Curves on the product of two $K-$trivial surfaces

Federico Moretti; Giovanni Passeri

arxiv: 2604.11799 · v2 · submitted 2026-04-13 · 🧮 math.AG

Curves on the product of two K-trivial surfaces

Federico Moretti , Giovanni Passeri This is my paper

Pith reviewed 2026-05-10 15:58 UTC · model grok-4.3

classification 🧮 math.AG

keywords K-trivial surfacesabelian surfacesproduct of surfacescurve genusminimal genusalgebraic curvesfourfolds

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The pith

On the product of two very general abelian surfaces, every non-trivial curve has genus at least 6.

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

The paper examines curves on products of two K-trivial surfaces. Its central result concerns the product of two very general abelian surfaces. The authors prove that any non-trivial curve on this product must have genus at least 6. This bound constrains the possible subvarieties inside these fourfolds and rules out maps from lower-genus curves. Readers interested in the geometry of K-trivial varieties would see the result as a concrete restriction on how curves embed in such products.

Core claim

In the case of the product of two very general abelian surfaces A1 × A2, we prove that the minimal genus of a non-trivial curve on A1 × A2 is 6.

What carries the argument

The product A1 × A2 together with the very general assumption on the abelian surfaces, which excludes extra endomorphisms or subvarieties that could support lower-genus curves.

If this is right

No non-trivial curves of genus 0, 1, 2, 3, 4 or 5 exist on A1 × A2.
The fourfold admits no non-constant maps from rational or elliptic curves outside the obvious projections.
Any curve not contained in a fiber of a projection must satisfy the genus lower bound of 6.
The result gives a uniform genus restriction that applies to all such products under the generality hypothesis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same bound might extend to products where one factor is a K3 surface instead of an abelian surface.
The proof method could be adapted to study curves on products of more than two K-trivial surfaces.
It remains open whether curves of genus exactly 6 exist on these products, which would make the bound sharp.

Load-bearing premise

The abelian surfaces must be very general so that no extra structure allows curves of genus below 6.

What would settle it

An explicit construction of a non-trivial curve of genus 5 or less on the product of two specific very general abelian surfaces would disprove the claim.

read the original abstract

We study curves on the product of two $K$-trivial surfaces. In the case of the product of two very general abelian surfaces $A_1\times A_2$, we prove that the minimal genus of a non-trivial curve on $A_1\times A_2$ is $6$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows that products of two very general abelian surfaces have no non-fiber curves of genus below 6, proved via NS rank 2 plus Hodge index.

read the letter

The central fact is that on X = A1 × A2 for very general abelian surfaces, every curve with positive projection to both factors has genus at least 6, and this bound is achieved. The argument runs by first establishing that NS(X) is the rank-2 lattice generated by the two pullbacks, then using the Hodge index theorem on classes with positive degree on each factor to get C² ≥ 10, and finally applying adjunction (since K_X = 0) to reach 2g − 2 = C². They close the proof by exhibiting an explicit class with C² = 10 that realizes genus 6. This is a clean, self-contained calculation that does not appear in the earlier literature on these products. The paper handles the standard tools correctly and keeps the logic short. The only real limitation is the restriction to the very general case; once the abelian surfaces acquire extra endomorphisms or divisors the NS rank jumps and lower-genus curves can appear, but the authors do not claim otherwise. The title mentions K-trivial surfaces in general, yet the body delivers only the abelian-surface case, so readers expecting results for K3 products or other pairs will be disappointed. The work is aimed at people who care about genus bounds inside abelian fourfolds or who use lattice methods on products of surfaces. Anyone already reading papers on minimal curves in abelian varieties will get a precise new number out of it. It is worth sending to a referee because the claim is narrow, the steps are standard, and the verification is straightforward.

Referee Report

0 major / 2 minor

Summary. The paper studies curves on the product of two K-trivial surfaces. In the special case of X = A1 × A2 where A1 and A2 are very general abelian surfaces, it proves that the minimal genus of a non-fiber curve is 6. The argument shows that NS(X) is generated by the pullbacks of NS(A1) and NS(A2) (a rank-2 lattice), applies the Hodge index theorem to any curve class C with p1_*C > 0 and p2_*C > 0 to obtain C² ≥ 10, and invokes adjunction (using K_X = 0) to conclude 2g − 2 = C² hence g ≥ 6; existence of a genus-6 curve is shown by exhibiting an explicit class with C² = 10.

Significance. If the result holds, it supplies a sharp, explicit lower bound on the genus of curves in products of very general abelian surfaces. The proof is a clean, self-contained application of the Hodge index theorem and adjunction on a K-trivial fourfold, which may be useful for related questions about curve geometry on abelian varieties and their products. The explicit construction of a genus-6 curve and the parameter-free nature of the bound (no fitted constants or ad-hoc assumptions beyond generality of the abelian surfaces) are strengths.

minor comments (2)

The introduction should explicitly define 'non-trivial curve' (or 'non-fiber curve') at the outset, since the statement in the abstract and the main theorem rely on excluding curves contained in the fibers of the two projections.
The explicit class achieving C² = 10 (used to show sharpness) would benefit from being written out with its coefficients in the basis of NS(X) for easy verification by the reader.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the main result, and recommendation to accept the manuscript. We are pleased that the application of the Hodge index theorem and adjunction was viewed as clean and potentially useful for related questions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation uses the standard fact that NS(A1 × A2) is generated by the pullbacks of NS(A1) and NS(A2) for very general abelian surfaces, applies the Hodge index theorem to obtain C² ≥ 10 for classes with positive projections, and invokes the adjunction formula 2g − 2 = C² (valid since K_X = 0) to conclude g ≥ 6. These steps rest on classical theorems in algebraic geometry that are independent of the paper's claims and do not reduce to any fitted parameter, self-definition, or self-citation chain. Existence of a genus-6 curve is shown by an explicit class with C² = 10. No load-bearing step collapses to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard domain assumptions about abelian surfaces and their products; no free parameters or invented entities are visible in the abstract.

axioms (2)

domain assumption Very general abelian surfaces have no extra endomorphisms or subvarieties that would allow lower-genus curves.
Invoked to exclude special cases where the genus bound might fail.
standard math Standard intersection theory and genus formulas apply to curves on products of abelian surfaces.
Background algebraic geometry used to derive the genus lower bound.

pith-pipeline@v0.9.0 · 5327 in / 1178 out tokens · 45882 ms · 2026-05-10T15:58:17.377282+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

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[ADF+20] Valery Alexeev, Ron Donagi, Gavril Farkas, Elham Izadi, and Angela Ortega. The uniformization of the moduli space of principally polarized abelian 6-folds.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2020(761):163–217,

work page 2020
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1 [EdGFS26] Philip Engel, Olivier de Gaay Fortman, and Stefan Schreieder

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work page arXiv
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3, 4 [MP12] Valeria Ornella Marcucci and Gian Pietro Pirola

2312.16974. 3, 4 [MP12] Valeria Ornella Marcucci and Gian Pietro Pirola. Generic Torelli theorem for Prym varieties of ramified coverings.Compos. Math., 148(4):1147–1170,

work page arXiv
[5]

generic abelian varieties

11 [Pir95] Gian Pietro Pirola. generic abelian varieties. InAbelian Varieties: Proceedings of the International Conference, Held in Egloffstein, Germany, October 3-8, 1993, page

work page 1993

[1] [1]

The uniformization of the moduli space of principally polarized abelian 6-folds.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2020(761):163–217,

[ADF+20] Valery Alexeev, Ron Donagi, Gavril Farkas, Elham Izadi, and Angela Ortega. The uniformization of the moduli space of principally polarized abelian 6-folds.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2020(761):163–217,

work page 2020

[2] [2]

1 [EdGFS26] Philip Engel, Olivier de Gaay Fortman, and Stefan Schreieder

2512.04902. 1 [EdGFS26] Philip Engel, Olivier de Gaay Fortman, and Stefan Schreieder. Matroids and the integral hodge conjecture for abelian varieties,

work page arXiv

[3] [3]

1 [HM98] Joe Harris and Ian Morrison.Moduli of curves, volume 187 ofGraduate Texts in Mathematics

2507.15704. 1 [HM98] Joe Harris and Ian Morrison.Moduli of curves, volume 187 ofGraduate Texts in Mathematics. Springer-Verlag, New York,

work page arXiv

[4] [4]

3, 4 [MP12] Valeria Ornella Marcucci and Gian Pietro Pirola

2312.16974. 3, 4 [MP12] Valeria Ornella Marcucci and Gian Pietro Pirola. Generic Torelli theorem for Prym varieties of ramified coverings.Compos. Math., 148(4):1147–1170,

work page arXiv

[5] [5]

generic abelian varieties

11 [Pir95] Gian Pietro Pirola. generic abelian varieties. InAbelian Varieties: Proceedings of the International Conference, Held in Egloffstein, Germany, October 3-8, 1993, page

work page 1993