Divisors on surfaces isogenous to a product of mixed type with $p_g=0$

Davide Frapporti; Kyoung-Seog Lee

REVIEW 3 major objections 5 minor 14 references

Divisors on surfaces isogenous to a product of mixed type with $p_g=0$

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Every surface isogenous to a product of mixed type with p_g=0 is a Mori dream surface, and its three divisor cones coincide.

desk verdict Solid extension proving the Mori dream property for mixed-type surfaces isogenous to a product with p_g=0, but families 2-5 rest on unarchived MAGMA output and one Lemma 1.9 application is mistyped. read the letter →

arxiv 1908.09330 v2 pith:IB4E2MHA submitted 2019-08-25 math.AG

Davide Frapporti , Kyoung-Seog Lee This is my paper · ORCID

classification math.AG MSC 14J2914E3014J5014H3714L3014Q10

keywords Moridreamspacefakequadricssurfacesisogenoustoaproductmixedtypeeffectiveconenefsemiampleorbitdivisors

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 proves a uniformity statement about a class of surfaces of general type: those obtained as free quotients of a product of a curve with itself by a mixed group action, with geometric genus zero. For each such surface $S$ the effective, nef, and semiample cones in the real space $N^1(S)$ are not merely nested but equal, and they are generated by the classes of two orbit divisors. The interest is that these surfaces include all reducible fake quadrics—smooth minimal surfaces with the same Hodge diamond as a quadric, $p_g=q=0$ and $K^2=8$—whose divisor theory was largely open. A Mori dream surface is one whose birational geometry is governed by finitely many polyhedral cones and a finitely generated Cox ring.

What carries the argument

The load-bearing construction is the orbit divisor induced by a subgroup $H$ of $\mathrm{Aut}(C)$. For $f\in H$, the graph $\Delta_f=\{(x,f(x))\}\subset C\times C$ is moved by the ambient group $G$; the reduced sum of the $G$-orbit of $\Delta_f$ descends to an irreducible effective divisor on $S$. The paper computes intersection numbers of these divisors by counting fixed points of automorphisms through the Riemann Existence Theorem, which relates fixed points to branch data, and then applies a numerical lemma: four distinct effective irreducible divisors $D_1,\dots,D_4$ with $D_i^2=0$, $D_1\cdot D_4=D_2\cdot D_3=0$, and the four remaining mixed intersections positive and equal, force $\mathrm{Eff}(S)=\mathrm{Nef}(S)=\mathrm{SAmp}(S)=\mathbb{R}_{\geq 0}\langle D_1,D_2\rangle$. In the five-point branch case the subgroup $H$ equal to the diagonal part $G_0$ suffices; in the three-point branch cases the symmetries of the three branch points are lifted to a larger automorphism group of $C$.

What would settle it

Recompute the intersection numbers among the orbit divisors for families 2–5 and check whether the asserted numerical pattern holds: $D_2^2=0$, $D_7^2=0$, $D_2\cdot D_7=16$, $D_1\sim (D_2+D_7)/2$, $D_5\sim D_2+D_7$, and $D_3\sim 2(D_2+D_7)$. A single failure would remove the four divisors required by Lemma 1.9, and the cone equality would no longer follow.

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Extended reading notes

Core claim

Let $S=(C\times C)/G$ be a surface isogenous to a product of mixed type with $p_g=q=0$, where $G$ is a finite group acting freely and exchanging the two factors. The paper establishes that $\mathrm{Eff}(S)=\mathrm{Nef}(S)=\mathrm{SAmp}(S)$ and that this common cone is $\mathbb{R}_{\geq 0}\langle D_1,D_2\rangle$ for two explicit orbit divisors. Hence $S$ is a Mori dream surface and carries no negative curves. Since the unmixed case was already settled, every reducible fake quadric is a Mori dream surface. The proof is case-by-case over the five classified families, using orbit divisors—sums of graphs of automorphisms of $C$ pushed down to $S$—whose intersection products are computed from branch data and verified computationally.

Load-bearing premise

The printed intersection numbers among the orbit divisors for families 2–5 are correct, even though the paper shows the verifying script only for family 1 and states that the other four families give analogous output.

Editorial extensions

If this is right

Every surface isogenous to a product of mixed type with p_g=0 is a Mori dream surface, so its Cox ring is finitely generated.
Combined with the previously known unmixed case, all reducible fake quadrics are Mori dream surfaces.
For each of the five families the divisor cone is the single chamber spanned by two semiample divisor classes; there are no negative curves and no nontrivial Mori chamber decomposition.
The cone equality means that inside this cone every effective divisor class is also semiample, so every effective divisor eventually has a base-point-free multiple.
The same numerical pattern—four orbit divisors satisfying the intersection conditions of Lemma 1.9—recurs across all five families, despite their different group-theoretic data.

Reading between the lines

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

The four-divisor criterion of Lemma 1.9 could serve as a cheap numerical test for the Mori dream property in other surfaces of general type with Picard number two: if four irreducible effective divisors with exactly that intersection pattern can be exhibited, the cone equality follows without further geometric input.
The number of orbit divisors (four in the first family, fifteen in the other four) is likely not essential; only the existence of two numerically equivalent pairs of divisors matters, so one could search for such witnesses in other constructions.
The success of lifting branch-point symmetries in the three-point cases suggests a general recipe: when the branch configuration of a cover has extra symmetries, those symmetries may enlarge the automorphism group and produce the orbit divisors needed to span the cones.
The authors raise whether every fake quadric, including irreducible ones, is a Mori dream surface; Lemma 1.9 gives a concrete way to attack that question by looking for four divisors with the same intersection table.

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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 headline is that this is a genuine extension, not a paradigm shift. Frapporti and Lee prove that every mixed-type surface isogenous to a product with p_g=0 is a Mori dream surface with Eff=Nef=SAmp, completing the reducible fake quadrics story when combined with the unmixed case in [KL19]. The new idea is the orbit-divisor construction with a subgroup H of Aut(C) properly containing G0, which is needed exactly for the bimodule cases where G0 alone gives only one divisor class. Lemma 1.9 is a clean reduction: once you find four irreducible divisors with the right intersection pattern, the cone equality follows formally. The theoretical framework is sound. Lemma 3.2 and 3.4 are standard, the application of Riemann Existence is correct, and the reduction to the classification in [BCG08, Fra13] is legitimate. Family 1 is fully verified by the printed MAGMA script, which is reproducible in principle. The construction of H for families 2-5 is explicit and credible, and the numerical equivalences for Case 2 are consistent with the stated intersection numbers—modulo the typo below.

The soft spots are real but not fatal. First, the verification for families 2-5 depends on MAGMA output that is not shipped. The appendix prints only the Family 1 script; for the rest the authors point to a personal webpage without a stable archive and summarize the output as 'analogous'. That is a reproducibility gap, and it is load-bearing for the main theorem. A referee cannot check that four divisors satisfying Lemma 1.9 actually exist without re-running an unarchived computation. Second, the specific application of Lemma 1.9 in Theorem 3.5 is mistyped: the ordered quadruple (D2,D9,D7,D14) does not satisfy the lemma's hypothesis, because D2.D14 and D9.D7 equal D2.D7=16, not 0. The stress-test note is right about this. The fix is trivial—use (D2,D7,D14,D9) instead—but as printed the proof contains a false assertion. Third, the phrase 'in the other 3 cases we have an analogous output' is too compressed for a serious referee.

None of this undermines the central theorem; the argument is well-structured and the errors are easily corrected. The paper deserves a serious referee. I would send it to peer review and ask for: an archive of all five MAGMA scripts (or a stable repository), the full intersection matrices for families 2-5, and a corrected Lemma 1.9 ordering. The reader's soundness score of 6 is proportionate.

Referee Report

3 major / 5 minor

Summary. The paper studies surfaces isogenous to a product of mixed type with p_g = p_q = 0, which form five irreducible families (Table 1). The authors construct G-invariant divisors on C × C by taking orbits of graphs of automorphisms, compute their intersection numbers via fixed-point counts from Riemann Existence (Lemmas 3.1–3.4), and then use a numerical criterion (Lemma 1.9) to conclude that for every such surface S one has Eff(S) = Nef(S) = SAmp(S). Consequently S is a Mori dream surface, and in particular every reducible fake quadric is a Mori dream surface. The proof is case-by-case: Family 1 is supported by a printed MAGMA script, while Families 2–5 rely on MAGMA computations summarized in the text and accessible from a personal webpage.

Significance. If the result is correct, it is a substantial contribution: it gives the first uniform statement that all reducible fake quadrics are Mori dream spaces, and it provides an explicit description of their effective, nef, and semiample cones. The method is elegant: it avoids Cox-ring computations by using orbit divisors and the elementary Lemma 1.9, and the fixed-point counting in Lemma 3.4 is a clean application of the Riemann Existence Theorem. The paper also ships a machine-checkable script for Family 1, which is a strength. The main weakness is that the computational verification for Families 2–5 is not fully shipped or archived, and one application of Lemma 1.9 is stated with an incorrect ordering. These issues are local and fixable, but they are load-bearing because the main theorem depends on them.

major comments (3)

[Section 3, Theorem 3.5, Families 2–5] The proof for Families 2–5 depends entirely on MAGMA computations that are not included in the manuscript. The text states that in the other three cases one obtains an analogous output, and the supporting material is hosted on a personal webpage that is not a stable archive. The full intersection matrix for the 15 orbit divisors in each family is never printed, so the existence of four divisors satisfying the hypotheses of Lemma 1.9 cannot be verified from the manuscript alone. Since a single incorrect intersection number would invalidate the cone equality, I ask that the complete computations (code, output tables, or a permanent DOI) be provided for all five families, not just Family 1.
[Section 3, Theorem 3.5, Case 2] The sentence 'Applying Lemma 1.9 to D2, D9, D7, D14' is inconsistent with the numerical equivalences stated in items 1 and 2. With the ordered quadruple (D1,D2,D3,D4) = (D2,D9,D7,D14), Lemma 1.9 requires D1.D4 = 0, but the stated equivalences give D2.D14 = D2.D7 = 16, and similarly D9.D7 = D2.D7 = 16. The correct ordering is (D2,D7,D14,D9), for which D2.D9 = D2^2 = 0 and D7.D14 = D7^2 = 0. Please correct the ordering and verify the analogous ordering in Families 3–5 as well.
[Section 2.1 and Theorem 3.5] The construction of the large automorphism group H = G(768, 1085341) and the claim that the covering c : C → P^1 is Galois are justified by a MAGMA script available only at an external URL. This is a separate, load-bearing computational step: without it, the existence of the orbit divisors for Families 2–5 is not established. The script and its output should be archived, or the relevant group-theoretic checks (e.g., the uniqueness of H and the lifting of generating vectors) should be reproduced in the appendix.

minor comments (5)

[Appendix A] The printed script computes self-intersections and pairings but does not label the orbit divisors D_i in a way that lets the reader match the numerical output to the names used in Theorem 3.5. Adding a table that maps the orbit index to the divisor name would make the verification transparent.
[Introduction] There is a typo: 'contruct extremal rays' should be 'construct extremal rays'.
[Lemma 3.3] The proof asserts that Jac(f_1^{-1} f_2)_{x0} ≠ 1 for a non-identity automorphism fixing x0. This is true for a finite-order automorphism of a germ, but the justification should be stated explicitly, since Cartan's Lemma alone gives the linearization, not the non-identity of the linear part.
[Theorem 3.5, Family 1] The notation KX is used for the canonical divisor, whereas elsewhere the surface is denoted S and the canonical divisor KS; please unify the notation.
[Table 1] The first data row of Table 1 appears garbled in the arXiv typeset version; please ensure the journal version renders the H1(S,Z) entry correctly.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the cone equality follows from orbit-divisor computations on a classified list of groups; remaining issues are verification gaps, not circularity.

full rationale

The main theorem is proved by taking the classification of mixed-type surfaces with p_g=0 from Theorem 2.6, quoted from [BCG08] and [Fra13], constructing G-invariant orbit divisors from graphs of automorphisms (Lemmas 3.1-3.3), computing their intersection numbers via Lemma 3.4 and MAGMA, and then applying Lemma 1.9, a criterion proved inside the paper. There is no fitted parameter, no normalization chosen to force the cone equality, and no divisor class is renamed as a prediction. The classification [Fra13] overlaps with the first author, but it is an external published theorem used as an input, not a claim whose content equals the target result; moreover it is paired with the external [BCG08]. The unmixed-type result [KL19] overlaps with the second author, but it is only motivational and not used in the mixed-type proof. The unarchived MAGMA scripts and the apparent mismatch in the printed ordered quadruple (D2,D9,D7,D14) versus Lemma 1.9 are correctness and verifiability concerns: the numerical data for families 2-5 are not fully reproducible from the paper, and the stated quadruple does not literally satisfy the hypothesis D1.D4=D2.D3=0 (a valid ordering would be (D2,D7,D14,D9)). These concerns do not amount to a circular step, because no part of the derivation reduces by construction or by self-citation to its own input. The only reason the score is not 0 is the presence of self-citations that are not load-bearing for the derivation itself; the central claim retains independent content.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the prior classification of mixed-type surfaces, on standard results in surface theory, and on MAGMA computations that are only partially shipped. No free parameters are fitted. No new entities are invented.

assumptions (4)

standard math Riemann Existence Theorem for compact Riemann surfaces
Used in Theorem 2.5 to realize group actions via generating vectors and to count fixed points in Lemma 3.4.
domain assumption Classification of surfaces isogenous to a product of mixed type with p_g=0 into the five families of Table 1
Theorem 2.6 is cited to [BCG08, Fra13] and not reproved. The main theorem is proved case by case over exactly these five families; if the classification were incomplete, the statement would not cover all mixed-type surfaces with p_g=0.
domain assumption Existence of the automorphism group H = G(768,1085341) of the curve C of order 768 containing G0, with the specific generating-vector properties, established by MAGMA
Section 2.1 uses a supporting MAGMA script at an external URL to assert that such an H exists and that it is unique. This is load-bearing for families 2-5 because the orbit divisors are induced by H, not just by G0.
domain assumption Correctness of the MAGMA computations of intersection numbers among the orbit divisors for all five families
Section 3 and Theorem 3.5 rely on these computations to exhibit the four divisors satisfying Lemma 1.9. The paper provides only a partial script for Family 1 in Appendix A and asserts analogous output for families 3-5.

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Pith. "Pith review of Divisors on surfaces isogenous to a product of mixed type with $p_g=0$." pith.science (2026). https://pith.science/paper/IB4E2MHA

@misc{pith2026190809330,
  author       = {Pith},
  title        = {Pith review of: Divisors on surfaces isogenous to a product of mixed type with $p_g=0$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IB4E2MHA}},
  note         = {Machine review of arXiv:1908.09330}
}

abstract

In this paper, we study effective, nef and semiample cones of surfaces isogenous to a product of mixed type with $p_g=0$. In particular, we prove that all reducible fake quadrics are Mori dream surfaces.

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Reference graph

Works this paper leans on

14 extracted references · 13 canonical work pages

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Pith tools

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

[1] [1]

Arzhantsev, U

I. Arzhantsev, U. Derenthal, J. Hausen, and A. Laface. Cox rings , volume 144 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2015

work page 2015

[2] [2]

Artebani, J

M. Artebani, J. Hausen, and A. Laface. On C ox rings of K 3 surfaces. Compos. Math. , 146(4):964--998, 2010

work page 2010

[3] [3]

Artebani and A

M. Artebani and A. Laface. Cox rings of surfaces and the anticanonical I itaka dimension. Adv. Math. , 226(6):5252--5267, 2011

work page 2011

[4] [4]

Bauer, F

I. Bauer, F. Catanese, and F. Grunewald. The classification of surfaces with p_g=q=0 isogenous to a product of curves. Pure Appl. Math. Q. , 4(2):547--586, 2008

work page 2008

[5] [5]

Bosma, J

W. Bosma, J. Cannon, and C. Playoust. The M agma algebra system. I . T he user language. J. Symbolic Comput. , 24(3-4):235--265, 1997. Computational algebra and number theory (London, 1993)

1997

[6] [6]

Catanese

F. Catanese. Fibred surfaces, varieties isogenous to a product and related moduli spaces. American Journal of Mathematics , 122(1):1--44, 2000

work page 2000

[7] [7]

Cancian and D

N. Cancian and D. Frapporti. On semi-isogenous mixed surfaces. Math. Nachr. , 291(2-3):264--283, 2018

work page 2018

[8] [8]

Carnovale and F

G. Carnovale and F. Polizzi. The classification of surfaces with p_g = q = 1 isogenous to a product of curves. Advances in Geometry , 9(2):233--256, 2009

work page 2009