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REVIEW 3 major objections 4 minor 28 references

On K-stability of $\mathbb P^3$ blown up along a smooth genus $2$ curve of degree $5$

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that infinitely many Fano threefolds, obtained by blowing up P^3 along smooth genus-2 degree-5 curves, are K-stable and hence admit Kähler–Einstein metrics.

desk verdict Genuine new result, plausible proof, but the written computation has a clear typo and several unshown integrals; needs a computational appendix before I would rely on it. read the letter →

arxiv 2412.18317 v1 pith:KSRNIVSY submitted 2024-12-24 math.AG

Tiago Duarte Guerreiro , Luca Giovenzana , Nivedita Viswanathan This is my paper · ORCID
classification math.AG MSC 14J4532Q20
keywords K-stabilitystabilitythresholddelta-invariantFanothreefoldsblow-upgenus2curvefamily2.19Kähler–Einsteinmetrics
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

This paper studies the Fano threefolds obtained by blowing up $\mathbb P^3$ along a smooth genus-2 curve of degree 5 that lies on a quadric, which is family 2.19 in the standard classification of Fano threefolds. Its main theorem states that for every such blow-up $X$, the local stability threshold satisfies $\delta_p(X)>1$ for every point $p$ outside the subset $E\setminus \widetilde Q$, where $E$ is the exceptional divisor and $\widetilde Q$ is the strict transform of the quadric. Since a Fano variety is K-stable exactly when its global threshold $\delta(X)>1$, this local estimate leaves only the curve $E\setminus \widetilde Q$ as a possible obstruction. Combining the theorem with an equivariant K-stability criterion, the paper proves K-stability, and hence existence of Kähler–Einstein metrics, for infinitely many explicit members of the family.

What carries the argument

The computational engine is the admissible-flag estimate (Theorems 3.7 and 3.8 in [ACC+21]): for a flag $p\in Z\subset Y\subset X$, the local threshold $\delta_p(X)$ is bounded below by a minimum of three terms involving volumes and orders of vanishing of the positive and negative parts of the Zariski decomposition of $-K_X-uY$. For $p\in\widetilde Q$ the flag uses one of the two rulings of the quadric, and at inflexion points a $(1,3)$-weighted blow-up is inserted; for $p\notin E\cup\widetilde Q$ the flag uses the strict transform $\widetilde S$ of a hyperplane section containing a secant or the tangent line to $C$. The sharp bounds $S_X(\widetilde Q)=10/13$, $S(V)=183/208$, and the final lower bounds $52/49$ and $208/205$ come from explicit Zariski decompositions and volume integrals.

What would settle it

Recompute Proposition 4.6, Case 2 with correct bookkeeping for the hyperplane section: $S\cap C$ consists of the doubled point $p_1$ plus three further points $p_2,p_3,p_4$, so the displayed class $2h-e_3-e_4-e_5$ should be $2h-e_2-e_3-e_4$. If the resulting $1/S(W_{\bullet,\bullet,\bullet}^{\widetilde S,\widetilde L};p)$ is at least $208/205$, the bound survives; if it is smaller, the proof of Theorem 1.1 fails for points with no secant through $C$.

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Extended reading notes

Core claim

The load-bearing result, Theorem 1.1, asserts that for any smooth Fano threefold $X=\operatorname{Bl}_C\mathbb P^3$ with $C$ a smooth genus-2, degree-5 curve lying on a smooth quadric, one has $\delta_p(X)>1$ for all $p\notin E\setminus \widetilde Q$. This is established pointwise: on the strict transform $\widetilde Q$ of the quadric, Proposition 4.4 gives $\delta_p(X)\ge 52/49$; outside $E\cup\widetilde Q$, Proposition 4.6 gives $\delta_p(X)\ge 208/205$; and for curves in $E$ disjoint from $E\cap\widetilde Q$, Lemma 4.2 gives a local bound above 1. The remaining points lie on $E\setminus\widetilde Q$, and the paper shows they cannot obstruct K-stability when $C$ satisfies a fixed-point-free automorphism condition (Corollary 1.2) or the rational-roots condition $s_0^2f_3(t_0,t_1)+s_1^2g_3(t_0,t_1)$ with $f_3,g_3$ having no rational solutions (Corollary 1.3). The concrete payoff is an infinite family of explicit curves, for instance $s_0^2(t_0^3-\eta_0t_1^3)+s_1^2(t_0^3-\eta_1t_1^3)=0$ with $\eta_0\neq\eta_1$ non-cubes, whose blow-ups are K-stable Fano threefolds in family 2.19.

Load-bearing premise

The load-bearing premise is that the Zariski decompositions and the numerical constants derived from them in Propositions 4.4, 4.6, and Lemma 4.2 are computed correctly; if any sharp constant, in particular $208/205$ in Case 2 of Proposition 4.6, is smaller than claimed, the strict inequality $\delta_p(X)>1$ outside $E\setminus\widetilde Q$ can fail and Theorem 1.1 collapses.

Editorial extensions

If this is right

  • For any member of family 2.19 whose automorphism group $\operatorname{Aut}(\mathbb P^3,C)$ has no fixed points on $C$, the threefold is K-stable (Corollary 1.2), and because $\operatorname{Aut}(X)$ is finite this means $X$ admits a Kähler–Einstein metric.
  • Curves of the form $s_0^2f_3(t_0,t_1)+s_1^2g_3(t_0,t_1)$ with $f_3,g_3$ homogeneous cubics over $\mathbb Q$ having no rational solutions give K-stable threefolds (Corollary 1.3).
  • Choosing $\eta_0\neq\eta_1$ non-cube integers in $s_0^2(t_0^3-\eta_0t_1^3)+s_1^2(t_0^3-\eta_1t_1^3)=0$ produces infinitely many K-stable examples (Example 1.4).
  • Pointwise, the paper establishes explicit lower bounds $\delta_p(X)\ge 52/49$ on $\widetilde Q$ and $\delta_p(X)\ge 208/205$ outside $E\cup\widetilde Q$, so the only possible failure locus for $\delta>1$ is the curve $E\setminus\widetilde Q$.

Reading between the lines

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

  • Beyond the paper, if $\delta_p(X)>1$ could also be proven on $E\setminus\widetilde Q$, then every smooth member of family 2.19, not just those with fixed-point-free automorphism action, would be K-stable.
  • The arithmetic criterion that $f_3$ and $g_3$ have no rational solutions suggests a broader mechanism: for Fano threefolds defined over $\mathbb Q$, the absence of rational points on the blown-up curve can eliminate equivariant divisors over the exceptional locus and reduce K-stability to pointwise $\delta$ estimates; the neighbouring families 2.22 and 2.25 are natural test cases.
  • Because the constants $52/49$ and $208/205$ sit close to 1, an independent computer-algebra check of the volume integrals and Zariski decompositions in Propositions 4.4 and 4.6 would be a decisive verification of the proof's numerical core.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies K-stability of Fano threefolds obtained by blowing up P3 along a smooth genus 2 curve of degree 5 lying on a smooth quadric, a family that corresponds to rank 2, degree 26 Fano threefolds in the Mori–Mukai list (family 2.19). The main theorem (Theorem 1.1) asserts that the local stability threshold δp(X) is strictly greater than 1 for all points p outside the set E \ eQ, where E is the exceptional divisor and eQ is the strict transform of the quadric. Using this estimate together with the equivariant K-stability criterion of Zhuang, the authors prove K-stability for infinitely many members of the family (Corollaries 1.2 and 1.3, Example 1.4). The proof is based on the Abban–Zhuang admissible flag technique and consists of a long series of Zariski decomposition and intersection computations.

Significance. If the main theorem is correct, the paper makes a valuable contribution to the classification of K-stable Fano threefolds by providing a unified estimate of the local stability threshold for a previously open family, and it produces an infinite family of explicit K-stable examples. The argument is self-contained modulo standard references and contains no fitted parameters; the numerical outputs (208/205, 52/49, etc.) are fixed intersection-theoretic quantities. The paper also gives a clean criterion based on the absence of fixed points of Aut(P3, C) on C. However, the lack of detail in several key computations and an internal inconsistency in Proposition 4.6 currently prevent full verification of the main claim.

major comments (3)
  1. [Section 4.3, Proposition 4.6, Case 2] The surface S in Case 2 is obtained from P2 by blowing up p1 with weights (1,2) and p2,p3,p4 ordinarily, so its Picard group has four exceptional divisors: e1 (weighted), e2, e3, e4. Nevertheless, the displayed Zariski decomposition for u ∈ [1,2] contains the classes 2h - e3 - e4 - e5 and (6-3u-2v)(2h - e3 - e4 - e5), involving a fifth exceptional divisor e5 that does not exist in this setup. If the intended class is 2h - e2 - e3 - e4, as the preceding notation suggests, then every volume integral in the definition of S(V) and S(W) must be recomputed; the printed values S(V)=183/208 and S(W)=205/208 are therefore unsupported. The final bound δp ≥ 208/205 leaves a margin of only 3/208, so a corrected computation could easily change the strict inequality. Since Case 2 covers all points through which no secant to C passes, this inconsistency is load-bearing for Theorem 1.1.
  2. [Section 4.1, Lemma 4.2] The key estimate 1/S(V^E_{•,•}, Z) = 468n/241 is asserted as a 'direct computation' with no intermediate steps. This lemma is used in the proofs of Corollaries 2.1 and 2.3 to handle divisors with centre a curve in E\eQ, and without a verifiable computation the reader cannot confirm that the inequality δ_Z(X) > 1 holds. The paper should supply the derivation or at least the explicit volume integrals that lead to this value.
  3. [Section 4.3, Proposition 4.6] The numerical values S(V)=183/208 and S(W)=205/208 in both Case 1 and Case 2 are quoted without showing the actual integrals or the intermediate steps that produce them. In a computation-heavy paper whose final margin is 3/208, such black-box evaluations make it impossible for the referee or reader to check the strict inequality. The authors should include the computed integrals (e.g., a table of contributions per interval in u and v) or provide a reproducible calculation.
minor comments (4)
  1. [Section 2, proof of Corollary 2.3] The quadric equation is written as 'Q = (x0x3 − x1x3 = 0)', which factors as two planes; the correct Segre embedding equation is x0x3 − x1x2 = 0.
  2. [Section 4.3, Proposition 4.6] The notation for the strict transform of the tangent line is inconsistent: eL, \tilde L, and \widetilde L are all used for the same object.
  3. [Section 4.1, Lemma 4.2] In the displayed Zariski decomposition of P(u)|_E − vZ, the interval for v in the second branch is malformed: it appears as 'v ∈ [2−2u/n]' and should be 'v ∈ [0, (2−2u)/n]' (and similarly in the first branch).
  4. [Section 3.3 and Corollary 2.3] In Corollary 2.3, the group G is not explicitly identified; the proof appears to use G = Gal(Q̄/Q), but this should be stated clearly when applying [Zhu21, Corollary 4.14].

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's delta-invariant bounds are fixed intersection-theoretic computations, and the load-bearing criteria are external theorems.

full rationale

I walked the derivation chain from Theorem 1.1 back to its inputs. The global criterion δ(X) > 1 ⇔ K-stability is quoted from the external papers [FO18, BJ20], and the local stability threshold machinery is quoted from [AZ22] and, in the form used, from [ACC+21, Theorem 1.112 and Remark 1.113]. Although one present author is a coauthor of [ACC+21], the cited items are general theorems of the Abban-Zhuang method rather than a uniqueness claim or an unverified premise that itself asserts the target result, and the original framework [AZ22] is cited as independent support. The actual estimates in Propositions 4.4 and 4.6 are explicit computations: S_X(eQ)=10/13, S_X(eS)=57/104, S(V)=183/208, S(W)=25/26 or 205/208, and the bounds 52/49 and 208/205 are minima of the reciprocals of computed quantities. There are no fitted parameters, no normalization chosen to force δ_p>1, and no quantity is defined in terms of the inequality it is meant to prove. Corollaries 1.2 and 1.3 use Theorem 1.1 together with external reduction criteria [Zhu21, Corollary 4.14], [Fuj16, Theorem 10.1], and [CPS19, Corollary 12.1]; they are applications, not reimports of the conclusion. The passage in Proposition 4.6, Case 2, that displays the class 2h-e3-e4-e5 after only four blown-up points pi (with e1 doubled and e2,e3,e4 ordinary) is an apparent computational inconsistency, and the paper also omits intermediate integrals for values such as S(V)=183/208 and S(W)=205/208. I flag these as correctness and completeness risks, not as circularity: the claimed lower bound is not assumed in the computation, and a correction to the Zariski decomposition would change a numerical output rather than reveal that the output was an input. Remark 4.5 likewise sketches, rather than fully proves, the existence of curves with no 2-secants through a general point; this is a missing-detail issue, not a circular one. I find no self-definitional step, no fitted input renamed as a prediction, and no load-bearing self-citation chain.

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

The proof is built entirely from fixed geometric data and external theorems; no parameters are fitted and no new objects are postulated. The only 'inputs' are the curve C and the ambient quadric, both given by the problem.

assumptions (5)
  • domain assumption X is K-stable if and only if δ(X) > 1; local thresholds δ_p determine δ(X) as the infimum over p.
    Stated in Section 3.1, from [FO18, BJ20].
  • domain assumption Abban-Zhuang inequality (Theorems 3.7 and 3.8, from [ACC+21, Theorem 1.112 and Remark 1.113]) lower-bounds δ_p(X) in terms of admissible flags.
    Section 3.6; used for all flag computations.
  • domain assumption Zhuang's equivariant criterion (Corollary 4.14 of [Zhu21]) reduces K-polystability to checking G-invariant prime divisors.
    Proofs of Corollaries 2.1 and 2.3.
  • domain assumption Fujita's theorem that S_X(E) < 1 for the exceptional divisor of a divisorially K-stable Fano variety ([Fuj16, Theorem 10.1]).
    Lemma 4.2 relies on this to conclude δ_Z(X) > 1.
  • domain assumption Blanc-Lamy classification: smooth Fano threefolds of family 2.19 are exactly blow-ups of P3 along a smooth genus 2 degree 5 curve on a quadric ([BL12, Proposition 3.1]).
    Sets up the geometry in Section 4.

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Pith. "Pith review of On K-stability of $\mathbb P^3$ blown up along a smooth genus $2$ curve of degree $5$." pith.science (2026). https://pith.science/paper/KSRNIVSY

@misc{pith2026241218317,
  author       = {Pith},
  title        = {Pith review of: On K-stability of $\mathbb P^3$ blown up along a smooth genus $2$ curve of degree $5$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KSRNIVSY}},
  note         = {Machine review of arXiv:2412.18317}
}
read the original abstract

We prove K-stability for infinitely many smooth members of the family 2.19 of the Mukai-Mori classification.

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