REVIEW 3 major objections 5 minor 7 references
Enumerating log rational curves on some toric varieties
T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read The paper computes all genus-0 logarithmic Tevelev degrees of the toric projective bundles $X_{r,s,a}$ and gives a counterexample to a tropical conjecture for blow-ups of $\mathbb{P}^2$.
desk verdict Solid computation paper: proves the Cela–Iribar López conjecture for the projective bundle family and gives a real counterexample to the blow-up conjecture; the one compressed proof step checks out. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the moduli space $\mathcal{Q}_\Gamma(X)$ of naive log quasimaps: a smooth projective tower of two projective bundles over a product of copies of $\mathbb{P}^1$ that parametrize the positions of the boundary intersection points $q_{j,v}$. A quasimap is given by sections $g_0,\dots,g_{r+s+1}$ of prescribed line bundles on $\mathbb{P}^1$ vanishing along universal divisors $D_j=\sum_v \mu_{j,v}D_{j,v}$, and the incidence locus $V(p,x)$ for a general point $x$ in the interior is cut out by exactly $r+s$ equations, with cycle class $\zeta_1^r\zeta_2^s$. The argument computes the top intersection of $n$ such classes by Segre classes, and separately establishes enumerativity through two twisting operations, (T1) and (T2), that remove base points of the map; the key point is that the diagonal constraints introduced by twisting at different points act on pairwise disjoint sets of $\mathbb{P}^1$ factors, so the dimension drop is the sum of the individual drops.
What would settle it
Take the projective bundle $X_{1,1,1}=\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}\oplus\mathcal{O}(-1))$ with tangency profile $\mu_0=(1,1,1)$, $\mu_3=(1,1,1)$, and $\mu_1=\mu_2$ empty, so that $m_0+m_3=6>s(n-1)=3$ and the theorem predicts $\log\mathrm{Tev}=0$. Computing the degree of $\tau$ on the actual moduli space of log stable maps by a degeneration to a union of toric surfaces, or via the tropical correspondence theorem, gives a concrete test: any nonzero count would falsify the vanishing statement.
Extended reading notes
Core claim
The central claim is Theorem 4.2.1: in the situation where the invariant is defined, if the multiplicities satisfy $m_j\le n-1$ for $j=1,\dots,r+s+1$, $\sum_{j=r+2}^{r+s+1}m_j\ge (s-1)(n-1)$, and $m_0+\sum_{j=r+2}^{r+s+1}m_j\le s(n-1)$, then $$\log\mathrm{Tev}^{X_{r,s,a}}_\Gamma=\left(\prod_{j=0}^{r+s+1}m_j!\right)\left(\prod_{j=0}^{r+s+1}\prod_{v=1}^{m_j}\mu_{j,v}\right)$a^{{k_0-m_0}}$\binom{k_0}{m_0},$$ where $k_0=s(n-1)-\sum_{j=r+2}^{r+s+1}m_j$; otherwise the invariant is zero. The blow-up section shows that the analogous conjectured formula for $\mathrm{Bl}_{p_1,p_2}(\mathbb{P}^2)$ does not always hold: for the tangency profile $\mu_1=\mu_2=(1)$, $\mu_3=(1,1,1,1)$, $\mu_4=\mu_5=(5)$, the actual logarithmic Tevelev degree is $2400$, not the value $5400$ predicted by the tropical conjecture. The proof works by showing that, whenever the naive quasimap intersection fails to be enumerative, either the count is zero or the excess intersection formula corrects it.
Load-bearing premise
The argument that the naive quasimap intersection is enumerative assumes that the base-point-twisting operations (T1) and (T2), applied at different marked or unmarked points, impose diagonal constraints on pairwise disjoint sets of the $\mathbb{P}^1$ factors, so the total dimension drop is the sum of the individual drops; if these constraints ever overlapped, the dimension count would be too optimistic and the enumerativity statement could fail.
Editorial extensions
If this is right
- The genus-0 logarithmic Tevelev degrees of every bundle $\mathbb{P}_{\mathbb{P}^r}(\mathcal{O}^s\oplus\mathcal{O}(-a))$ are now known in closed form whenever the dimension condition $n=m/(r+s)+1$ holds.
- The case $s=1$ proves the earlier conjecture for these bundles and specializes to the known Hirzebruch surface counts; taking $a=0$ recovers the product formula for $\mathbb{P}^r\times\mathbb{P}^s$.
- The counterexample on $\mathrm{Bl}_{p_1,p_2}(\mathbb{P}^2)$ shows that tropical predictions for fixed-domain log counts on blow-ups of projective space are not reliable outside the range where the naive quasimap intersection is enumerative.
- The vanishing criteria in the theorem make explicit the exact boundary of the enumerative range, so the formula can be applied without checking individual cases.
Reading between the lines
- The same tower-of-projective-bundles construction should extend to iterated projective bundles or toric varieties with a similar flag of fibrations, yielding closed formulas by the same Segre-class computation.
- The factor $a^{k_0-m_0}$ suggests interpreting the count as a weighted count in which intersections with the $\mathcal{O}(-a)$-summand divisor contribute a factor of $a$; a degeneration proof might make this weight visible and generalize it to other toric weights.
- One could test whether the excess intersection correction in the blow-up example can be packaged as a universal formula in the multiplicities, which would predict exactly which tropical conjectures fail and by how much.
- The enumerativity failure in the blow-up case may be tied to the existence of quasimaps where two of the three sections vanish; a natural next question is whether every non-enumerative profile has a canonical excess locus with a closed-form contribution.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies genus-zero logarithmic Tevelev degrees of toric varieties, i.e. the degree of the forgetful map from a moduli space of pointed log maps with prescribed boundary tangency to M_{0,n} x X^n, in the balanced case n = m/dim(X)+1. The first main result, Theorem 4.2.1, gives a closed formula for these degrees when X = P_{P^r}(O^s oplus O(-a)) and explicit inequalities on the multiplicities hold, and proves vanishing otherwise; the s=1 case verifies [CIL24, Conjecture 14]. The second main result, Theorem 5.2.2, computes an example on the blow-up of P^2 at two points where the logarithmic Tevelev degree is 2400, while the integral of the corresponding tautological class on the naive quasimap space is 5400, thereby disproving [CIL24, Conjecture 15]. The proofs are based on moduli spaces of naive log quasimaps and direct Segre-class computations rather than tropical methods.
Significance. If correct, the projective-bundle formula is a substantial and parameter-free result: it completely determines a family of logarithmic Tevelev degrees and confirms a conjecture of Cela and Iribar Lopez. The counterexample to Conjecture 15 is also significant because it shows that the tropical prediction fails precisely in a range not covered by the vanishing criteria, and it demonstrates the power of excess-intersection arguments on quasimap spaces. The approach is a useful algebro-geometric alternative to tropical enumerations. The main computation in Section 4.2 is clean and convincing, and the paper is honest about the distinction between its normalization of the Tevelev degree and that of [CIL24]. However, several load-bearing auxiliary claims are only sketched, especially the classification of the excess locus in Theorem 5.2.2 and the full statement of Proposition 5.2.1, so the paper needs revision before acceptance.
major comments (3)
- [5.2, Theorem 5.2.2 and Lemma 5.2.3] The proof of the classification of the excess locus Z is compressed into a 'straightforward analysis' and an 'easy incidence correspondence'. Since the disproof of [CIL24, Conjecture 15] rests on this, the authors should give a complete argument that V cap (Q_Gamma(X) - Q_Gamma^{ne 0}(X)) is contained in the locus g_4 = g_5 = 0 and that the incidence conditions force q_{1,1} = q_{2,1} = p_i for a single i and D_3 equal to the complementary four points. In particular, the vanishing conditions are on twisted line bundles: 'vanishes at p_i as a section of O(6)(-D_1)' means the order of vanishing exceeds the multiplicity of p_i in D_1, and the resulting five pairs of inequalities should be written out. Lemma 5.2.3 should also spell out the first-order calculation, including the four incidence conditions on the pair [gamma_4(z-q_4)^5 : gamma_5(z-q_5)^5]. The claimed classification is plausible and the alternative configurations mentioned by a skeptical reader appear to be excluded by degree reasons, but as written the proof is not complete.
- [3.2, Proposition 3.2.1] The proof relies on the assertion that the diagonal constraints introduced by twisting at distinct points are imposed on pairwise disjoint subsets of the factors of B. This is stated without proof, and the dynamic re-indexing of the points q_{j,v} in Algorithm 3.2.2 makes the assertion nontrivial. Since Proposition 3.0.1(iii), and hence the enumerativity of the main formula in Theorem 4.2.1, depends on this dimension count, the authors should either prove the disjointness/independence claim directly or replace it with a global dimension estimate that does not require tracking individual factors. Relatedly, the sentence concluding that V' 'must dominate' (P^1)^n x (X^circ)^n from existence at one general point should be justified using constructibility or by choosing the points very generally.
- [5.1-5.2, Proposition 5.2.1] Proposition 5.2.1 collects six nontrivial assertions: reducedness of the intersection on M_Gamma(X), emptiness on the non-bpf and vanishing-section loci, the vanishing criteria for logTev, and the closed-form integral. It is stated that these follow by adapting previous arguments, but no proofs are supplied. The counterexample in Theorem 5.2.2 and the range in which logTev equals the predicted integral both depend on this proposition. The authors should provide full proofs or precise proposition-by-proposition reductions, especially for parts (iv) and (v), whose vanishing-section analysis is specific to the blow-up and is not literally identical to the projective-bundle case.
minor comments (5)
- [5.2, Theorem 5.2.2] The comparison with [CIL24, Conjecture 15] should explicitly state the normalization issue: the present logTev omits the factor by which [CIL24] divide their Tevelev degree. In the example that factor is 4! = 24, so the contradiction survives after dividing both the computed value and the predicted value, but this should be said explicitly.
- [2.1, Equation (2)] The two C^*-actions defining X_{r,s,a} both use the symbol lambda; using different letters for the two scaling parameters would avoid ambiguity.
- [4.2, after Equation (7)] Several displayed formulas use the character 'Z' instead of the integral sign, for example in the display preceding Equation (7). Readers would benefit from having these rewritten with proper integral signs. There is also a stray 'n' in the display below Equation (7) that should be removed.
- [5.2, after Proposition 5.2.1(vi)] The numerical example would be easier to follow if the paper explicitly wrote 24*5*5*4 = 2400 and 24*25*9 = 5400 before comparing the two values.
- [2.3, Definition 2.3.1] The displayed definition of Q_Gamma(X_{r,s,a}) is visually ambiguous: the base of the projective tower is not separated from the description of the vector bundle. A small diagram with P(E_1) and Q on separate lines would improve readability.
Circularity Check
No circularity: logTev is independently defined from the forgetful map, and the main formula is derived from an intersection computation on a moduli space constructed in this paper.
full rationale
The logarithmic Tevelev degree logTev^X_Gamma is defined in Definition 1.2.2 as deg(tau) for tau: M_{Gamma,n}(X) to M_{0,n} x X^n, a quantity independent of the later quasimap spaces. The main theorem (Theorem 4.2.1) is obtained in two independent stages: Proposition 3.0.1 (with Propositions 3.1.1, 3.2.1, 3.3.1) proves that, under the stated inequalities, the cycle V = cap V(p_i,x_i) is supported on M_Gamma(X) and is reduced of dimension 0, so that its degree equals logTev; then Section 4.2 computes the integral of product [V(p_i,x_i)] on Q_Gamma(X_{r,s,a}) by standard Chern and Segre class manipulations. No parameter is fitted to the invariant being computed, and no term in the displayed formula is inserted as an input; the polynomial factor (product m_j!)(product mu_{j,v}) a^{k_0-m_0} binom{k_0}{m_0} arises from extracting the coefficient of product H_{j,v} in a Segre-class expansion. The Cela and Iribar Lopez conjectures are used only as a comparison target, and the paper's self-citations to [CL23a, CL25a] are contextual, describing similar moduli spaces for blow-ups of P^r; the construction here is given directly and the cited results are not the source of the enumerativity or the formula. The proof of Theorem 5.2.2 contains an abbreviated classification of the excess locus, and the dimension-count hypothesis in the twisting argument that diagonal constraints are imposed on pairwise disjoint subsets of factors of B is asserted rather than fully expanded; these are potential correctness gaps, not circular reductions. No step of the derivation defines the output in terms of itself or uses a fitted or renamed copy of the output as an input, so the circularity score is 0.
Assumptions & free parameters
assumptions (6)
- domain assumption Moduli space of genus 0 log stable maps M_{Γ,n}(X) is irreducible of expected dimension (Ran17, Proposition 3.3.5).
- domain assumption Morphisms from P^1 to a toric variety are parameterized by tuples of sections of line bundles, with tangency data encoded by zero divisors of the sections (Cox 1995).
- standard math Standard degeneracy locus formula for maps of line bundles: when the expected codimension is reached, the class is the corresponding power of the difference of first Chern classes.
- standard math Cohomology and Base Change: degree 0 line bundles on P^1 fibers push forward to line bundles on B.
- standard math Excess intersection formula.
- standard math Generic smoothness and dimension counting for maps between smooth varieties (Hartshorne Corollary 10.7).
Cite this review
Pith. "Pith review of Enumerating log rational curves on some toric varieties." pith.science (2026). https://pith.science/paper/RZOBEWU2
@misc{pith2026250613975,
author = {Pith},
title = {Pith review of: Enumerating log rational curves on some toric varieties},
year = {2026},
howpublished = {\url{https://pith.science/paper/RZOBEWU2}},
note = {Machine review of arXiv:2506.13975}
}
read the original abstract
The genus 0, fixed-domain log Gromov-Witten invariants of a smooth, projective toric variety X enumerate maps from a general pointed rational curve to a smooth, projective toric variety passing through the maximal number of general points and with prescribed multiplicities along the toric boundary. We determine these invariants completely for the projective bundle X=P_{P^r}(O^s+O(-a)), proving a conjecture of Cela--Iribar L\'opez. A different conjecture when X is the blow-up of P^r at r points is disproven. Whereas the conjectures were predicted using tropical methods, we give direct intersection-theoretic calculations on moduli spaces of "naive log quasimaps."
Figures
Reference graph
Works this paper leans on
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[4]
[CL23a] A. Cela and C. Lian. Fixed-domain curve counts for blow-ups of projective space.arXiv 2303.03433,
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Enumerative Geometry of Quantum Periods
arXiv 2502.19408. [GS13] M. Gross and B. Siebert. Logarithmic Gromov-Witten invariants.J. Amer. Math. Soc., 26(2):451– 510,
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Higher genus Gromov-Witten invariants from projective bundles on smooth log Calabi-Yau pairs
arXiv 2503.17713. W ashington University in St. Louis, Department of Mathematics, 1 Brookings Drive St. Louis, MO 63130 Email address:clian@wustl.edu Tulane University, Department of Mathematics, 6823 St. Charles A ve New Orleans, LA 70118 Email address:nsakran@tulane.edu
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[2021]
[CD24] R. Cavalieri and E. Dawson. Tropical Tevelev degrees.arXiv 2407.20025,
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[2024]
[BP21] A. Buch and R. Pandharipande. Tevelev degrees in Gromov-Witten theory.arXiv 2112.14824,
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Reviewed August 15, 2026 · model on record in the stance chip above.
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