Pointed Evaluation Fibers of Rational Curves on del Pezzo Manifolds

Ari Krishna

arxiv: 2606.30605 · v1 · pith:YF3ASMXRnew · submitted 2026-06-29 · 🧮 math.AG

Pointed Evaluation Fibers of Rational Curves on del Pezzo Manifolds

Ari Krishna This is my paper

Pith reviewed 2026-06-30 03:05 UTC · model grok-4.3

classification 🧮 math.AG

keywords del Pezzo manifoldKontsevich moduli spacerational curveevaluation morphismgeometric irreducibilitystable mapPicard rank one

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The pith

The generic fiber of the one-pointed evaluation map from the moduli space of rational curves on a Picard-rank-one del Pezzo manifold to the manifold is geometrically irreducible for every degree d at least 1.

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

The paper proves that for a Picard-rank-one del Pezzo manifold X of dimension n at least 4 over an algebraically closed field of characteristic zero, the evaluation morphism from the one-pointed Kontsevich space of degree-d stable maps to X has geometrically irreducible generic fiber in every degree d at least 1. In the cases where the anticanonical class is very ample, the two-pointed evaluation morphism to the product X times X likewise has geometrically irreducible generic fiber for all d at least 2. This refines the known irreducibility of the unpointed moduli spaces by controlling how the curves behave when one or two marked points are fixed. A reader would care because the result determines whether the families of rational curves passing through one or two general points remain irreducible in the moduli space.

Core claim

For every d greater than or equal to 1 the one-pointed evaluation morphism from the moduli space of stable maps of degree d with one marked point to X has geometrically irreducible generic fiber. In the very ample cases where the self-intersection of the hyperplane class equals 3, 4 or 5, the two-pointed evaluation morphism from the two-marked moduli space to X times X has geometrically irreducible generic fiber for every d greater than or equal to 2.

What carries the argument

The one-pointed and two-pointed evaluation morphisms on the Kontsevich moduli spaces of stable maps of degree d.

If this is right

The unpointed moduli spaces remain irreducible after fixing one or two general points.
Families of rational curves through one or two general points on X form irreducible components in the moduli space.
The result gives a pointed refinement of the irreducibility theorem for unpointed Kontsevich spaces on these manifolds.

Where Pith is reading between the lines

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

The irreducibility would imply that the locus of degree-d rational curves through a general point is itself irreducible.
Similar fiber-irreducibility statements might hold for other Fano manifolds if the same deformation techniques apply.
The result could be used to study the birational geometry of the moduli spaces themselves when points are fixed.

Load-bearing premise

X must be a Picard-rank-one del Pezzo manifold of dimension at least 4 over an algebraically closed field of characteristic zero.

What would settle it

An explicit example of such an X and some d where the generic fiber of the evaluation map over a general point of X (or pair of points) has more than one irreducible component would falsify the claim.

read the original abstract

Let $X$ be a Picard-rank-one del Pezzo manifold of dimension $n\geq 4$ over an algebraically closed field of characteristic zero. Okamura proved that the unpointed Kontsevich spaces $\overline{M}_{0,0}(X,d)$ are irreducible of the expected dimension for every $d\geq 1$. We refine this result by studying pointed evaluation fibers. First, we prove that for every $d\geq 1$, the one-pointed evaluation morphism $\overline{M}_{0,1}(X,d)\to X$ has geometrically irreducible generic fiber. Second, in the very ample cases $H^n=3,4,5$, we prove that for every $d\geq 2$, the two-pointed evaluation morphism $\overline{M}_{0,2}(X,d)\to X\times X$ has geometrically irreducible generic fiber.

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

This paper refines Okamura's unpointed irreducibility result to show geometric irreducibility of generic fibers for the one-pointed evaluation map on all such del Pezzo manifolds and for the two-pointed map in the low-degree very ample cases.

read the letter

The core advance is extending the known irreducibility of the unpointed Kontsevich spaces to the generic fibers of the evaluation morphisms at one or two points. The one-pointed statement holds for every d ≥ 1 on any Picard-rank-one del Pezzo manifold of dimension at least 4 in characteristic zero. The two-pointed statement is restricted to the cases where the ample generator has self-intersection 3, 4, or 5 and d ≥ 2. Both are stated as geometric irreducibility over the algebraically closed base field.

This is a direct, incremental refinement rather than a wholesale new direction. It uses the unpointed result as the base and adds the pointed data without introducing visibly new hypotheses on the moduli spaces themselves. The setup stays inside the standard framework for these Fano threefolds and higher-dimensional del Pezzo manifolds.

The main limitation is that the abstract alone does not let us inspect the actual arguments. We cannot yet see how the pointed fibers are handled—whether by deformation theory, by reducing to the unpointed case via forgetful maps, or by some other technique. If the proofs turn out to be short and clean, the paper will be a useful reference for anyone computing enumerative invariants or studying rational curve families on these varieties. If they rely on heavy case-by-case analysis or unstated boundedness results, the contribution shrinks.

The work is aimed at specialists in moduli of stable maps to Fano varieties. A reader already familiar with Okamura's paper will get the most out of it. The claims are internally consistent with the cited background and do not appear to overreach. I would send it to a referee who knows the unpointed case well; the referee can check whether the pointed extensions are as straightforward as they look.

Referee Report

0 major / 2 minor

Summary. The paper refines Okamura's theorem on the irreducibility of unpointed Kontsevich spaces ČM_{0,0}(X,d) for Picard-rank-one del Pezzo manifolds X of dimension n≥4 (char 0, algebraically closed field). It proves that the generic fiber of the one-pointed evaluation morphism ČM_{0,1}(X,d)→X is geometrically irreducible for every d≥1, and that in the very ample cases H^n=3,4,5 the generic fiber of the two-pointed evaluation morphism ČM_{0,2}(X,d)→X×X is geometrically irreducible for every d≥2.

Significance. If the results hold, the refinement supplies geometrically irreducible generic fibers for pointed evaluation maps, strengthening the geometric understanding of rational curve moduli on del Pezzo manifolds and potentially supporting further work on rational connectedness or deformation theory in this setting. The work directly extends a prior irreducibility theorem without adding visibly fragile hypotheses.

minor comments (2)

[Introduction] §1 (Introduction): the expected dimension of ČM_{0,1}(X,d) is stated only implicitly via the unpointed case; an explicit formula citing the degree of the ample generator H would clarify the fiber dimension claims.
The transition from Okamura's unpointed result to the pointed generic-fiber statements would benefit from a short paragraph outlining the precise deformation-theoretic or valuative criterion used to pass from irreducibility of the total space to geometric irreducibility of the generic fiber.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript, the positive assessment of its significance as a refinement of Okamura's theorem, and the recommendation for minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation refines Okamura's independent unpointed irreducibility theorem (different author) to geometrically irreducible generic fibers of pointed evaluation morphisms. The abstract and claim structure present the pointed results as direct extensions without any reduction of the new statements to the prior theorem by definition, fitting, or self-citation chain. No self-definitional steps, fitted inputs renamed as predictions, ansatzes smuggled via citation, or renaming of known results appear in the provided derivation outline. The setup uses standard hypotheses on del Pezzo manifolds and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available; no free parameters or invented entities visible. Relies on standard domain assumptions of algebraic geometry and a cited prior theorem.

axioms (2)

domain assumption The base field is algebraically closed of characteristic zero.
Explicitly stated in the abstract as the setting.
domain assumption X is a Picard-rank-one del Pezzo manifold of dimension n≥4.
Defines the class of varieties under study.

pith-pipeline@v0.9.1-grok · 5669 in / 1285 out tokens · 77781 ms · 2026-06-30T03:05:40.124443+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references

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F. Okamura,The irreducibility of the spaces of rational curves on del Pezzo manifolds, Int. Math. Res. Not. IMRN2024, no. 12, 9893–9909. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA Email address:akrishna@college.harvard.edu 12

[1] [1]

Abdelkerim and I

R. Abdelkerim and I. Coskun,Parameter spaces of Schubert varieties in hyperplane sections of Grass- mannians, J. Pure Appl. Algebra216(2012), no. 4, 800–810

2012

[2] [2]

Coskun and J

I. Coskun and J. Starr,Rational curves on smooth cubic hypersurfaces, Int. Math. Res. Not. IMRN2009, no. 24, 4626–4641

[3] [3]

Fulton and R

W. Fulton and R. Pandharipande,Notes on stable maps and quantum cohomology, Algebraic geometry– Santa Cruz 1995, Proc. Sympos. Pure Math.62, Part 2, Amer. Math. Soc., 1997, 45–96

1995

[4] [4]

Harris, M

J. Harris, M. Roth, and J. Starr,Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571(2004), 73–106

2004

[5] [5]

V. A. Iskovskikh and Yu. G. Prokhorov,Fano varieties, Algebraic Geometry V, Encyclopaedia Math. Sci. 47, Springer, 1999

1999

[6] [6]

Kollár,Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3

J. Kollár,Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 32, Springer, 1996

1996

[7] [7]

Lehmann and S

B. Lehmann and S. Tanimoto,Geometric Manin’s conjecture and rational curves, Compos. Math.155 (2019), no. 5, 833–862

2019

[8] [8]

Okamura,The irreducibility of the spaces of rational curves on del Pezzo manifolds, Int

F. Okamura,The irreducibility of the spaces of rational curves on del Pezzo manifolds, Int. Math. Res. Not. IMRN2024, no. 12, 9893–9909. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA Email address:akrishna@college.harvard.edu 12