Complete quasimaps to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$

1. Problem Statement and Motivation

The central problem addressed is the computation of geometric Tevelev degrees ($Tev_{g,n,\beta}^X$), which count the number of maps $f: C \to X$ from a fixed general curve $C$ of genus $g$ to a target variety $X$, subject to incidence conditions $f(p_i) = x_i$ for $n$ general points.

• The Obstacle: Traditional compactifications of the moduli space of maps, such as the moduli space of stable maps ($\overline{M}_{g,n}(X, \beta)$) or quasimaps ($Q_\beta(C, X)$), often fail to provide enumerative counts. While they allow for intersection-theoretic calculations, the resulting integrals typically yield "virtual" counts. These virtual counts include contributions from maps with singular domains or maps that degenerate in ways that do not correspond to actual geometric curves satisfying the incidence conditions (excess intersection at the boundary).
• The Goal: The authors aim to construct a specific compactification of the moduli space of maps where the intersection of incidence cycles is transverse and supported entirely on the locus of honest, non-degenerate maps. This would allow tautological intersection numbers to equal the actual geometric count of curves.

The specific target variety studied is the blow-up of projective space along a linear subspace:
$$X = Bl_s(\mathbb{P}^r)$$
where $s \leq r-2$.

2. Methodology and Construction

The authors propose a construction based on Complete Quasimaps, extending previous work on quasimaps to $\mathbb{P}^r$ (Lian, 2024). The methodology proceeds in three main stages:

A. Augmented Quasimaps and Linear Series Inclusions

Instead of starting with the standard quasimap space $Q_\beta(C, X)$, the authors introduce a refined space of augmented quasimaps, denoted $Q^+_\beta(C, X)$.

• Underlying Geometry: A non-degenerate map $f: C \to X$ corresponds to an Inclusion of Linear Series (ILS). Specifically, the map is determined by a line bundle $L$ of degree $d$, an effective divisor $D$ of degree $k$, and a subspace inclusion $V \subset W$ of sections, where $W \subset H^0(C, L)$ has dimension $r+1$ and $V \subset H^0(C, L(-D))$ has dimension $r-s$.
• The Space: $Q^+_\beta(C, X)$ is constructed as a projective bundle over the moduli space of such inclusions, $Gr^{r-s-1}_{d,k}(C)$. This space is known to be of the expected dimension and pure.

B. The Blow-up Strategy (Complete Quasimaps)

The space $Q^+_\beta(C, X)$ still contains loci where the incidence conditions fail to be transverse. These failures occur at strata where the sections defining the map become linearly dependent (low "bi-rank").

• Bi-rank: Defined as $(r_1, r_2) = (\dim \langle u_1, \dots, u_{r-s} \rangle, \dim \langle u_1, \dots, u_{r+1} \rangle)$.
• Construction: The authors define the moduli space of complete quasimaps, $\widetilde{Q}_\beta(C, X)$, as an iterated blow-up of $Q^+_\beta(C, X)$.
  • The blow-ups are performed along the closures of strata of augmented quasimaps with low bi-rank, ordered first by $r_1$ and then by $r_2$.
  • This process resolves the excess intersections by "blowing up" the components of larger-than-expected dimension out of existence, effectively compactifying the space of non-degenerate maps.

C. Incidence Loci

The authors define incidence loci $Inc(p, \Lambda) \subset \widetilde{Q}_\beta(C, X)$ as the proper transforms of the loci where $f(p) \in \Lambda$ (for a linear subspace $\Lambda \subset X$). The core conjecture is that the intersection of $n$ such general incidence loci is:

1. Generically smooth.
2. Pure of the expected codimension.
3. Supported entirely on the open locus of honest, non-degenerate maps.

3. Key Contributions and Results

Main Conjecture

Conjecture 1.1.2 / 2.7.3: For $X = Bl_s(\mathbb{P}^r)$, the geometric Tevelev degree is given by the integral of the product of incidence classes over the moduli space of complete quasimaps:
$$Tev_{g,n,\beta}^X = \int_{\widetilde{Q}_\beta(C, X)} [Inc(p, x)]^n$$
This holds provided $n \geq r+1$ (ensuring non-degeneracy).

Theorem 1.1.3 (Dimension 2)

The authors prove the conjecture in the case where $X = Bl_q(\mathbb{P}^2)$ (the blow-up of $\mathbb{P}^2$ at a point).

• Key Input: The proof relies on a Brill-Noether theorem for curves on toric surfaces (Theorem 4.1.1, from their previous work [CL25a]). This theorem guarantees that the moduli space of maps to certain toric blow-ups of $\mathbb{P}^2$ is generically smooth and of the expected dimension, provided the image curve has a sufficiently high anti-canonical degree.
• Transversality: By analyzing the strata of the blow-up (specifically the bi-rank $(1,1)$ and $(1,2)$ loci) and using the Brill-Noether results, they demonstrate that the incidence loci intersect transversely and do not acquire excess components in the boundary.

Set-Theoretic Descriptions

The paper provides explicit set-theoretic descriptions of the points in $\widetilde{Q}_\beta(C, X)$ and the incidence loci.

• Points in the exceptional divisors are described via complete bi-collineations, a generalization of complete collineations. These involve sequences of linear maps between vector spaces derived from the sections of the map, resolving the degeneracy of the sections.
• For $X = Bl_q(\mathbb{P}^2)$, the authors explicitly characterize the intersection of incidence loci with the exceptional divisor, showing how the "vanishing" of sections at the marked points is resolved by the blow-up data.

Relation to Other Moduli Spaces

The authors compare $\widetilde{Q}_\beta(C, X)$ with:

1. Standard Quasimaps ($Q_\beta$): These may have components of unexpected dimension and virtual classes that do not match enumerative counts.
2. Naive Quasimaps ($P$): A space used in previous work [CL23a] for "large degree" calculations.

• Result: In the "large degree" regime ($n > d+1$), the intersection numbers on $\widetilde{Q}_\beta$, $Q^+_\beta$, and $P$ coincide and are enumerative. However, for small $n$ (where transversality fails in standard spaces), only the complete quasimap space $\widetilde{Q}_\beta$ is expected to yield the correct enumerative count.

4. Significance and Future Directions

• Resolution of Excess Intersection: The paper provides a concrete geometric construction to resolve the "excess intersection" problem in enumerative geometry for blow-ups of projective spaces. It moves beyond virtual counts to actual geometric counts for specific classes of targets.
• Brill-Noether Theory: The work highlights the necessity of new Brill-Noether theorems for maps to toric varieties. The proof in dimension 2 relies on specific properties of surfaces; the general case requires a suitable analog for higher-dimensional toric varieties.
• Future Work:
  • Higher Dimensions: The authors identify that proving the conjecture for $X = Bl_s(\mathbb{P}^r)$ with $r > 2$ requires a "pointed Brill-Noether theorem" for maps to higher-dimensional toric varieties (Question 5.2.2).
  • Calculations: While the existence of the moduli space and the transversality are established (in dim 2), the explicit computation of the integrals (the actual Tevelev degrees) is deferred to future work, likely utilizing degeneration techniques.

In summary, this paper introduces a robust moduli space of complete quasimaps that serves as a candidate for a "true" enumerative compactification for maps to blow-ups of projective spaces. It successfully proves this for surfaces by leveraging advanced Brill-Noether theory, offering a pathway to solve similar problems in higher dimensions.