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arxiv: 2606.18102 · v1 · pith:TQCK3ETBnew · submitted 2026-06-16 · 🧮 math.AG

A remark on rational quartic curves in prime Fano threefolds of degree 22

Kiryong Chung , Jeong-Seop Kim This is my paper

Pith reviewed 2026-06-26 22:32 UTC · model grok-4.3

classification 🧮 math.AG
keywords Fano threefoldrational quartic curveHilbert schemeSarkisov linkdel Pezzo threefoldV_22V_5
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The pith

The Hilbert scheme of rational quartic curves in a prime Fano threefold of degree 22 admits a generically two-to-one rational map to projective four-space.

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

This note uses the Sarkisov link between the prime Fano threefold V_22 and the quintic del Pezzo threefold V_5 to define a rational map on the Hilbert scheme of rational quartic curves lying in V_22. The map lands in P^4 and is shown to be generically two-to-one. A sympathetic reader would care because the result supplies an explicit parametrization of these curves that reduces their study to linear geometry on projective space. If the claim holds, questions about the curves on V_22 become equivalent to questions about pairs of points in P^4.

Core claim

Using the Sarkisov link between the prime Fano threefold V_22 of degree 22 and the quintic del Pezzo threefold V_5, the Hilbert scheme of rational quartic curves in V_22 admits a generically 2-to-1 rational map onto the projective space P^4.

What carries the argument

The Sarkisov link between V_22 and V_5, which induces the rational map on the Hilbert scheme of rational quartic curves.

Load-bearing premise

The Sarkisov link between V_22 and V_5 induces a well-defined rational map on the Hilbert scheme of rational quartic curves that is generically 2-to-1.

What would settle it

An explicit rational quartic curve on V_22 whose image under the induced map has no distinct partner curve mapping to the same point in P^4, or a direct computation showing the map degree differs from 2.

read the original abstract

In this short note, using the Sarkisov link between a prime Fano threefold $V_{22}$ of degree $22$ and the quintic del Pezzo threefold $V_5$, we prove that the Hilbert scheme of rational quartic curves in $V_{22}$ admits a generically $2$-to-$1$ rational map onto the projective space $\mathbb{P}^4$.

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.

Referee Report

2 major / 0 minor

Summary. The paper claims that the Sarkisov link between a prime Fano threefold V_{22} of degree 22 and the quintic del Pezzo threefold V_5 induces a generically 2-to-1 rational map from the Hilbert scheme of rational quartic curves on V_{22} onto \mathbb{P}^4.

Significance. If the claimed map is rigorously established, the result would give a simple birational parametrization of these curves, potentially simplifying questions about their moduli and incidence relations within the birational geometry of degree-22 Fano threefolds.

major comments (2)
  1. The manuscript asserts that the Sarkisov link produces the desired rational map on the Hilbert scheme but provides no explicit description of the induced correspondence between rational quartics on V_{22} and points of \mathbb{P}^4, nor any verification that the map is defined for a generic such curve (i.e., that a generic rational quartic avoids the exceptional loci of the link in a manner that makes the image curve well-defined).
  2. There is no argument establishing that the induced map is generically 2-to-1; the abstract states only that the link is used to prove the claim, without checking the degree of the correspondence or ruling out that the map collapses on a positive-dimensional locus.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments on our short note. We respond to each major comment below.

read point-by-point responses

  1. Referee: The manuscript asserts that the Sarkisov link produces the desired rational map on the Hilbert scheme but provides no explicit description of the induced correspondence between rational quartics on V_{22} and points of \mathbb{P}^4, nor any verification that the map is defined for a generic such curve (i.e., that a generic rational quartic avoids the exceptional loci of the link in a manner that makes the image curve well-defined).

    Authors: The induced correspondence sends a rational quartic C \subset V_{22} to the point in \mathbb{P}^4 that parametrizes the image of the proper transform of C under the Sarkisov link to V_5 (via the linear system that defines the link). The map is defined on a dense open subset of the Hilbert scheme because the exceptional loci of the link form a closed subset of codimension at least one in V_{22}; the condition that a curve meets this locus is a proper closed condition on the Hilbert scheme, which is irreducible. We will add an explicit paragraph describing the correspondence and this genericity argument. revision: yes

  2. Referee: There is no argument establishing that the induced map is generically 2-to-1; the abstract states only that the link is used to prove the claim, without checking the degree of the correspondence or ruling out that the map collapses on a positive-dimensional locus.

    Authors: The generically 2-to-1 property follows from the geometry of the Sarkisov link: the inverse construction from a general point of \mathbb{P}^4 produces a curve on V_5 whose proper transform on V_{22} yields exactly two rational quartics (accounting for the degree of the link in this case). Dominance is clear from the birationality of the link and the fact that the curves fill V_{22}; there is no positive-dimensional locus of collapse because the fiber dimension is zero on a dense open set by a standard dimension count on the incidence variety. We will insert a short verification of the degree in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity; proof invokes external Sarkisov link as independent input

full rationale

The derivation claims that the Sarkisov link V_22 -- V_5 induces a generically 2-to-1 rational map on the Hilbert scheme of rational quartics. This is presented as a consequence of standard birational geometry rather than a self-referential definition or a fitted parameter renamed as a prediction. No equations or steps reduce the claimed map to quantities already determined by the same Hilbert scheme data. The argument is therefore self-contained against external benchmarks in birational geometry; any gaps concern completeness of the exceptional-locus check, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof depends on the existence and basic properties of the Sarkisov link and on standard facts about Hilbert schemes of curves on Fano threefolds; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Existence and standard properties of the Sarkisov link between V_22 and V_5
    Invoked directly in the abstract as the tool that produces the map on the Hilbert scheme.
  • domain assumption Standard facts from algebraic geometry on prime Fano threefolds and del Pezzo threefolds
    Background assumptions required to define V_22, V_5, and their curve Hilbert schemes.

pith-pipeline@v0.9.1-grok · 5587 in / 1355 out tokens · 39574 ms · 2026-06-26T22:32:39.457509+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

24 extracted references · 1 linked inside Pith

  1. [1]

    Arend Bayer and Yuri I. Manin. ( S emi)simple exercises in quantum cohomology. In The Fano Conference , pages 143--173. University of Torino, Turin, 2004

    2004

  2. [2]

    Compactified moduli spaces of rational curves in projective homogeneous varieties

    Kiryong Chung, Jaehyun Hong, and Young-Hoon Kiem. Compactified moduli spaces of rational curves in projective homogeneous varieties. J. Math. Soc. Japan , 64(4):1211--1248, 2012

    2012

  3. [3]

    Geometry of moduli spaces of rational curves in linear sections of G rassmannian G r(2,5)

    Kiryong Chung, Jaehyun Hong, and SangHyeon Lee. Geometry of moduli spaces of rational curves in linear sections of G rassmannian G r(2,5). Journal of Pure and Applied Algebra , 222(4):868 -- 888, 2018

    2018

  4. [4]

    Desingularization of K ontsevich's compactification of twisted cubics in V_5

    Kiryong Chung. Desingularization of K ontsevich's compactification of twisted cubics in V_5 . Manuscripta Math. , 171(1-2):347--367, 2023

    2023

  5. [5]

    Rational curves in a quadric threefold via an SL (2, C ) -representation

    Kiryong Chung, Sukmoon Huh, and Sang-Bum Yoo. Rational curves in a quadric threefold via an SL (2, C ) -representation. arXiv , 2023

    2023

  6. [6]

    Hilbert scheme of rational cubic curves via stable maps

    Kiryong Chung and Young-Hoon Kiem. Hilbert scheme of rational cubic curves via stable maps. Amer. J. Math. , 133(3):797--834, 2011

    2011

  7. [7]

    Rational quartic curves in the M ukai- U memura variety

    Kiryong Chung, Jaehyun Kim, and Jeong-Seop Kim. Rational quartic curves in the M ukai- U memura variety. J. Pure Appl. Algebra , 229(11):Paper No. 108102, 20, 2025

    2025

  8. [8]

    Quartic curves in the quintic del P ezzo threefold, to appear in J

    Kiryong Chung, Jaehyun Kim, and Jeong-Seop Kim. Quartic curves in the quintic del P ezzo threefold, to appear in J. Math. Soc. Japan, 2025

    2025

  9. [9]

    In preparation

    Kiryong Chung and Dae-Won Lee. In preparation. 2026

    2026

  10. [10]

    Correspondence of D onaldson- T homas and G opakumar- V afa invariants on local C alabi- Y au 4-folds over V_5 and V_ 22

    Kiryong Chung, Sanghyeon Lee, and Joonyeong Won. Correspondence of D onaldson- T homas and G opakumar- V afa invariants on local C alabi- Y au 4-folds over V_5 and V_ 22 . arXiv , 2021

    2021

  11. [11]

    Mori's program for the moduli space of conics in G rassmannian

    Kiryong Chung and Han-Bom Moon. Mori's program for the moduli space of conics in G rassmannian. Taiwanese J. Math. , 21(3):621--652, 2017

    2017

  12. [12]

    Genus zero G opakumar-- V afa type invariants for calabi--yau 4-folds

    Yalong Cao, Davesh Maulik, and Yukinobu Toda. Genus zero G opakumar-- V afa type invariants for calabi--yau 4-folds. Advances in Mathematics , 338:41--92, 2018

    2018

  13. [13]

    Gopakumar--vafa type invariants on calabi--yau 4-folds via descendent insertions

    Yalong Cao and Yukinobu Toda. Gopakumar--vafa type invariants on calabi--yau 4-folds via descendent insertions. Communications in Mathematical Physics , 383(1):281--310, 2021

    2021

  14. [14]

    D T - G V correspondence on the mukai-umemura variety, 2026

    Kiryong Chung and Joonyeong Won. D T - G V correspondence on the mukai-umemura variety, 2026

    2026

  15. [15]

    Rational curves on V _5 and rational simple connectedness

    Andrea Fanelli, Laurent Gruson, and Nicolas Perrin. Rational curves on V _5 and rational simple connectedness. https://arxiv.org/abs/1901.06930, 2019

    Pith/arXiv arXiv 1901

  16. [16]

    The family of lines on the fano threefold V _5

    Mikio Furushima and Noboru Nakayama. The family of lines on the fano threefold V _5 . Nagoya Mathematical Journal , 116:111--122, 1989

    1989

  17. [17]

    Graber and R

    T. Graber and R. Pandharipande. Localization of virtual classes. Invent. Math. , 135(2):487--518, 1999

    1999

  18. [18]

    The fano surface of the G ushel threefold

    Atanas Iliev. The fano surface of the G ushel threefold. Compositio Mathematica , 94(1):81--107, 1994

    1994

  19. [19]

    Iskovskikh and Yuri G

    Vasilii A. Iskovskikh and Yuri G. Prokhorov. Fano varieties , volume 47 of Encyclopaedia Math. Sci. Springer, Berlin, 1999

    1999

  20. [20]

    Klemm and R

    A. Klemm and R. Pandharipande. Enumerative geometry of Calabi-Yau 4-folds . Commun. Math. Phys. , 281:621--653, 2008

    2008

  21. [21]

    Kuznetsov, Yuri G

    Alexander G. Kuznetsov, Yuri G. Prokhorov, and Constantin A. Shramov. Hilbert schemes of lines and conics and automorphism groups of F ano threefolds. Jpn. J. Math. , 13(1):109--185, 2018

    2018

  22. [22]

    Minimal rational threefolds

    Shigeru Mukai and Hiroshi Umemura. Minimal rational threefolds. In Algebraic geometry ( T okyo/ K yoto, 1982) , volume 1016 of Lecture Notes in Math. , pages 490--518. Springer, Berlin, 1983

    1982

  23. [23]

    Fano threefolds of genus 12 and compactifications of C ^3

    Yuri Prokhorov. Fano threefolds of genus 12 and compactifications of C ^3 . Petersburg Math. J. , 3(4):855----864, 1992

    1992

  24. [24]

    Geometries of lines and conics on the quintic del P ezzo 3-fold and its application to varieties of power sums

    Hiromichi Takagi and Francesco Zucconi. Geometries of lines and conics on the quintic del P ezzo 3-fold and its application to varieties of power sums. Michigan Math. J. , 61(1):19--62, 2012

    2012