{"id":"015fa7ba-1f85-451e-aa72-845ae11bb393","arxiv_id":"2607.27738","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a degree-q self-map of P^n, the number of totally invariant codimension-r subvarieties is at most (n+1 choose r), and the bound is optimal.","lead":"This paper proves that an algebraic self-map of projective n-space can have at most (n+1 choose r) totally invariant codimension-r subvarieties, with the bound achieved by the coordinate power map. This settles an open question in dynamics for the projective plane and bounds such invariant sets in every dimension.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorems rely on Lemma 2.2's unproved characteristic-free extension of Meng's eigenvalue criterion; if that extension fails, Theorem 1.1 and the P^n bound have no positive-characteristic proof.","rationale":"Good-faith reading: the central proof is a trace-splitting lemma plus uniform Fujita vanishing plus a Hilbert-function finite difference. Conditional on Lemma 2.2, I found Theorem 1.1 and Theorem 1.3 internally coherent: the nonzerodivisor choices, the induction for vanishing, the Koszul computation of B_{X,L,a,r}, and the length/degree identification all check out. Sharpness via the coordinate power map is correct, and the toric and completely-ramified applications follow from the same machinery. The single load-bearing point is Lemma 2.2: the eigenvalue criterion of [26] is quoted with the admission that it is stated in characteristic zero. The paper asserts the proof is purely cone-theoretic but gives no derivation. Since the theorem is stated over arbitrary algebraically closed fields with deg(f) invertible, this is not cosmetic. A finite-dimensional cone argument would likely justify it, but it is not written, so the main result should remain conditional on that verification. The reader's secondary concerns: the q^{ar} exponent is actually correct for the flat pullback of a codimension-r cycle, so I would not count that; Corollary 5.6's projection formula seems to be a c/f^* typo rather than a structural flaw. Thus the reader's CONDITIONAL verdict stands, unchanged.","tokens_in":16079,"tokens_out":28750,"duration_ms":342611,"concrete_test":"Independently re-derive Lemma 2.2 without using [26, Theorem 1.1]: let V = N^1(X)_R, C = ample cone, T = f^*|_V. Prove the finite-dimensional cone statement: if T(C) ⊂ C and T[M] − [M] ∈ C for some M ∈ C, then every complex eigenvalue of T has modulus > 1 (equivalently, S = T^{-1} contracts the nef cone to 0). This proof should use only convexity, compactness of a slice of the nef cone, and the invertibility of T, with no resolution or characteristic-zero input. If the proof succeeds, Lemma 2.2 is valid over every algebraically closed field and the main theorem stands; if it fails, exhibit an int-amplified endomorphism over an algebraically closed field of characteristic p, with deg(f) not divisible by p, whose f^* action has an eigenvalue of modulus ≤ 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof chain is: Theorem 1.3 ← Theorem 1.1 ← Proposition 2.3 ← Lemma 2.2. Lemma 2.2 invokes [26, Theorem 1.1 and Remark 1.2] to conclude that all eigenvalues of T = f^*|N^1(X)_R have modulus > 1, and the paper explicitly concedes that this criterion is stated in characteristic zero, adding only that the proof is 'purely cone-theoretic' and works over any algebraically closed field. No derivation is supplied. This is load-bearing because Theorem 1.1 is stated over arbitrary algebraically closed k and the trace splitting in Lemma 2.1 already requires deg(f) invertible, so positive characteristic is an intended regime. If the cited criterion secretly uses characteristic-zero tools (resolution of singularities, Hodge index, or a Mori cone theorem with char assumptions), then Lemma 2.2 is unsupported, Proposition 2.3's vanishing collapses, and the degree/cardinality bounds do not follow in char p. I do not see a counterexample, and a finite-dimensional cone argument may well prove the lemma, but the paper leaves it as an assertion. For calibration: I do not regard the q^{ar} exponent in Proposition 1.5(4) as an error—it is the correct factor for flat pullback of a codimension-r cycle—and the suspect projection formula in Corollary 5.6 looks like a c/f^* typo rather than a structural flaw.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves explicit upper bounds on the total degree and number of totally-periodic prime cycles of codimension r for int-amplified endomorphisms of smooth projective varieties, under the assumption that deg(f) is invertible in the ground field. The main mechanism is a normalized trace splitting for ideal sheaves of totally periodic subschemes (Lemma 2.1), followed by uniform Fujita-type vanishing (Proposition 2.3) and a finite-difference/Hilbert-function computation. In projective space the bound becomes binom(n+1,r), with sharpness attained by the coordinate power map; this answers Favre's question for P^2 by giving |T_∞(f)|≤3. The paper also proves toric analogues, studies the totally-periodic polynomial, and derives applications to completely ramified points, log canonical centers, and contractible extremal rays.","tokens_in":16376,"tokens_out":24042,"duration_ms":285379,"significance":"If the proof is completed at the points noted below, the paper is a significant contribution. Theorem 1.1 gives a closed-form, parameter-free bound that is uniform over all int-amplified endomorphisms once L and a are fixed, and the projective-space specialization is sharp with an explicit extremal example. The method is transparent, uses standard intersection-theoretic and vanishing-theoretic tools, and reproduces known codimension-one results as special cases. The paper also gives falsifiable numerical predictions and explicit constructions, including the coordinate power map and the F_1 example in Remark 4.2. The positive-characteristic scope is an intended feature, so the missing justification for the cone-theoretic criterion is a genuine gap rather than a stylistic issue.","major_comments":[{"comment":"This lemma is the only bridge between the int-amplified hypothesis and the numerical eigenvalue statement used in Proposition 2.3. The cited criterion [26, Theorem 1.1 and Remark 1.2] is stated in characteristic zero, and the manuscript's one-sentence assertion that the proof is 'purely cone-theoretic' and works over any algebraically closed field is not a proof. Since Theorem 1.1 is stated over an arbitrary algebraically closed field and only deg(f) invertible is imposed, positive characteristic is explicitly in scope. If this characteristic-free extension cannot be justified, Proposition 2.3, and hence Theorem 1.1 and the P^n bounds, do not follow in positive characteristic. Please provide a self-contained proof or a citation to a characteristic-free statement.","section":"Section 2, Lemma 2.2"},{"comment":"As printed, the proof uses the equality (f^a)_*V = q^{ar}V and defines the Chow morphism τ_{r,e,a} with target Chow_{n-r,q^{ar}e}. For an integral V of codimension r (dimension m=n-r) that is totally invariant, flat pushforward satisfies (f^a)_*[V] = q^{a m}[V] = q^{a(n-r)}[V], not q^{ar}[V]; for points (r=n) one gets 1·[V], not q^{an}[V]. The factor q^{ar} is the correct factor for flat pullback (f^a)^*[V]. Thus the displayed equality and the subsequent properness argument for C_{r,e,a} are invalid as written. If the intended operation is pullback, replace the pushforward stars with pullback stars throughout and check the parametrized family; if pushforward is intended, the degree factor and the target Chow degree must be changed. This does not affect Theorems 1.1–1.4, but it is load-bearing for the generic left-equality statement in Proposition 1.5(4).","section":"Section 4, proof of Proposition 1.5(4)"}],"minor_comments":[{"comment":"The expression Φ_{X,f}(t)-t^{n+1} includes the constant term 1. This is correct for the product inequality in (3), but it is easy to misread; a parenthetical reminder that the constant term is retained would help.","section":"Section 4, Proposition 1.5(3)"},{"comment":"The sentence 'The divisor h*H−H is a sum of pullbacks of the ample divisor f*H−H' is terse. Since h=f^b, the telescoping sum is immediate, but spelling it out would improve readability.","section":"Section 3, Proposition 2.3 proof"},{"comment":"For finite fields, 'q is invertible in k' means q is not divisible by p. Stating this explicitly would remove ambiguity in the example k=F_p.","section":"Remark 3.2"},{"comment":"The identity 'projection formula gives c deg(f|_Z)=deg(f)' is stated without derivation. A one-line computation of the cycle classes involved would help the reader verify the contradiction.","section":"Section 5.3, Corollary 5.6"},{"comment":"There are minor typographical issues, including inconsistent spacing in formulas such as T^r_∞(X,f) and Chow_{n-r,e}(P^n). These should be cleaned up during revision.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is likely correct, and the positive-characteristic gap in Lemma 2.2 appears fillable rather than fatal. The Section 4 pushforward/pullback mix-up is also local and fixable, but it currently invalidates the proof of Proposition 1.5(4). I recommend requesting a full proof of Lemma 2.2 (or a precise characteristic-free citation) and a careful correction of the cycle-formula notation before publication. The paper fits the journal's scope and the novelty is adequate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper deserves a serious referee. The headline result is the sharp bound |T∞(f)| ≤ 3 for endomorphisms of P^2, answering Favre's question, and more generally the binomial bound ≤ binom(n+1,r) for totally invariant codimension-r cycles in P^n. Theorem 1.1, a Hilbert-function bound for arbitrary smooth projective X, is new and cleanly proved via trace splitting, uniform Fujita vanishing, and a Koszul restriction. The toric refinement and the surjectivity result for evaluation maps are nice bonuses. The paper is honest: it cites prior work, and the bounds are derived, not fitted.\n\nThe main thing to fix is Lemma 2.2. The theorem is stated over any algebraically closed field, and the proof leans on Meng's cone-theoretic eigenvalue criterion, which is stated in characteristic zero. The paper says the proof is purely cone-theoretic and works in general, but gives no argument. That is load-bearing: if it fails, Theorem 1.1 and the P^n bound have no positive-characteristic proof. I suspect the extension is true—the criterion is about N^1 and eigenvalues, and a finite-dimensional cone argument should give it—but the authors need to spell that out or restrict to characteristic zero.\n\nThe reader flagged Proposition 1.5(4)'s exponent q^{ar} as an error. On reading, that is correct: for a codimension-r cycle, f^a pulls back to degree q^{ar}. So that concern disappears. There is a garbled projection formula in Corollary 5.6 that looks like a typo and should be rewritten. Proposition 1.5(4)'s equality characterization is plausible and the proof is dense but coherent.\n\nOne minor frustration: the paper packs in a lot of applications (regularity, log canonical thresholds, Fano blow-ups, extremal rays). Some of these read as sketches, and if the core is accepted, they should be checked separately. But they do not undermine the main result.\n\nFor a reading group: yes, the main theorem and its proof are worth studying. I would cite it if I worked in arithmetic dynamics or endomorphisms. Send it to peer review; the referee should insist on a proof or a characteristic-zero caveat for Lemma 2.2.","headline":"Resolves Favre's question with a sharp P^2 bound and proves a clean binomial bound in all codimensions for P^n; the main proof is solid, with one load-bearing unproved characteristic-free lemma to fix.","tokens_in":16906,"tokens_out":5827,"would_cite":true,"duration_ms":64144,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E05","14C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any self-map of projective n-space of degree q>1, the number of totally-invariant codimension-r cycles is at most the binomial coefficient n+1 choose r, and this bound is sharp.","keywords":["int-amplified endomorphism","totally invariant subvariety","projective space","degree bounds","Hilbert function","trace splitting","totally-periodic polynomial","completely ramified points"],"falsifier":"An endomorphism of P^n of degree q>1 invertible in k with total degree of its totally-invariant codimension-r cycles exceeding binom(n+1,r), or a smooth projective X with an int-amplified endomorphism of degree invertible in k where the Hilbert-function inequality in Theorem 1.1 fails, would refute the central claim. A simpler place to look: verify Lemma 2.2 in characteristic p>0 with p ∤ deg(f); if the ampleness of (f^s)^*L ⊗ L^{-1} fails for some large s, Proposition 2.3 collapses.","tokens_in":15931,"feed_emoji":"🔢","tokens_out":6530,"duration_ms":63521,"temperature":0.7,"pith_summary":"This paper establishes explicit, sharp upper bounds on how many subvarieties of a projective variety can be swept into themselves by an int-amplified endomorphism, i.e. a self-map whose pullback amplifies a line bundle. For projective n-space, it proves that the sum of degrees of all totally-invariant prime cycles of codimension r is at most binom(n+1,r), and that the number of such cycles is at most the same number; both are attained by the coordinate power map. The argument works in every codimension, for cycles that become invariant only after iteration, and over any algebraically closed field provided the map's degree is invertible. A corollary settles a question of Favre by showing that any endomorphism of the projective plane of degree at least two has at most three totally-invariant points.","feed_headline":"At most n+1 choose r totally-invariant cycles in projective n-space","feed_subtitle":"Total degree of codimension-r cycles a P^n self-map sweeps into itself never exceeds n+1 choose r — and the bound is attained.","key_machinery":"The paper's working object is the normalized trace of a finite locally free endomorphism. When a reduced subscheme Z satisfies (h^{-1}(Z))_red = Z, Lemma 2.1 splits the inclusion I_Z -> h_*I_Z by d^{-1}Tr_h, making H^i(X, I_Z ⊗ A) a direct summand of H^i(X, I_Z ⊗ h^*A). Combined with Lemma 2.2 (iterates of an int-amplified map make h^*L ⊗ L^{-1} ample) and uniform Fujita vanishing, this yields the vanishing H^i(X, I_Z ⊗ L^t) = 0 for all i,t>0 (Proposition 2.3). Successive very ample hyperplane sections then express the degree of Z as the length of a zero-dimensional scheme, bounded by the Hilbert function of a complete intersection, which in P^n evaluates to the binomial coefficient.","core_discovery":"The central claim is Theorem 1.3: for an endomorphism f of P^n with f^*O(1) = O(q), q>1 invertible in the field, the set of totally-periodic prime cycles of codimension r is finite and the sum of their degrees is at most binom(n+1,r), so there are at most binom(n+1,r) of them. This bound is sharp, realized by the coordinate power map, whose coordinate linear subspaces are the extremal cycles. The same mechanism yields a general bound on any smooth projective variety X: for an ample line bundle L, the total L-degree of codimension-r totally-periodic cycles is controlled by a finite difference of the Hilbert function of L, given in Theorem 1.1. The proof uses a normalized trace to split the id","pith_inferences":["If the paper's one-sentence characteristic-free assertion of the cone-theoretic criterion is correct, Theorem 1.1's bounds hold verbatim over every algebraically closed field of characteristic not dividing deg(f); checking a single int-amplified map over F_p whose degree is coprime to p would be a cheap test.","Remark 4.2 shows that the polynomial identity Φ = (1+t)^{n+1} does not characterize P^n as an underlying variety; a natural next step would be to classify all smooth projective X admitting an int-amplified f with maximal totally-periodic polynomial.","Because the proof of Proposition 2.3 only needs iterates to amplify L, the same Hilbert-function bound should apply to any self-map satisfying that amplification condition, not only to int-amplified endomorphisms.","The sharp bound on completely ramified points when q ≥ n+1 suggests that for low-degree maps (q < n+1) the linear-independence statement may fail; computing examples in degree 2 on P^2 would map the boundary of the sharp result."],"forward_implications":["Favre's question for P^2 is answered: |T_∞(f)| ≤ 3 for every degree-q>1 endomorphism of the projective plane, with equality for the coordinate power map.","In every codimension 1 ≤ r ≤ n of P^n, the coordinate power map attains the bound, so the binomial constants cannot be improved; any endomorphism achieving the full binomial polynomial must, after an iterate, be projectively conjugate to that map.","For completely ramified points, when q ≥ n+1, the paper obtains the sharp bound n+1 and projective linear independence, improving the previous plane-curve bound of nine to three for cubic and higher degree maps of P^2.","The general Hilbert-function bound applies to all smooth projective varieties admitting int-amplified endomorphisms of degree invertible in k, giving degree and cardinality bounds that are uniform over all such endomorphisms once L and a are fixed.","Applications include: blow-ups of P^n along smooth codimension-two totally-invariant cycles are Fano; log canonical centers of invariant boundaries satisfy the same binomial bound; and the number of contractible extremal rays is linearly controlled."],"supporting_citations":[{"why":"Supplies the definition of int-amplified endomorphism and the cone-theoretic eigenvalue criterion used in Lemma 2.2.","marker":"[26]"},{"why":"Supplies uniform Fujita vanishing, used to kill higher cohomology in Proposition 2.3.","marker":"[15]"},{"why":"Supplies the trace-base-change and miracle-flatness facts behind the splitting Lemma 2.1.","marker":"[31]"},{"why":"Supplies the finiteness of totally periodic Zariski closed subsets, the starting finiteness for T^r_∞(X,f).","marker":"[27]"},{"why":"Poses Favre's question for P^2 that Theorem 1.3 answers.","marker":"[13]"}],"fun_headline_variants":["Total invariant cycles capped at n+1 choose r","Sharp cap on invariant cycles: binom(n+1,r)","At most binom(n+1,r) invariant cycles in P^n","Invariant cycles bounded by binomial, and sharp","New bound: at most n+1 choose r invariant cycles"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof assumes that the cone-theoretic eigenvalue criterion for int-amplified endomorphisms, which the cited reference states in characteristic zero, remains valid over any algebraically closed field; the paper asserts this without proof.","fun_headline_variants_meta":{"raw":{"variants":["Total invariant cycles capped at n+1 choose r","Sharp cap on invariant cycles: binom(n+1,r)","At most binom(n+1,r) invariant cycles in P^n","Invariant cycles bounded by binomial, and sharp","New bound: at most n+1 choose r invariant cycles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000444,"raw_usage":{"total_tokens":2040,"prompt_tokens":657,"completion_tokens":1383,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":401,"completion_tokens_details":{"reasoning_tokens":1310}},"tokens_in":401,"tokens_out":1383,"duration_ms":11018,"temperature":1.0,"reasoning_tokens":1310,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T04:36:42.259326+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An endomorphism of P^n of degree q>1 invertible in k with total degree of its totally-invariant codimension-r cycles exceeding binom(n+1,r), or a smooth projective X with an int-amplified endomorphism of degree invertible in k where the Hilbert-function inequality in Theorem 1.1 fails, would refute the central claim. A simpler place to look: verify Lemma 2.2 in characteristic p>0 with p ∤ deg(f); if the ampleness of (f^s)^*L ⊗ L^{-1} fails for some large s, Proposition 2.3 collapses.","supporting_citations":[{"cited_title":"Meng,Building blocks of amplified endomorphisms of normal projective varieties, Math","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of int-amplified endomorphism and the cone-theoretic eigenvalue criterion used in Lemma 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the trace-base-change and miracle-flatness facts behind the splitting Lemma 2.1."},{"cited_title":"Meng and D.-Q","cited_arxiv_id":null,"evidence_quote":"Supplies the finiteness of totally periodic Zariski closed subsets, the starting finiteness for T^r_∞(X,f)."},{"cited_title":"Favre,Equidistribution problems in holomorphic dynamics inP2, inDynamical systems","cited_arxiv_id":null,"evidence_quote":"Poses Favre's question for P^2 that Theorem 1.3 answers."}],"review_version":2}