in the Weyl orbit of a line through $1-i$ points).

This result resolves the question of whether numerical invariants alone are sufficient to identify Weyl lines in $\mathbb{P}^3$, showing that they are, provided the strong projection inequality (reflecting irreducibility) is included.

4. Significance

• Extension of Mori Dream Space Theory: The paper extends the understanding of curve classes beyond the realm of Mori Dream Spaces, addressing the more complex infinite Weyl group scenarios.
• Bridge between Divisors and Curves: It draws a parallel between the theory of divisors (where similar numerical criteria exist) and the theory of curves, suggesting a unified Coxeter-theoretic framework for both.
• Algorithmic Reduction: The work provides a concrete, algorithmic method (via Cremona transformations) to determine if a curve class is a Weyl line, reducing the problem to checking numerical invariants and the projection inequality.
• Sharp Criteria for $\mathbb{P}^3$: By proving the Noether-type inequality for curves, the authors provide a definitive answer for the geometry of rational curves on blowups of $\mathbb{P}^3$, a fundamental space in algebraic geometry.

In summary, the paper successfully utilizes Coxeter theory to solve the classification problem for $(i)$-curves on blowups of projective space, providing necessary and sufficient numerical conditions that account for both the algebraic invariants and the geometric constraints of irreducibility.