Second main theorem for holomorphic curves into algebraic varieties intersecting moving hypersurfaces targets

Libing Xie; Tingbin Cao

REVIEW 3 major objections 3 minor 17 references

Second main theorem for holomorphic curves into algebraic varieties intersecting moving hypersurfaces targets

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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).

desk verdict 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. read the letter →

arxiv 1908.05844 v2 pith:36LDLFHG submitted 2019-08-16 math.CV

Libing Xie , Tingbin Cao This is my paper · ORCID

classification math.CV MSC 30D3532H30

keywords secondmaintheoremholomorphiccurvesalgebraicvarietiesmovinghypersurfacessubgeneralpositionwithindexvaluedistributiontheorydefectrelation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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 $(1+(N-\ell)/\max\{1,\min\{N-\ell,\kappa\}\})(\ell+1)$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $r$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

Now 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.

Is 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.

For 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.

Referee Report

3 major / 3 minor

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.

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 (3)

[Section 3, Lemma 3.3 and the paragraph immediately after it] 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 3, Definition 3.4 through Lemma 3.9 and the paragraph after (12)] 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 4, opening paragraphs of the proof of Theorem 1.5] 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.

minor comments (3)

[Section 3, after (14)] 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.
[Lemma 3.3, condition (ii)] 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.
[Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the index-kappa coefficient is derived from the geometric hypothesis via external lemmas, not assumed or fitted.

full rationale

I walked the derivation chain for Theorems 1.4 and 1.5. The new parameter kappa enters through Definition 1.3 as a geometric hypothesis (N-subgeneral position with index kappa), and it reaches the coefficient (1 + (N-l)/max{1,min{N-l,kappa}}) through the counting inequalities (6), (7), and (8), not by being inserted into the conclusion. Lemma 3.3 is used to construct auxiliary polynomials P_j from the Q_j; the subsequent sentence claiming that only finitely many choices of N+1 polynomials imply finitely many P'_j is not justified, because the coefficients in Lemma 3.3 are chosen pointwise and can vary with the point a. That is a real correctness gap in the uniformity/globalness of the construction, but it is not circular: the construction does not presuppose the desired second main theorem, and the final estimate is not a renamed form of the input. All heavy auxiliary ingredients (Lemma 2.2, Proposition 2.4, Theorem 2.5, the Corvaja-Zannier filtration lemmas, and the dimension estimates) are cited to external works [2, 4, 8, 10, 11, 17], not to the authors' own prior results, and none of those cited statements is identical to Theorem 1.4 or 1.5. The paper contains no fitted parameters called predictions, no uniqueness theorem imported from the authors, and no ansatz smuggled in by self-citation. The Acknowledgement explicitly says that Qiming Yan pointed out 'a hard to find but serious gap in the original version (arXiv:1908.05844v1)', which is an admitted defect statement about the earlier version; it does not establish circularity of the present derivation. Overall, the paper's central claim has content independent of its assumptions, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No data-dependent constants appear. The proof is a chain of inequalities with auxiliary parameters L and p0 chosen by size estimates, not fitted. The external results listed are load-bearing: Lemma 2.2 and the filtration dimension counts carry most of the weight. No new entities are introduced.

assumptions (5)

domain assumption Lemma 2.2: for moving hypersurfaces in N-subgeneral position, there exist h_1,h_2 in C_f such that h_2 ||f||^d <= max_{k=1..N+1} |Q_{j_k}(f)| <= h_1 ||f||^d.
Imported from Dethloff-Tan, Quang, and Yan-Yu; it is the main handle translating the geometric position condition into estimates involving the characteristic function.
standard math Theorem 2.5: Ru's second main theorem for hyperplanes, bounding the proximity sum over linearly independent hyperplane subsets by (n+1)T_f - N_W + o(T_f).
Used as a black box in both proofs to bound the proximity sum of the auxiliary map F.
standard math Lemma 3.1: Hilbert polynomial asymptotics, dim_{K_Q} V_L = Delta L^l/l! + rho(L) with rho(L)=O(L^{l-1}).
Provides the asymptotic dimension count needed to convert the filtration estimates into the final constant (l+1).
domain assumption Lemma 3.2: for all a outside a discrete set, dim V(a) = l, so moving hypersurfaces specialize to hypersurfaces in a variety of the same dimension.
Quoted from [3,8,17]; needed to apply Lemma 3.3 pointwise at the points where ordered estimates are evaluated.
standard math Lemma 4.1 and filtration lemmas 3.9-3.10 from Corvaja-Zannier and Ru: the quotient dimension by n general homogeneous polynomials of degree d equals d^n for large L, and the filtration dimension counts hold.
The core algebraic filtration machinery; assumed from the cited literature rather than re-proved.

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Pith. "Pith review of Second main theorem for holomorphic curves into algebraic varieties intersecting moving hypersurfaces targets." pith.science (2026). https://pith.science/paper/36LDLFHG

@misc{pith2026190805844,
  author       = {Pith},
  title        = {Pith review of: Second main theorem for holomorphic curves into algebraic varieties intersecting moving hypersurfaces targets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/36LDLFHG}},
  note         = {Machine review of arXiv:1908.05844}
}

read the original abstract

Since the great work on holomorphic curves into algebraic varieties intersecting hypersurfaces in general position established by Ru in 2009, recently there has been some developments on the second main theorem into algebraic varieties intersecting moving hypersurfaces targets. The main purpose of this paper is to give some interesting improvements of Ru's second main theorem for moving hypersurfaces targets located in subgeneral position with index.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Corvaja, U

P. Corvaja, U. Zannier, On a general Thues equation , Amer. J. Math. 126(2004), no.5, 1033- 1055

work page 2004

[2] [2]

Holomorphic curves into algebraic varieties intersecting moving hypersurface targets

G. Dethloﬀ, T. V. Tan, Holomorphic curves into algebraic varieties intersecting moving hypersurfaces targets, arXiv:1503.08801v3 [math.CV] 7 Nov 2018

work page Pith review arXiv 2018

[3] [3]

Dethloﬀ, T

G. Dethloﬀ, T. V. Tan, A second main theorem for moving hypersurface targets , Houston J. Math. 37(2011), 79-111

work page 2011

[4] [4]

Fujimoto, Non-integrated defect relation for meromorphic maps of com plete K ¨ahler man- ifolds into PN1 (C) ×

H. Fujimoto, Non-integrated defect relation for meromorphic maps of com plete K ¨ahler man- ifolds into PN1 (C) × . . .× PNk (C). Japanese journal of mathematics. New series, 1985, 11(2) : 233-264

work page 1985

[5] [5]

Q. Ji, Q. Yan, G. Yu, Holomorphic curves into algebraic varieties intersecting divisors in subgeneral position, Math. Ann. 373(2019), 1457-1483

work page 2019

[6] [6]

Levin, Generalizations of Siegels and Picards theorems , Ann

A. Levin, Generalizations of Siegels and Picards theorems , Ann. Math. 170(2009), no.2, 609- 655

work page 2009

[7] [7]

Noguchi and J

J. Noguchi and J. Winkelmann, Nevanlinna Theory in Several Complex Variables and Dio- phantine Approximation , Springer, Japan, 2014

work page 2014

[8] [8]

S. D. Quang, Second main theorem for meromorphic mappings with moving hy persurfaces in subgeneral position, J. Math. Anal. Appl. 465(2018), no.1, 604-623

work page 2018