On $(i)$-Curves in Blowups of $\mathbb{P}^r$

This paper investigates $(i)$-curves for $i \in \{-1, 0, 1\}$ in blowups of $\mathbb{P}^r$ by introducing a Weyl-invariant anticanonical curve class and a bilinear form to characterize the finiteness of these curves, the arithmetic definition of $(-1)$-curves, and the extremal rays of the movable cone, ultimately linking these properties to the Mori Dream Space condition.

The Gist

Imagine you are an architect working in a magical, multi-dimensional city called Projective Space ($\mathbb{P}^r$). In this city, the buildings are perfect, and the streets are straight lines. But sometimes, you need to remodel the city. You decide to pick specific spots (points) and "blow them up."

In the language of this paper, "blowing up" a point is like taking a tiny speck of dust and inflating it into a whole new neighborhood (an exceptional divisor). You do this at $s$ different locations. The resulting city is called $Y^r_s$.

The authors, Olivia Dumitrescu and Rick Miranda, are trying to understand the roads (curves) that exist in this new, complex city. Specifically, they are looking for three special types of roads, which they call $(i)$-curves, where $i$ can be $-1$, $0$, or$1$.

Here is the breakdown of their discovery using simple analogies:

1. The Three Types of Roads

Think of these roads as having different "traffic rules" or "stability":

• The $(-1)$-Curves (The Rigid, Lonely Roads):
  Imagine a road that is so perfectly balanced and rigid that it cannot move at all. If you try to wiggle it, it snaps. In math terms, these are "rigid" curves. They are the most famous type because they were studied in the 1990s by physicists and mathematicians (like Kontsevich) trying to count paths in string theory.

  • Analogy: A single, frozen bridge that is the only one of its kind in a specific spot.

• The $(0)$-Curves (The Flexible, Moving Roads):
  These roads are flexible. They can slide around a bit. They form families where you can move the road slightly without breaking it.

  • Analogy: A train track that can be shifted slightly left or right, but still connects the same two general areas.

• The $(1)$-Curves (The Super-Flexible Roads):
  These are even more flexible. They can move freely in large groups.

  • Analogy: A highway system where you can drive in many different lanes and directions.

2. The Great Question: Finite or Infinite?

The big mystery the authors solve is: If you blow up enough points in your city, do you end up with a finite number of these special roads, or an infinite number?

• The "Mori Dream Space" (The Organized City):
  If you don't blow up too many points, your city remains organized. The number of special roads is finite. The city is a "Mori Dream Space." It's like a city with a finite number of unique landmarks.
• The Chaos (The Unorganized City):
  If you blow up too many points (specifically, if the number of points $s$ gets too large relative to the dimension of the city), the city becomes chaotic. Suddenly, you can generate an infinite number of these special roads. The city is no longer a "Mori Dream Space."

The Golden Rule: The paper proves that the city is "organized" (a Mori Dream Space) if and only if the number of these special flexible roads ($0$and$1$) is finite.

3. The Magic Mirror (Cremona Transformations)

How do they find all these roads? They use a magical tool called the Cremona Transformation.

• The Analogy: Imagine a funhouse mirror that distorts the city. It takes a straight line and bends it into a curve, or takes a complex curve and straightens it out.
• The Weyl Group: This is the name of the "mirror system." The authors discovered that if you keep looking in these mirrors, you can generate new roads from old ones.
  • If the mirror system is finite (it eventually cycles back to the start), you have a finite number of roads.
  • If the mirror system is infinite (it keeps generating new, unique distortions), you have an infinite number of roads.

4. The "Anticanonical" Compass

The authors introduce a special compass called the Anticanonical Class ($F$).

• Think of $F$ as the "gravity" of the city.
• They created a new way to measure roads using a bilinear form (a special math formula that acts like a ruler and a protractor combined).
• By measuring a road against this compass ($F$) and checking its "self-interaction" (how much it twists on itself), they can mathematically predict:
  1. Is this road a $(-1)$, $(0)$, or $(1)$ curve?
  2. Is the city organized or chaotic?

5. The Big Discovery (The Main Results)

Here is what they found, translated into plain English:

• For Organized Cities (Mori Dream Spaces):
  If the city is organized, every $(-1)$ road is actually a "Weyl line" (a road you can find by looking in the magic mirrors). You can count them all, and there is a specific formula to find them.

  • Surprise: Even in some chaotic cities (like $Y^5_9$), the number of rigid $(-1)$ roads might still be finite, but the flexible ones explode to infinity.

• For Chaotic Cities:
  If you have too many points, the flexible roads ($0$and$1$) become infinite. This proves the city is not a "Mori Dream Space."

  • The Twist: The authors used this to prove a famous result by Shigeru Mukai: If the "gravity" of the city ($F^2$) is weak or negative, the city is chaotic.

6. Why Does This Matter?

You might ask, "Why do we care about counting roads in a blown-up city?"

• For Mathematicians: It solves a puzzle about the "Cox Ring," which is like the master blueprint of the city. If the blueprint is finite, the city is easy to study. If it's infinite, it's a nightmare. This paper gives a clear test to see if the blueprint is finite.
• For Physicists: These curves relate to the shape of the universe in String Theory (Calabi-Yau manifolds). Understanding when these shapes are "finite" or "infinite" helps physicists understand the fundamental laws of nature.
• The Method: They used "movable curves" (roads that can slide) to solve problems that were previously only solvable by looking at "divisors" (walls). It's like solving a puzzle by looking at the shadows instead of the objects themselves.

Summary

The paper is a guidebook for a magical city. It tells us:

1. Blow up too many points? The city becomes chaotic with infinite special roads.
2. Blow up just the right amount? The city is organized, and we can count every special road.
3. How do we know? We use a magic mirror system (Weyl Group) and a special compass (Bilinear Form) to measure the roads. If the compass says the city is "stable," the roads are finite. If it says "unstable," the roads go on forever.

This work bridges the gap between abstract algebra, geometry, and the physical intuition of how shapes behave when we poke holes in them.

Technical Summary

Here is a detailed technical summary of the paper "On (i)-Curves in Blowups of $\mathbb{P}^r$" by Olivia Dumitrescu and Rick Miranda.

1. Problem Statement and Context

The paper investigates the geometry of $(i)$-curves on the blowup of projective space $\mathbb{P}^r$ at $s$ general points, denoted as $Y^r_s$.

• Definition: An $(i)$-curve is a smooth, irreducible rational curve whose normal bundle splits as $\mathcal{O}(i)^{\oplus (r-1)}$. The authors focus on $i \in \{-1, 0, 1\}$.
  • $i=-1$: Generalizes the classical $(-1)$-curves on surfaces (Del Pezzo surfaces) and rigid curves on Calabi-Yau threefolds.
  • $i=0, 1$: New generalizations relevant to higher-dimensional birational geometry.
• Core Question: Under what conditions are the sets of $(i)$-curves finite? How do these curves relate to the structure of the Cox ring and the classification of Mori Dream Spaces (MDS)?
• Historical Motivation:
  • $(-1)$-curves were central to early mirror symmetry (Kontsevich) and Hilbert's 14th problem (Nagata).
  • Nagata showed that for $s \ge 9$ points in $\mathbb{P}^2$, the Cox ring is not finitely generated due to the infinity of $(-1)$-curves.
  • The authors aim to generalize this relationship to higher dimensions ($r \ge 3$) and other normal bundles ($i=0, 1$).

2. Methodology

The authors employ a combination of intersection theory, Cremona transformations, and Coxeter group theory.

• The Chow Ring and Intersection Forms:
  • They work within the Chow ring $A^*(Y^r_s)$, specifically focusing on divisor classes $A^1$ and curve classes $A^{r-1}$.
  • They introduce a bilinear form $\langle -, - \rangle$ on curve classes, derived from the quadratic invariant $q_{r-1}(dh - \sum m_i e_i) = d^2 - (r-1)\sum m_i^2$.
  • They define a unique anticanonical curve class $F = (r+1)h - \sum e_i$, which is invariant under the Weyl group.
• Weyl Group and Cremona Transformations:
  • The standard Cremona transformation (centered at $r+1$ points) generates a group action on the Chow ring.
  • The group generated by these transformations and the symmetric group of points is the Weyl Group ($W$).
  • The authors analyze the orbits of curve classes under $W$. They define $(i)$-Weyl lines as curves in the $W$-orbit of the proper transform of a line passing through $1-i$ base points.
• Numerical Invariants:
  • They define numerical $(i)$-classes based on linear and quadratic invariants:
    • Linear: $\langle c, F \rangle = 2 + i(r-1)$ (equivalent to $(-K_Y \cdot c)$).
    • Quadratic: $\langle c, c \rangle$ takes specific values (e.g., $3-2r$for$i=-1$).
  • They investigate whether numerical conditions are sufficient to characterize actual geometric curves (the Conjecture 2.2 that every $(i)$-Weyl line is an $(i)$-curve).

3. Key Contributions and Results

A. Characterization of Mori Dream Spaces

The paper establishes a precise equivalence between the finiteness of curve classes and the Mori Dream Space property.

• Theorem 1.2: For $Y = Y^r_s$, the following are equivalent:
  1. $Y$ is a Mori Dream Space.
  2. The Weyl group is finite.
  3. $F^2 = \langle F, F \rangle > 0$ (which corresponds to specific bounds on $r$ and $s$: $r=2, s\le 8$; $r=3,4, s\le r+4$; $r\ge 5, s\le r+3$).
  4. There are finitely many $(0)$-curves (or $(0)$-Weyl lines).
  5. There are finitely many $(1)$-curves (or $(1)$-Weyl lines).
• Significance: This provides a numerical criterion for the MDS property using curve classes, extending Nagata's result from surfaces to higher dimensions.

B. Classification of $(i)$-Curves

• Theorem 1.3: In Mori Dream Spaces (and the specific case $Y^5_9$), a curve $C$ is a $(-1)$-curve if and only if it is a $(-1)$-Weyl line, which is equivalent to satisfying specific numerical invariants ($\langle c, c \rangle = 3-2r$ and $\langle c, F \rangle = 3-r$).
  • Note: The authors prove that for MDS, numerical $(-1)$-classes correspond to unique geometric curves.
• Exceptions: They identify specific cases where numerical classes do not correspond to Weyl lines or where irreducibility fails (e.g., the class $F$ in $Y^3_7$ or $2F$in$Y^4_8$for$i=0$).
• Conjecture 2.2: They prove that every $(i)$-Weyl line is indeed an $(i)$-curve for $i \in \{0, 1\}$ for all $r, s$, and for $i=-1$ when $s \le r+4$.

C. Infinity of Curves and Non-MDS

• Theorem 5.1 & 5.9: If $Y^r_s$ is not a Mori Dream Space (i.e., $s$ is large enough such that $F^2 \le 0$), there are infinitely many classes of $(0)$- and $(1)$-curves.
• Corollary 5.14: If $s \ge r+5$, there are infinitely many $(-1)$-Weyl lines (and thus $(-1)$-curves).
• Surprising Result: The space $Y^5_9$ is not a Mori Dream Space (infinite Weyl group), yet it has only finitely many $(-1)$-curves. This demonstrates that the finiteness of $(-1)$-curves alone is not sufficient to characterize MDS in higher dimensions; the behavior of $(0)$ and $(1)$ curves is the critical differentiator.

D. Applications to Effective Cones

• Theorem 1.4 & 6.7: For $Y^r_{r+3}$, the extremal rays of the cone of movable curves are exactly the $(0)$-Weyl lines and $(1)$-Weyl lines.
• Theorem 6.13: If $Y$ is not an MDS, the $(0)$- and $(1)$-Weyl lines provide an infinite set of necessary conditions for a divisor to be effective.
• Re-proof of Mukai's Result: The authors use the theory of movable curves to re-prove that if $F^2 \le 0$, then $Y^r_s$ is not a Mori Dream Space.

4. Significance and Impact

1. Unification of Dimensions: The paper successfully generalizes the theory of $(-1)$-curves from surfaces ($r=2$) to arbitrary dimensions, introducing the concepts of $(0)$ and $(1)$ curves to fill the gap in higher-dimensional birational geometry.
2. Numerical Criteria for MDS: It provides a robust numerical test (finiteness of $(0)$ and $(1)$ classes) to determine if a blowup of $\mathbb{P}^r$ is a Mori Dream Space, linking the geometry of curves directly to the finite generation of the Cox ring.
3. Movable Cone Geometry: By identifying the extremal rays of the movable cone of curves with Weyl lines, the paper offers a new tool for analyzing the effective cone of divisors, which is crucial for understanding the birational geometry of these varieties.
4. Refinement of Nagata's Legacy: It clarifies the limits of Nagata's counterexample logic, showing that while $(-1)$-curves are infinite in non-MDS cases for $s \ge r+5$, the "detecting" curves for the MDS property in higher dimensions are actually the $(0)$ and $(1)$ curves.

In summary, Dumitrescu and Miranda establish a comprehensive framework connecting the arithmetic of curve classes, the action of the Weyl group, and the birational classification of blowups of projective space, resolving long-standing questions about the finiteness of curves and the structure of Mori Dream Spaces in higher dimensions.