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

Alessio Cela; Carl Lian

arxiv: 2505.14672 · v4 · submitted 2025-05-20 · 🧮 math.AG

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

Alessio Cela , Carl Lian This is my paper

Pith reviewed 2026-05-22 13:43 UTC · model grok-4.3

classification 🧮 math.AG

keywords complete quasimapsmoduli spacesblow-upsenumerative geometrytautological intersectionsBrill-Noether theoremtoric surfaces

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The pith

A moduli space of complete quasimaps to blown-up projective space is built by blowing up standard quasimaps at linear dependence loci, with the conjecture that its tautological intersections count curves of fixed complex structure under inc

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs moduli spaces of complete quasimaps to the blow-up Bl of P^r by P^s. The construction blows up the Ciocan-Fontanine-Kim quasimap space at the loci where sections of the tautological line bundles become linearly dependent. The central conjecture states that the tautological intersection numbers on these spaces enumerate curves of fixed complex structure on the target satisfying general incidence conditions. This differs from classical moduli spaces of maps, which permit the complex structure on the curve to vary. Purity of the expected dimension follows from a theorem of Farkas, and the conjecture is established in dimension two by invoking a Brill-Noether theorem for general curves on toric surfaces.

Core claim

The authors define complete quasimaps to Bl_{P^s}(P^r) by successive blow-ups of the quasimap space at points of linear dependence among sections. They conjecture that the tautological intersections on the resulting moduli spaces compute enumerative counts of curves with fixed complex structure on the blown-up space, subject to general incidence conditions, and prove this conjecture when the target has dimension two.

What carries the argument

The moduli space of complete quasimaps, constructed by blowing up the Ciocan-Fontanine-Kim quasimap space at loci where sections of the relevant line bundles are linearly dependent.

If this is right

The spaces provide a compactification in which the complex structure on the domain curve stays fixed while still allowing intersection theory.
Tautological classes on these spaces directly yield enumerative invariants for curves on blown-up projective spaces.
In dimension two the conjecture supplies explicit counts via the Brill-Noether theorem on toric surfaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same blow-up procedure could be applied to quasimaps into other toric targets to obtain analogous enumerative theories.
Comparison with stable-map or stable-quasimap compactifications may reveal how fixing the complex structure alters the resulting curve counts.
Higher-dimensional cases would become accessible once Brill-Noether-type statements are established for curves on higher-dimensional toric varieties.

Load-bearing premise

The blown-up moduli spaces remain pure of the expected dimension.

What would settle it

A concrete mismatch, in dimension three or higher, between the tautological intersection numbers on the complete quasimap space and independently known enumerative counts of fixed-complex-structure curves on the blown-up target.

read the original abstract

We introduce a moduli space of ``complete quasimaps'' to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$. The construction, following previous work for curves on projective spaces, essentially proceeds by blowing up Ciocan-Fontanine--Kim's space of quasimaps at loci where sections of line bundles are linearly dependent. We conjecture that tautological intersection numbers on these moduli spaces give enumerative counts of curves of fixed complex structure on $X$ subject to general incidence conditions, in contrast with traditional compactifications of the moduli spaces of maps. A result of Farkas guarantees that these spaces are pure of expected dimension. The conjecture is proven in dimension 2, where the main input is a Brill-Noether theorem for general curves on toric surfaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper builds a new complete quasimap space to blow-ups of projective space by blowing up the standard quasimap space and conjectures that its intersections count fixed-complex-structure curves, with a proof only in dimension 2.

read the letter

Hi, The key point with this paper is the construction of a moduli space of complete quasimaps to the blow-up of projective space, obtained by blowing up the standard quasimap space at linear dependence loci, together with a conjecture that its tautological intersections count curves of fixed complex structure on the blow-up. They do a good job extending the Ciocan-Fontanine-Kim framework in a natural way. The use of Farkas' purity result to control the dimension is a solid move, and the proof of the conjecture in dimension 2 via Brill-Noether theory on toric surfaces gives concrete evidence that the idea works at least in low dimensions. The abstract is straightforward about what is conjectural and what is proven. The potential issue is whether the blow-up step preserves the purity and the expected dimension without extra components. The stress-test note raises this, and it would be good to see an explicit check that the centers are of the right codimension and that the resulting space still meets the conditions for Farkas' theorem. Without that, the claim that the intersections are enumerative in the usual sense rests on an assumption that might need more justification in higher dimensions. The conjecture itself is not circular, as it introduces new objects. This work is for specialists in enumerative algebraic geometry who deal with moduli of curves and maps to projective varieties or their blow-ups. A reader looking for new compactifications that track complex structure more precisely would find value here. It has enough substance and novelty to merit a serious referee report rather than a desk rejection. I would recommend putting it through peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a moduli space of complete quasimaps to Bl_{P^s}(P^r) obtained by blowing up the Ciocan-Fontanine--Kim quasimap space at loci of linear dependence among sections. It conjectures that tautological intersection numbers on this space enumerate curves of fixed complex structure on the target subject to general incidence conditions. Purity of expected dimension is asserted via a result of Farkas, and the conjecture is proved in dimension 2 using a Brill-Noether theorem for general curves on toric surfaces.

Significance. If the conjecture holds, the construction supplies a new compactification suited to enumerative problems with fixed complex structure, offering a contrast to traditional moduli of maps. The explicit proof in dimension 2 and the appeal to Farkas' purity constitute concrete strengths that ground the approach in existing results.

major comments (1)

[Construction section] Construction (the blow-up step following Ciocan-Fontanine--Kim): the loci of linear dependence are taken as blow-up centers, yet no explicit computation or reference verifies that these centers have the expected codimension, that the resulting space remains irreducible, or that it satisfies the precise hypotheses (mild singularities, quasimap data form) of Farkas' purity theorem. This verification is load-bearing for the claim that the spaces are pure of expected dimension and that ordinary intersection theory yields enumerative counts.

minor comments (2)

[Abstract] The abstract states that the conjecture is proven in dimension 2 but does not name the Brill-Noether theorem as the main input; adding this would improve immediate clarity.
Notation for the linear-dependence loci and the resulting exceptional divisors could be introduced with a displayed equation or short diagram to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We respond point by point to the major comment below.

read point-by-point responses

Referee: [Construction section] Construction (the blow-up step following Ciocan-Fontanine--Kim): the loci of linear dependence are taken as blow-up centers, yet no explicit computation or reference verifies that these centers have the expected codimension, that the resulting space remains irreducible, or that it satisfies the precise hypotheses (mild singularities, quasimap data form) of Farkas' purity theorem. This verification is load-bearing for the claim that the spaces are pure of expected dimension and that ordinary intersection theory yields enumerative counts.

Authors: We agree that the manuscript does not currently contain an explicit verification of the codimension of the linear dependence loci or a direct confirmation that the blown-up space meets all hypotheses of Farkas' purity theorem. In the revision we will add a paragraph in the Construction section supplying these details. The codimension computation will follow from the expected rank of the evaluation map on the Ciocan-Fontanine--Kim space, using the same bundle data as in the projective-space case. Irreducibility of the resulting space will be argued by observing that the centers lie in the boundary divisor and the main component remains dominant after blow-up. For Farkas' theorem we will note that the blow-up centers are regularly embedded and that the quasimap data (including the sections and the stability condition) are preserved, yielding at worst mild singularities compatible with the theorem's hypotheses. This addition will make the purity claim fully grounded. revision: yes

Circularity Check

0 steps flagged

No circularity: construction and conjecture are self-contained with external inputs

full rationale

The paper defines the moduli space of complete quasimaps to the blow-up by blowing up the Ciocan-Fontanine--Kim quasimap space at linear-dependence loci, following prior independent work on projective space. It invokes an external result of Farkas to guarantee purity of expected dimension, enabling ordinary intersection theory. The central conjecture equates tautological intersections to enumerative counts of fixed-complex-structure curves and is proven in dimension 2 via a Brill-Noether theorem on toric surfaces. No equations or steps reduce the claimed counts or purity to fitted parameters, self-definitions, or load-bearing self-citations; all load-bearing steps rely on independent external theorems or new constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the purity result of Farkas and on the Brill-Noether theorem in dimension 2; no free parameters or new invented entities are introduced.

axioms (1)

domain assumption A result of Farkas guarantees that these spaces are pure of expected dimension.
Invoked to ensure the moduli space has the correct dimension for intersection theory.

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