{"id":"8c1c8c8c-8fca-4eb5-a25a-19f4fad4cd71","arxiv_id":"2506.13975","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"All genus 0 logarithmic Tevelev degrees of X=P_{P^r}(O^s⊕O(-a)) are determined by a closed formula, and the separate conjecture for blow-ups of P^2 at two points is disproven.","lead":"The paper computes logarithmic Tevelev degrees, which count rational curves with prescribed boundary tangency passing through fixed general points, for projective bundles over projective space, proving one conjecture and disproving another. The computation is done by intersection theory on new moduli spaces of naive log quasimaps rather than by tropical enumeration.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.2.2's excess locus classification rests on an unproved 'straightforward analysis'; a missed support configuration would change the counterexample.","rationale":"The paper's main projective bundle formula (Theorem 4.2.1) is supported by a lengthy but essentially complete argument: the twisting algorithm in Proposition 3.2.1 has a dimension count whose asserted disjointness of diagonal constraints is plausible because distinct base points q cannot force the same point factor q_{j,v} to equal two different values, so independence holds. The integral computation in Section 4.2 is straightforward once the class [V(p,x)] = ζ_1^r ζ_2^s is accepted. The residue of uncertainty is in the disproof of Conjecture 15. The proof of Theorem 5.2.2 splits V into the enumerative part (logTev) and an excess locus Z, then computes logTev = 2400 by subtracting the excess contribution from the integral 5400. The entire subtraction depends on the asserted classification of Z as 24·5 components each isomorphic to P^1×P^1. The paper's 'straightforward analysis' is an omitted proof, and the subtlety of the twisted-jet incidence conditions makes it easy to miss configurations, as a naive check with q_{1,1}=p_i, q_{2,1} free initially appears to work but fails on the twisted section. A complete derivation of the classification is therefore the single check that would settle the counterexample. If it goes through, the reader's CONDITIONAL verdict can stand; if it reveals missing components, the disproof of Conjecture 15 collapses. I do not find an independent error in the main theorem's enumerativity argument, so the verdict remains CONDITIONAL rather than REJECT.","tokens_in":25433,"tokens_out":39242,"duration_ms":349111,"concrete_test":"Derive the classification in full: for a point of V with g_4=g_5=0, write div(g_3)=D_1+D_2+D_3 (degree 6) and impose, for each i, ord_{p_i}(g_3)>ord_{p_i}(D_1) and ord_{p_i}(g_3)>ord_{p_i}(D_2). Show the resulting system of five pairs of inequalities forces D_1=D_2=p_{i0} for a single i0 and D_3=sum of the other four p_i. Explicitly test the potentially missing case q_{1,1}=p_i, q_{2,1} not among {p_j}, D_3=sum of the other four p_i, and verify it fails because the twisted section O(6)(−D_1) does not vanish at p_i when ord_{p_i}(D_1)=1 and ord_{p_i}(g_3)=1. Recompute the component count and the excess contribution afterward.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The counterexample in Theorem 5.2.2 depends on the completeness of the classification of the excess locus Z ⊂ V ∩ (Q_Γ(X) − Q^{≠0}_Γ(X)). The proof asserts: 'A straightforward analysis as in the proof of Proposition 3.3.1 shows that ... we must have g_4 = g_5 = 0, and the g_3 must vanish at all of p_1,...,p_5 when viewed either as a section of O(6)(−D_1) or O(6)(−D_2). Concretely, this means D_3 + q_{1,1} = D_3 + q_{2,1} = p_1+...+p_5.' This step is not derived. The subtlety is that the incidence conditions are imposed on twisted line bundles: 'vanishes at p_i as a section of O(6)(−D_1)' means the order of vanishing of g_3 at p_i exceeds the multiplicity of p_i in D_1, not merely that g_3(p_i)=0. A complete proof must show that the five pairs of inequalities (b_i+c_i ≥ 1 and a_i+c_i ≥ 1 for i=1,...,5, where a_i,b_i,c_i are the multiplicities of p_i in D_1,D_2,D_3) force exactly one index i0 with q_{1,1}=q_{2,1}=p_{i0} and D_3 = sum of the remaining p_i. If an alternative configuration survives (e.g., q_{1,1}=p_5, q_{2,1} free, D_3=p_1+...+p_4), the component count 24·5 and the excess contribution 3000 would change, altering the claimed logTev = 2400 and invalidating the disproof of [CIL24, Conjecture 15].","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":1711,"tokens_out":2064,"duration_ms":257268,"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":[{"comment":"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.","section":"5.2, Theorem 5.2.2 and Lemma 5.2.3"},{"comment":"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.","section":"3.2, Proposition 3.2.1"},{"comment":"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.","section":"5.1-5.2, Proposition 5.2.1"}],"minor_comments":[{"comment":"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.","section":"5.2, Theorem 5.2.2"},{"comment":"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.","section":"2.1, Equation (2)"},{"comment":"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.","section":"4.2, after Equation (7)"},{"comment":"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.","section":"5.2, after Proposition 5.2.1(vi)"},{"comment":"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.","section":"2.3, Definition 2.3.1"}],"recommendation":"major_revision","confidential_remarks":"This is a well-motivated paper in the scope of math.AG. The main projective-bundle computation is convincing and likely correct, and the counterexample to Conjecture 15 is valuable. The obstacles to acceptance are local but load-bearing: the excess-locus classification and Proposition 5.2.1 need complete proofs, and the independence claim in Proposition 3.2.1 needs justification. I see no reason for rejection; the issues are fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the projective bundle case is done: Theorem 4.2.1 gives a complete closed formula for log Tevelev degrees of X_{r,s,a}, and the s=1 case proves Conjecture 14 of Cela–Iribar López. The computation is honest intersection theory on a tower of projective bundles, with no tropical input and no fitted constants. Second, the blow-up counterexample is real. I went through the 'straightforward analysis' in Theorem 5.2.2 that the stress-test flagged, and it works: the five incidence inequalities force D1 and D2 to coincide with the same p_i, and D3 to be the remaining four points. The alternative configuration the stress-test worries about dies because it would require all five indices to lie in the support of D3 when |C|=4. So the counterexample stands.\n\nWhat the paper does well: the naive quasimap space Q_Γ is simple enough to compute on, and the authors are careful about exactly when the tautological intersection number is enumerative. The vanishing Proposition 4.1.1 is a clean dimension argument, and the specializations to Hirzebruch surfaces and blow-ups of P^{r+s} check out against known results.\n\nSoft spots, in proportion. The proof of Theorem 5.2.2 is genuinely compressed at the point the stress-test identifies: the excess locus classification is asserted in one sentence, and Lemma 5.2.3's tangent-vector argument is a bit ad hoc. A referee should ask for the classification to be written out; it is a few lines of case analysis, and I believe it is correct. Proposition 3.2.1's disjointness claim for diagonal constraints is stated without proof, but it is fine: each factor q_{j,v} can be constrained to equal at most one point, so the supports are disjoint. Neither issue is load-bearing.\n\nThis paper is for enumerative geometers and anyone tracking Tévelev degrees or log Gromov-Witten invariants of toric varieties. It deserves a serious referee; with the excess locus step expanded, I would accept it as is.","headline":"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.","tokens_in":26339,"tokens_out":6420,"would_cite":true,"duration_ms":55780,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N35","14M25","14N10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $\\mathbb{P}^2$.","keywords":["logarithmic Gromov-Witten invariants","Tevelev degrees","toric varieties","projective bundles","quasimaps","tropical curve counting","enumerative geometry","log curves"],"falsifier":"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.","tokens_in":25183,"feed_emoji":"📐","tokens_out":8818,"duration_ms":78282,"temperature":0.7,"pith_summary":"The paper determines the genus-zero, fixed-domain logarithmic Gromov-Witten invariants of the toric projective bundles $X_{r,s,a}=\\mathbb{P}_{\\mathbb{P}^r}(\\mathcal{O}^s\\oplus\\mathcal{O}(-a))$: these invariants count maps from a general pointed rational curve to $X_{r,s,a}$ passing through the maximal number of general interior points while meeting each toric boundary divisor with prescribed multiplicities. For every such bundle it proves a closed formula, and it shows the count vanishes outside a precisely described range of the tangency data. The formula confirms a conjecture that had been made from tropical curve-counting, and a second result produces a counterexample to a different tropical conjecture on the blow-up of $\\mathbb{P}^2$ at two points. The paper's route is direct intersection theory on moduli spaces of 'naive log quasimaps,' bypassing tropical combinatorics.","feed_headline":"Closed formula counts log curves on toric projective bundles","feed_subtitle":"Direct quasimap intersection theory proves a conjectured count and finds a counterexample on blow-ups of P^2.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"States the two conjectures the paper proves and disproves, and supplies the tropical correspondence theorem that the conjectures rest on.","marker":"[CIL24]"},{"why":"Provides the descendant log Gromov-Witten tropical correspondence from which fixed-domain log counts can be recovered.","marker":"[MR20]"},{"why":"Introduces stable toric quasimaps, the model for the paper's naive log quasimap spaces.","marker":"[CFK10]"},{"why":"Constructs similar naive quasimap moduli spaces for blow-ups of projective space, adapted in Section 5.","marker":"[CL23a]"},{"why":"Defines stable logarithmic maps, the theory within which the invariants are formulated.","marker":"[AC14a]"},{"why":"Establishes irreducibility and expected dimension for genus-0 log stable maps, justifying the degree-of-forgetful-map definition of logTev.","marker":"[Ran17]"}],"fun_headline_variants":["Closed formula for log curve counts on toric bundles","Tropical log-count conjecture disproved on blow-ups","Excess intersection fixes naive quasimap counts","Cela-Iribar López conjecture: proven and overturned"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Closed formula for log curve counts on toric bundles","Tropical log-count conjecture disproved on blow-ups","Excess intersection fixes naive quasimap counts","Cela-Iribar López conjecture: proven and overturned"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000585,"raw_usage":{"total_tokens":2775,"prompt_tokens":998,"completion_tokens":1777,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":1714}},"tokens_in":614,"tokens_out":1777,"duration_ms":14563,"temperature":1.0,"reasoning_tokens":1714,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:56:39.271589+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":2}