Fixed-domain curve counts for blow-ups of projective space

Imagine you are an architect trying to build a specific type of house (a "curve") on a very specific piece of land (a "variety").

In the world of mathematics, specifically Algebraic Geometry, there are two main ways to count how many of these houses you can build:

The "Virtual" Count (The Theoretical Blueprint): This is a calculation based on the rules of the universe. It asks, "If the laws of physics and geometry were perfectly flexible, how many houses should fit here?" It's a number derived from complex formulas, often ignoring the messy reality of whether the houses actually touch or overlap in weird ways.
The "Geometric" Count (The Actual Construction): This is the real-world answer. It asks, "If I actually try to build these houses on this specific land, how many distinct, non-overlapping houses can I fit?"

For a long time, mathematicians knew that for simple, flat lands (like a standard sheet of paper, or Projective Space), these two numbers usually agreed if you were building enough houses. But what happens if you take that flat land and punch holes in it, or blow it up into a mountainous terrain with craters? This is what the paper "Fixed-Domain Curve Counts for Blow-Ups of Projective Space" investigates.

Here is the breakdown of their discovery, using some everyday analogies.

1. The Setup: Blowing Up the Land

Imagine Projective Space ( $\mathbb{P}^r$ ) as a giant, perfectly flat, infinite parking lot.

Blowing up a point on this lot is like taking a drill and creating a deep, circular crater (an "exceptional divisor") at that spot.
The authors are studying what happens when you drill several of these craters into the parking lot at random spots.
The Goal: They want to draw a specific path (a curve) that starts at a fixed shape (the "fixed domain") and passes through specific points on this cratered parking lot.

2. The Big Question: Do the Blueprints Match Reality?

The paper tackles three questions:

What does the Virtual Count say? (The theoretical number).
What does the Geometric Count say? (The actual number).
Do they match?

In the old days, for simple flat lands, the answer was usually "Yes, they match." But the authors found that when you start drilling craters (blowing up points), things get messy.

The "Strong Asymptotic Enumerativity" (SAE) Test

The authors introduced a concept called SAE. Think of this as a "Stability Test."

The Test: If you keep adding more and more points (constraints) to your parking lot, does the Virtual Count eventually settle down and match the Geometric Count?
The Discovery:
- For 2D and 3D lands with few craters: Yes! The blueprints match the reality. Even if the land isn't perfectly "nice" (mathematically called "Fano"), the counts agree.
- For 4D+ lands with 2 or more craters: No! The blueprints lie. The Virtual Count predicts a certain number of paths, but the actual Geometric Count is different. The "craters" create hidden traps where paths can get stuck or overlap in ways the theoretical formula didn't predict.

Analogy: Imagine a GPS (Virtual Count) telling you there are 100 routes to a destination. But because of a new construction zone (the craters), the actual roads (Geometric Count) are blocked or loop back on themselves. The GPS says "100," but you can only physically drive 50. In high dimensions, the GPS is unreliable.

3. How They Solved the "Real" Count (Geometric)

Since the GPS (Virtual Count) was lying for these complex terrains, the authors had to build a new way to count the real paths.

They used a clever trick involving Jacobian varieties.

The Analogy: Imagine the parking lot is a giant stage. Instead of trying to draw the path directly on the stage, they moved the problem to a "control room" (the Jacobian).
In this control room, the problem of drawing a path becomes an integral (a fancy way of summing up areas).
They derived a massive formula (Theorem 1.11) that acts like a super-calculator. If you plug in the number of craters, the size of the path, and the number of points you need to hit, this formula spits out the exact number of real paths.
They tested this on simple cases (like 1 crater) and found beautiful, clean formulas involving binomial coefficients (like combinations in a lottery).

4. How They Solved the "Virtual" Count

For the theoretical count, they used Quantum Cohomology.

The Analogy: This is like using a "Quantum Simulator." Instead of looking at the parking lot as it is, the simulator allows the paths to "tunnel" through the craters or interact in magical ways defined by the rules of quantum physics (in math terms).
They calculated the "Quantum Euler Class" (a special number that summarizes the shape of the land) and used it to predict the Virtual Count.
The Result: They found that for the specific case of one crater, the Virtual Count formula actually matches the Geometric Count formula!
- Wait, didn't we just say they don't match?
- The Catch: They match only when the path is "long enough" or the constraints are loose enough. If the path is too short or the constraints too tight, the Virtual Count (the simulator) still gives the wrong answer compared to reality.

5. The "Permutahedral" Twist

To prove that their "Real Count" was actually correct and that the paths didn't get stuck in weird loops, they used a shape called a Permutahedral Variety.

The Analogy: Imagine the parking lot is a flat sheet of paper. To see all the hidden folds and wrinkles, they blew the paper up into a complex, multi-faceted crystal (the Permutahedron).
By lifting their problem onto this crystal, they could prove that the paths they counted were "honest" (they didn't have hidden overlaps). It was like using X-ray vision to ensure the houses weren't phasing through each other.

Summary of the Takeaway

The Problem: Counting paths on a landscape with holes (blow-ups).
The Surprise: In high dimensions with multiple holes, the "theoretical" count (Virtual) stops matching the "real" count (Geometric). The universe gets too complex for the simple formulas to work.
The Solution: The authors built a new "calculator" (Integral Formula) to get the real number and a "Quantum Simulator" (Quantum Cohomology) to get the theoretical number.
The Conclusion: Sometimes the simulator is right, sometimes it's wrong. But now, we have the tools to know exactly when it's lying and how to find the truth.

In short, this paper is about learning when to trust your GPS and when to look out the window, specifically when driving through a city that has been under heavy construction.

Here is a detailed technical summary of the paper "Fixed-Domain Curve Counts for Blow-Ups of Projective Space" by Alessio Cela and Carl Lian.

1. Problem Statement

The paper investigates the problem of counting holomorphic maps from a fixed smooth curve $(C, p_1, \dots, p_n)$ of genus $g$ to a target variety $X$ , where the map passes through $n$ general points in $X$ . This is known as the fixed-domain Tevelev degree problem.

Context: Traditionally, curve counting is formulated via Gromov–Witten (GW) theory, which integrates against a virtual fundamental class $[M_{g,n}(X, \beta)]^{vir}$ . This yields "virtual counts" ( $vTev$ ).
The Fixed-Domain Twist: Instead of varying the complex structure of the domain curve, the authors fix the domain curve $(C, p_1, \dots, p_n)$ to be general. This restricts the moduli space to the fiber of the forgetful map $\tau: M_{g,n}(X, \beta) \to M_{g,n} \times X^n$ .
The Core Questions:
1. What is the virtual count $vTev_{g,n,\beta}^X$ ?
2. What is the geometric count $Tev_{g,n,\beta}^X$ (the actual number of such maps)?
3. Do these counts agree? (i.e., Is the virtual count enumerative?)

The authors focus on $X$ being the blow-up of projective space $\mathbb{P}^r$ at $\ell$ general points ( $X = \text{Bl}_{q_1, \dots, q_\ell}(\mathbb{P}^r)$ ).

2. Methodology

The authors employ a dual approach, combining geometric enumerative techniques with quantum cohomology calculations.

A. Enumerativity and Asymptotic Behavior

Strong Asymptotic Enumerativity (SAE): They analyze whether the virtual count equals the geometric count for sufficiently large $n$ (and thus large degree $\beta$ ).
Strategy: They utilize the boundary stratification of the moduli space $M_{g,n}(X, \beta)$ . By analyzing the dimension of loci where the domain curve degenerates (e.g., acquiring rational tails or singularities), they determine if the general fiber of the evaluation map $\tau$ is contained in the locus of smooth domains.
Tools: They use Brill-Noether theory, properties of Fano varieties, and specific intersection theory on blow-ups to establish criteria for when SAE holds or fails.

B. Geometric Counts (Integral Formulas)

For the case where the number of blown-up points $\ell \le r+1$ , the authors derive explicit formulas for the geometric count.

Parametrization: A map $f: C \to X$ $f : C \to X$ of class $\beta = dH^\vee + \sum k_i E_i^\vee$ $β = d H^{\lor} + \sum k_{i} E_{i}^{\lor}$ is parametrized by:
- A line bundle $L$ of degree $d$ on $C$ .
- Divisors $D_i$ of degree $k_i$ on $C$ (mapping to the exceptional points).
- Sections of $L$ vanishing on specific combinations of $D_i$ .
Moduli Space: They construct a parameter space $S = \text{Jac}^d(C) \times \text{Sym}^{k_1}(C) \times \dots \times \text{Sym}^{k_\ell}(C)$ and a projective bundle $\mathcal{P} \to S$ representing the sections.
Intersection Theory: The count is expressed as an integral over $\mathcal{P}$ of a cohomology class representing the condition that the map hits the $n$ general points.
Pushforward: Using the Grothendieck-Riemann-Roch (GRR) theorem, they push this integral down to the base $S$ . This involves computing Chern characters of vector bundles over products of Jacobians and symmetric products, utilizing the cohomology of $\text{Sym}^k(C)$ (Macdonald's description) and the theta divisor on the Jacobian.

C. Virtual Counts (Quantum Cohomology)

To compute the virtual Tevelev degrees, the authors utilize the Quantum Cohomology Ring $QH^*(X)$ .

Formula: They rely on the result that fixed-domain virtual counts can be computed as coefficients in the quantum product involving the quantum Euler class $\Delta$ .
Calculation: For $X = \text{Bl}_q(\mathbb{P}^r)$ , they explicitly compute the quantum Euler class and the relevant quantum products using the toric structure of the blow-up and the relations in $QH^*(X)$ .

3. Key Contributions and Results

I. Enumerativity (SAE)

The paper provides a nuanced picture of when virtual counts are enumerative:

Negative Result (Theorem 1.8): If $r \ge 4$ and $\ell \ge 2$ , the blow-up $X$ fails SAE. The authors construct counterexamples using negative rational curves (strict transforms of lines between blown-up points) that cause the general fiber to contain maps with singular domains.
Positive Results (Theorem 1.9):
- Del Pezzo Surfaces ( $r=2, \ell \le 8$ ): SAE holds.
- Single Point Blow-up ( $\ell=1$ ): SAE holds for any $r$ .
- Dimension 3 ( $r=3, \ell \le 4$ ): SAE holds. Notably, for $2 \le \ell \le 4 $,$ X $is **not Fano** (it is$ (-K_X)$-nef), making this the first known example of a non-Fano variety satisfying SAE.

II. Geometric Counting Formulas

The authors derive explicit integral formulas for $Tev_{g,n,\beta}^X$ when $\ell \le r+1$ :

General Formula (Theorem 1.11): An integral over $S$ involving elementary symmetric functions and exponential terms derived from GRR.
Genus 0 Specialization (Theorem 1.12): For $g=0$ $g = 0$ , the formula simplifies to a product of binomial coefficients (or a finite sum thereof).
- Example: For $X = \text{Bl}_q(\mathbb{P}^r)$ with $\ell=1$ , $Tev_{0,n,\beta} = \binom{n-d-1}{k}$ .
Arbitrary Genus with One Blow-up (Theorem 1.13): For $X = \text{Bl}_q(\mathbb{P}^r)$ , they provide a closed-form summation:
$Tev_{g,n,\beta} = \sum_{m=0}^g (2r)^{g-m}(1-r)^m \binom{g}{m} \binom{n-d+g-m-1}{k}$

III. Virtual Counts and Comparison

Virtual Formula (Theorem 1.14): Using quantum cohomology, they compute the virtual count for $X = \text{Bl}_q(\mathbb{P}^r)$ . Remarkably, the formula is identical to the geometric formula derived in Theorem 1.13.
Range of Validity: The virtual formula holds under the condition $n-d \ge 1$ , whereas the geometric formula requires the stricter condition $n-d \ge g+1$ (to ensure transversality and avoid base-point issues).
Discrepancy: In the "low degree" regime (where $n-d$ is small), the geometric count can be zero (the map is not dominant), while the virtual count remains non-zero. This highlights that while the formulas match in the "large degree" limit, enumerativity fails in low degrees for these specific blow-ups.

4. Significance

Resolution of the Fano Conjecture: The paper challenges the speculation that all Fano varieties satisfy SAE by providing counterexamples for blow-ups of $\mathbb{P}^r$ ( $r \ge 4, \ell \ge 2$ ).
Non-Fano SAE: It establishes the first examples of non-Fano varieties (specifically $(-K_X)$ -nef blow-ups of $\mathbb{P}^3$ ) that satisfy SAE, expanding the scope of enumerative geometry beyond Fano targets.
Explicit Computations: The derivation of explicit geometric formulas for fixed-domain counts on blow-ups is a significant technical achievement, bridging the gap between abstract GW theory and concrete enumerative geometry.
Methodological Innovation: The use of the permutohedral variety (a higher blow-up of $\mathbb{P}^r$ ) to prove transversality and the application of GRR on products of Jacobians and symmetric products provide powerful new tools for studying fixed-domain problems.

In summary, Cela and Lian provide a comprehensive analysis of fixed-domain curve counts on blow-ups of projective space, distinguishing clearly between virtual and geometric counts, establishing precise conditions for enumerativity, and delivering explicit formulas that reveal the asymptotic agreement between the two counts in the Fano and certain non-Fano cases.