{"id":"d7e636ad-639d-4822-a4b2-f435107e15f3","arxiv_id":"1908.05844","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors establish second main theorems with improved constants for holomorphic curves in algebraic varieties intersecting moving hypersurfaces in N-subgeneral position with index kappa.","lead":"This paper proves refined second main theorems, a core tool of Nevanlinna theory, for holomorphic curves intersecting slowly moving hypersurfaces under a generalized position condition called subgeneral position with index. The new bounds generalize and improve earlier results by Ru, Dethloff-Tan, and Quang, giving sharper constants when the moving hypersurfaces enjoy extra transversality.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.3 is proved pointwise but is then used as if it supplied global coefficients; the missing uniform/global choice of P_j in K_Q is the load-bearing gap.","rationale":"The reader correctly identifies Lemma 3.3 and the uniformity of the construction as the weakest point of the proof. I agree that this is the most load-bearing step: all subsequent inequalities (8), (12), and (13) depend on having the polynomials P_j available with controlled coefficients, and Definition 3.4 requires them to live in K_Q[x] for the filtration to make sense. However, I would sharpen the reader's formulation. The real difficulty is not merely that the constant C is not proved to be uniform; pointwise coefficients can be normalized to the unit sphere, which would give a crude uniform bound of 1 for the coefficients if pointwise choices were all that were needed. The deeper problem is that Lemma 3.3 chooses coefficients separately at each point a, with no argument that they can be chosen as elements of K_Q (or as functions whose logarithmic growth is o(T_f)). The sentence 'the total number of such P'_j s is finite' is false for the stated pointwise construction, since a fixed N+1-tuple gives infinitely many linear combinations; what is finite is only the number of N+1-tuples. A possible repair is to choose, for each N+1-tuple, a generic constant coefficient vector once and for all, and to show that the bad set of points a where the common-zero condition fails is discrete or at least of measure zero. Such an argument would justify a single constant C and would make the P_j constants, hence trivially in K_Q. The theorem is likely correct and the gap may be repairable, but the manuscript as written does not supply the required argument, so the reader's conditional verdict is appropriate. I do not see a more serious flaw elsewhere: the Hilbert-polynomial filtration, the product-to-sum estimate, and the truncation computation in Theorem 1.5 follow the established Ru/Quang machinery once the global P_j are available.","tokens_in":21029,"tokens_out":20211,"duration_ms":214496,"concrete_test":"Test the missing global choice in the minimal nontrivial case V=P^2, l=2, kappa=1, N=3. Let Q_1=x_0, Q_2=x_1, Q_3=x_2, Q_4=x_0+phi(z)x_1+x_2, with phi slowly moving, and try to choose fixed constants a,b,c,d such that P_1=x_0, P_2=a x_1+b x_2, P_3=c x_0+d x_1+x_2 have empty common zero on P^2 for every z outside a discrete set. Compute the exceptional set E(a,b,c,d)={z: P_1,P_2,P_3 have a common zero on V} for phi(z)=z and phi(z)=e^z. If for every constant vector the exceptional set has nonempty interior, the 'finitely many choices, hence constant C' claim fails. If a generic constant vector makes E discrete, the gap is repairable and the conditional verdict should stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.4 hinges on Lemma 3.3 and on the paragraph immediately after it. Lemma 3.3 is proved for a fixed point a: for each a satisfying the three stated conditions, one chooses complex coefficients c_tj by avoiding a countable union of proper subspaces that depend on a. The paper then says that because there are only finitely many choices of N+1 polynomials among Q_1,...,Q_q, the total number of polynomials P'_j is finite, so a single constant C bounds the coefficients for all z. This inference is not justified: for a fixed N+1-tuple the coefficients c_tj vary continuously with a, and the admissible set is an open cone, so there are infinitely many possible P'_j, not finitely many. What the subsequent argument actually needs is a global construction: for each N+1-tuple, polynomials P_1,...,P_l with coefficients in K_Q (or at least in the field of small functions) whose common-zero condition holds on a Zariski open set. This is required because Definition 3.4 uses P_1(z),...,P_l(z) to define a filtration of the K_Q-vector space V_L; if the P_j are only pointwise choices with arbitrary complex coefficients, they need not lie in K_Q[x], and the quotient spaces W_i^*/W_{i'}^* and Lemma 3.9 are not well defined. Normalizing the pointwise coefficients gives a uniform bound |c_tj|≤1, but it does not give membership in K_Q, and the paper supplies no algebraic selection theorem. Consequently inequality (8), the filtration step (12), and the final constant in Theorem 1.4 are unsupported as written. The theorem may be repairable by a generic-choice argument that selects, for each N+1-tuple, one coefficient vector working outside a discrete exceptional set, but that argument is absent.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Nevanlinna theory for holomorphic curves from C^m into an irreducible algebraic variety V ⊂ P^n(C) intersecting slowly moving hypersurfaces. It introduces the index κ of subgeneral position into the moving-target setting. Theorem 1.4 states that for an algebraically nondegenerate f and q moving hypersurfaces Q_i in N-subgeneral position with index κ, the counting functions satisfy (q − (1 + (N−ℓ)/max{1,min{N−ℓ,κ}})(ℓ+1) − ε)T_f(r) ≤ Σ (1/d_i) N(r, f^*Q_i) + o(T_f(r)) outside a set of finite Lebesgue measure. Theorem 1.5 gives the analogous result for V = P^n with truncated counting functions and an explicit truncation level L0. The proofs combine the Dethloff–Tan moving-hypersurface technique, Quang's subgeneral-position handling, the Corvaja–Zannier filtration, and Ru's hyperplane second main theorem.","tokens_in":21385,"tokens_out":12842,"duration_ms":124027,"significance":"The statement is a natural and potentially useful unification: the coefficient (1 + (N−ℓ)/max{1,min{N−ℓ,κ}})(ℓ+1) interpolates between the general-position coefficient ℓ+1 and the subgeneral-position coefficient, and Theorem 1.5 recovers Quang's Theorem 1.2 when κ = 1. The use of index κ is motivated by Ji–Yan–Yu. However, both main theorems currently rest on an unproved pointwise-to-global step in Lemma 3.3. The paper also contains useful explicit estimates for the truncation level in Theorem 1.5. If the gap in Lemma 3.3 can be repaired, the results would be a solid contribution to the area.","major_comments":[{"comment":"The proof of Lemma 3.3 is not carried out for the point a fixed in the statement: it begins by using the definition of N-subgeneral position to assert the existence of a point a for which the Q_i(a) have only the trivial common zero, which proves the conclusion at best for that particular point and not for every a satisfying (i)–(iii). More importantly, the conclusion of the lemma is pointwise: the coefficients c_tj depend on a. The paragraph after the lemma claims that because there are only finitely many (N+1)-tuples among Q_1,...,Q_q, the number of P'_j is finite, and hence a uniform constant C exists. This inference is invalid: for a fixed tuple, the admissible coefficient vectors form an infinite open cone, and the proof gives no continuous or algebraic selection. The acknowledgement states that the original version had a serious gap; this is precisely the point that remains unproved. Consequently the uniform bound used in the display after Lemma 3.3 and in inequality (8) is not established.","section":"Section 3, Lemma 3.3 and the paragraph immediately after it"},{"comment":"The filtration must be a single filtration of the fixed K_Q-vector space V_L, with a fixed basis and a fixed finite set of linear forms L_s, in order to apply Theorem 2.5 in (13)–(14). Lemma 3.3 supplies, for each point a, polynomials P_t(a) ∈ C[x] ⊂ K_Q[x]; even if one regards these as elements of K_Q[x], the resulting ideals I^i_L and quotients W_i^*/W_{i'}^* depend on a. The proof does not show that the P_t(a) can be chosen so that the filtration, and hence the linear forms L_s, is independent of z. Thus the filtration step (12) and the declaration after (12) that the collection of all possible linear forms L_s is finite are not justified.","section":"Section 3, Definition 3.4 through Lemma 3.9 and the paragraph after (12)"},{"comment":"The proof introduces P_{I1},...,P_{I(n+1)} as 'the moving hypersurfaces obtained in Lemma 3.3' and then asserts that there exist functions h, g0, g ∈ C_f, independent of I and z, such that the displayed estimates hold. This requires both a global version of Lemma 3.3 with polynomials in K_Q[x] and a uniform bound over all I and all z. Neither is proved. The gap in Lemma 3.3 therefore propagates directly to Theorem 1.5; the estimates (16)–(20) and the subsequent truncation argument rest on this unproved uniformity.","section":"Section 4, opening paragraphs of the proof of Theorem 1.5"}],"minor_comments":[{"comment":"The sentence 'Take L large enough such that ε < (...) o(1)' is not meaningful as written, since ε is a fixed positive constant and o(1) depends on r and tends to 0. The intended asymptotic choice of L should be reformulated as a limit as L → ∞ followed by r → ∞ outside exceptional sets.","section":"Section 3, after (14)"},{"comment":"The condition 'have no non-trivial common zeros' appears to mean no common zeros in P^n, whereas N-subgeneral position in V only guarantees empty intersection with V(a). The proof only needs the latter, and the discrepancy should be clarified.","section":"Lemma 3.3, condition (ii)"},{"comment":"The paper contains many typographical and grammatical errors (e.g., 'the set of the set of the defining homogeneous polynomials', 'componet', 'respecitively', 'tow sides', 'Jensens fomular'). A careful proofreading is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is an honest attempt and the authors acknowledge an earlier gap, but the current version still contains the pointwise-versus-global problem in Lemma 3.3. I recommend major revision and ask that the authors provide a global version of Lemma 3.3 with polynomials in K_Q[x], or a detailed algebraic selection argument showing the coefficients can be chosen uniformly. Without that, the central claims of Theorems 1.4 and 1.5 are not proved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely useful: it adapts Ji–Yan–Yu's notion of subgeneral position with index to moving hypersurfaces and proves second main theorems (Theorems 1.4 and 1.5) that unify and improve the Dethloff–Tan and Quang results. For kappa=1 or N=l it recovers known theorems; for other kappa the constant is new and, on its face, better than what a naive Nochka-weight argument would give. The proof follows the established Dethloff–Tan–Quang–Yan–Yu machine, with the Corvaja–Zannier filtration and Ru's hyperplane SMT as black boxes. The authors also honestly acknowledge that v1 had a serious gap and thank Yan for pointing it out. That is a good sign.\n\nNow the soft spot, and it is not minor. Lemma 3.3 is proved pointwise: for each admissible a you choose coefficients c_tj in C that avoid a countable union of proper subspaces depending on a. The paragraph immediately after the lemma asserts that because there are only finitely many N+1-tuples of Q's, the total number of P's is finite, so a single constant C bounds the coefficients uniformly. That inference is wrong. For a fixed tuple, the coefficients vary continuously with a; the admissible set is an open cone, not a finite set. Worse, the subsequent filtration (Definition 3.4) requires the P_j to lie in K_Q[x], not just in C[x] with pointwise coefficients. The pointwise construction gives no membership in K_Q, and no algebraic selection theorem is supplied. Inequality (8), the filtration step (12), and the final constant in Theorem 1.4 all rest on this missing uniform/global choice. The stress-test concern lands.\n\nIs the theorem repairable? Probably yes, by a generic-choice argument: for each N+1-tuple, the bad coefficient vectors form a proper algebraic variety, and one can likely select a single coefficient vector in K_Q that works on a Zariski open set. But that argument is absent, and the current text does not even state the uniformity claim as a lemma. There are also smaller issues: the statement of Lemma 3.2 is garbled (it reads like a lead-in to the lemma rather than the lemma itself), and the proof of Theorem 1.5 has a few typo-level slips. These are secondary.\n\nFor a reader working in Nevanlinna theory or Diophantine approximation, this is worth a careful look: the index-kappa position is a natural concept and the constants are interesting. But I would not cite the theorem as proved until the uniformity gap is closed. A serious referee should flag Lemma 3.3 and the paragraph after it as the key point. This deserves peer review, not desk rejection.","headline":"A plausible extension of second main theorems to moving hypersurfaces with index, but the proof has a load-bearing uniformity gap around Lemma 3.3 that needs a real fix before the result can be trusted.","tokens_in":21950,"tokens_out":2211,"would_cite":false,"duration_ms":25196,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["30D35","32H30"],"pacs":[],"model":"deepseek-v4-flash","headline":"For holomorphic curves in algebraic varieties, moving hypersurfaces in N-subgeneral position with index kappa satisfy a second main theorem with defect coefficient (1+(N-l)/max{1,min{N-l,kappa}})(l+1).","keywords":["second main theorem","holomorphic curves","algebraic varieties","moving hypersurfaces","subgeneral position with index","value distribution theory","defect relation"],"falsifier":"In the case $V=\\mathbb{P}^2$, $N=3$, $\\kappa=1$, consider a one-parameter family of four moving lines in 3-subgeneral position with index 1, and for each parameter compute the smallest norm of the coefficient vector $(c_2,c_3,c_4)$ that Lemma 3.3 needs for $P_2,P_3$ to avoid the forbidden component. If that minimal norm is unbounded as the parameter approaches a degeneracy while the subgeneral-position assumptions still hold, the uniform constant $C$ used immediately after the lemma does not exist and the proof's inequality collapses; this calculation is directly checkable from the lemma's own construction.","tokens_in":20844,"feed_emoji":"📐","tokens_out":18393,"duration_ms":155981,"temperature":0.7,"pith_summary":"The paper aims to prove a second main theorem for nonconstant meromorphic maps $f:\\mathbb{C}^m\\to V\\subset\\mathbb{P}^n(\\mathbb{C})$ into an irreducible algebraic variety of dimension $\\ell$, when the target hypersurfaces are slowly moving and lie in $N$-subgeneral position with index $\\kappa$ on $V$. For an algebraically nondegenerate map and any $\\epsilon>0$, the theorem gives $$(q - (1 + \\frac{N-\\ell}{\\max\\{1,\\min\\{N-\\ell,\\kappa\\}\\}})(\\ell+1) - \\epsilon)T_f(r) \\le \\sum_{i=1}^q \\frac{1}{d_i} N(r,f^*Q_i) + o(T_f(r))$$ outside a set of finite Lebesgue measure. If true, this interpolates between the earlier general-position case ($N=\\ell$, $\\kappa=1$) and the earlier subgeneral-position case ($\\kappa=1$), showing that a larger index systematically shrinks the defect coefficient. A companion theorem for $V=\\mathbb{P}^n(\\mathbb{C})$ replaces the counting functions by explicitly truncated counting functions, which is the form relevant for uniqueness problems.","feed_headline":"Defect bound shrinks as the index of subgeneral position grows","feed_subtitle":"Moving hypersurface targets obey a second main theorem whose error term improves with the index.","key_machinery":"The load-bearing construction is Lemma 3.3: from any $N+1$ of the moving hypersurfaces in $N$-subgeneral position with index $\\kappa$ on $V$, one builds $\\ell+1$ homogeneous polynomials $P_1=Q_1,\\dots,P_\\kappa=Q_\\kappa$, $P_{\\kappa+1},\\dots,P_{\\ell+1}$, where each $P_t$ with $t\\ge\\kappa+1$ is a $\\mathbb{C}$-linear combination of $Q_{\\kappa+1},\\dots,Q_{N-\\ell+t}$, and the $\\ell+1$ polynomials have empty common zero set on $V$. The proof feeds this geometric construction into a lexicographic filtration of the space of homogeneous polynomials of degree $L$ modulo the ideal of $V$, together with a product-to-sum estimate for hyperplanes. The filtration's successive quotients have dimensions governed by the standard polynomial growth of projective linear systems, producing the factor $L^{\\ell+1}/(\\ell+1)!$ and the final coefficient $(1+(N-\\ell)/\\max\\{1,\\min\\{N-\\ell,\\kappa\\}\\})(\\ell+1)$.","core_discovery":"The central discovery is that the index $\\kappa$ of $N$-subgeneral position controls the defect coefficient of a second main theorem in a precise, explicit way. For slowly moving hypersurfaces $Q_1,\\dots,Q_q$ of degrees $d_1,\\dots,d_q$ in $N$-subgeneral position with index $\\kappa$ on an $\\ell$-dimensional variety $V$, and for $f$ algebraically nondegenerate over the field $K_Q$ generated by ratios of the coefficients of the $Q_i$, the paper proves the displayed inequality with coefficient $(1+(N-\\ell)/\\max\\{1,\\min\\{N-\\ell,\\kappa\\}\\})(\\ell+1)$. The counting function $N(r,f^*Q_i)$ counts the zeros of $Q_i(f)$ in the ball of radius $r$, and the inequality holds for all $r$ outside a set of finite Lebesgue measure. For $V=\\mathbb{P}^n(\\mathbb{C})$ the same coefficient appears with $N^{[L_0]}(r,f^*Q_i)$ in place of the full counting function, where $L_0$ is an explicit integer built from $n$, the degrees, $\\kappa$, $N$, and $\\epsilon$. This unifies and improves the two known endpoint theorems.","pith_inferences":["The exact form of the coefficient suggests a sharpness problem: for each $\\kappa$, one might try to construct hypersurface families attaining the predicted defect, which would show the index-dependence is not an artifact of the proof method.","The unproved uniform boundedness in the geometric lemma points to a concrete repair: a compactness argument over the projective space of admissible coefficient choices, away from the degeneracy locus of the moving hypersurfaces, would turn the pointwise construction into a uniform one.","The same filtration and product-to-sum machinery may extend to divisors with multiplicities or to maps of finite order; the authors' own open question asks whether algebraic nondegeneracy can be dropped, which would require a different argument."],"forward_implications":["If the theorem is correct, an algebraically nondegenerate curve can asymptotically avoid at most $(1+(N-\\ell)/\\max\\{1,\\min\\{N-\\ell,\\kappa\\}\\})(\\ell+1)$ of the slowly moving hypersurfaces; the total defect is bounded by that coefficient.","The endpoint cases are recovered: $N=\\ell$, $\\kappa=1$ gives the general-position coefficient $\\ell+1$, and $\\kappa=1$ gives the earlier subgeneral-position coefficient $(N-\\ell+1)(\\ell+1)$.","For $V=\\mathbb{P}^n(\\mathbb{C})$, the truncated counting functions at the explicit level $L_0$ make the theorem directly usable for uniqueness problems for meromorphic mappings.","Because the inequalities hold outside a set of finite Lebesgue measure, the defect relation is an asymptotic statement valid along almost every sphere of radius $r$."],"supporting_citations":[{"why":"Establishes the moving-hypersurface second main theorem on algebraic varieties in general position that Theorem 1.4 directly extends.","marker":"[2]"},{"why":"Sets up slowly moving hypersurfaces and gives the two-sided bound on target values used in Lemma 2.2.","marker":"[3]"},{"why":"Introduces the index of subgeneral position, the exact geometric hypothesis used in both main theorems.","marker":"[5]"},{"why":"Provides the subgeneral-position technique and the truncated theorem for P^n with kappa=1 that Theorem 1.5 generalizes.","marker":"[8]"},{"why":"Supplies the First Main Theorem for moving hypersurfaces used to convert proximity integrals into counting functions.","marker":"[9]"},{"why":"Supplies the lexicographic filtration and the multiplicity identity (11) that produce the L^(l+1)/(l+1)! factor.","marker":"[10]"},{"why":"Supplies the product-to-sum estimate for hyperplanes (Theorem 2.5) used to bound sums of proximity functions.","marker":"[11]"},{"why":"Establishes the foundational second main theorem for algebraic varieties intersecting hypersurfaces in general position that this paper improves.","marker":"[12]"},{"why":"Supplies the polynomial-growth fact for dimensions of spaces of degree-L polynomials modulo the ideal of V.","marker":"[15]"},{"why":"Provides recent estimates for moving hypersurfaces in projective space and the refined filtration arguments used in Section 4.","marker":"[17]"}],"fun_headline_variants":["Subgeneral position index tightens second main theorem bound","Defect coefficient shrinks as subgeneral position index grows","Index of subgeneral position improves defect bound for moving targets","Explicit defect bound from moving hypersurface index","Second main theorem: index sharpens error term"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that the coefficient vectors that build the auxiliary polynomials in Lemma 3.3 can be chosen with one uniform bound over all base points and all choices of $N+1$ targets; the lemma proves pointwise existence but not that uniform bound.","fun_headline_variants_meta":{"raw":{"variants":["Subgeneral position index tightens second main theorem bound","Defect coefficient shrinks as subgeneral position index grows","Index of subgeneral position improves defect bound for moving targets","Explicit defect bound from moving hypersurface index","Second main theorem: index sharpens error term"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000626,"raw_usage":{"total_tokens":2852,"prompt_tokens":859,"completion_tokens":1993,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":475,"completion_tokens_details":{"reasoning_tokens":1917}},"tokens_in":475,"tokens_out":1993,"duration_ms":13453,"temperature":1.0,"reasoning_tokens":1917,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:03:44.626866+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the case $V=\\mathbb{P}^2$, $N=3$, $\\kappa=1$, consider a one-parameter family of four moving lines in 3-subgeneral position with index 1, and for each parameter compute the smallest norm of the coefficient vector $(c_2,c_3,c_4)$ that Lemma 3.3 needs for $P_2,P_3$ to avoid the forbidden component. If that minimal norm is unbounded as the parameter approaches a degeneracy while the subgeneral-position assumptions still hold, the uniform constant $C$ used immediately after the lemma does not exist and the proof's inequality collapses; this calculation is directly checkable from the lemma's own construction.","supporting_citations":[{"cited_title":"Holomorphic curves into algebraic varieties intersecting moving hypersurface targets","cited_arxiv_id":"1503.08801","evidence_quote":"Establishes the moving-hypersurface second main theorem on algebraic varieties in general position that Theorem 1.4 directly extends."},{"cited_title":"Dethloff, T","cited_arxiv_id":null,"evidence_quote":"Sets up slowly moving hypersurfaces and gives the two-sided bound on target values used in Lemma 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the index of subgeneral position, the exact geometric hypothesis used in both main theorems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the subgeneral-position technique and the truncated theorem for P^n with kappa=1 that Theorem 1.5 generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the First Main Theorem for moving hypersurfaces used to convert proximity integrals into counting functions."},{"cited_title":"Ru, A defect relation for holomorphic curves intersecting hype rsurfaces, Amer","cited_arxiv_id":null,"evidence_quote":"Supplies the lexicographic filtration and the multiplicity identity (11) that produce the L^(l+1)/(l+1)! factor."},{"cited_title":"Ru, On the general form of the second main theorem , Trans","cited_arxiv_id":null,"evidence_quote":"Supplies the product-to-sum estimate for hyperplanes (Theorem 2.5) used to bound sums of proximity functions."},{"cited_title":"Ru, Holomorphic curves into algebraic varieties , Ann","cited_arxiv_id":null,"evidence_quote":"Establishes the foundational second main theorem for algebraic varieties intersecting hypersurfaces in general position that this paper improves."},{"cited_title":"Sombra, Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz , Algorithms for algebra (Eindhoven,1996)","cited_arxiv_id":null,"evidence_quote":"Supplies the polynomial-growth fact for dimensions of spaces of degree-L polynomials modulo the ideal of V."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides recent estimates for moving hypersurfaces in projective space and the refined filtration arguments used in Section 4."}],"review_version":1}