Algebraic hyperbolicity of adjoint linear systems on spherical varieties

Haesong Seo; Minseong Kwon

arxiv: 2508.09414 · v2 · submitted 2025-08-13 · 🧮 math.AG

Algebraic hyperbolicity of adjoint linear systems on spherical varieties

Minseong Kwon , Haesong Seo This is my paper

Pith reviewed 2026-05-18 23:43 UTC · model grok-4.3

classification 🧮 math.AG

keywords algebraic hyperbolicityadjoint linear systemsspherical varietiessmooth orbit closureshorospherical varietiestoroidal varieties

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The pith

Very general elements of high-degree adjoint linear systems on spherical varieties with smooth orbit closures are algebraically hyperbolic.

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 spherical varieties with smooth orbit closures, when the dimension is n and an ample line bundle is given, taking m at least three times n plus one makes very general members of the adjoint linear system algebraically hyperbolic. This is significant because algebraic hyperbolicity means the hypersurface contains no rational curves and only curves of sufficiently high genus, providing concrete examples of varieties with restricted curve geometry. The result extends the known cases to a broad class of varieties that include many interesting examples like horospherical and toroidal ones. It also shows a version of the property that holds away from the boundary orbits for more general spherical varieties.

Core claim

For a spherical variety X with smooth orbit closures, an ample line bundle A and an integer m at least three times the dimension plus one, very general elements of the adjoint linear system given by the canonical bundle tensor A to the power m are algebraically hyperbolic.

What carries the argument

Smooth orbit closures on the spherical variety, which allow control of the geometry and application of hyperbolicity criteria to the adjoint linear system.

If this is right

The result applies directly to horospherical varieties.
The result applies directly to toroidal spherical varieties.
For general spherical varieties the hyperbolicity holds outside the complement of an open dense orbit.

Where Pith is reading between the lines

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

Resolving any singularities in orbit closures could extend the result to all spherical varieties without the smoothness assumption.
The method might apply to other varieties with similar decompositions into orbits under group actions.
Direct verification in low dimensions could test if the bound on m is optimal.

Load-bearing premise

The spherical variety has smooth orbit closures.

What would settle it

Finding a spherical variety with smooth orbit closures where for m equal to three times dimension plus one a very general adjoint hypersurface contains a rational curve would disprove the claim.

read the original abstract

Moraga and Yeong conjectured that for a smooth complex projective variety $X$ of dimension $n$, an ample line bundle $A$ on $X$ and an integer $m \ge 3 n + 1$, very general elements of the adjoint linear system $|\omega_{X} \otimes A^{\otimes m}|$ are algebraically hyperbolic. We prove the conjecture for spherical varieties with smooth orbit closures. As a corollary, we conclude that the conjecture holds for horospherical varieties, and for toroidal spherical varieties. Furthermore, for any spherical variety, we show that the conjecture holds modulo the complement of an open dense orbit.

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

Kwon and Seo prove the Moraga-Yeong conjecture for spherical varieties with smooth orbit closures, giving corollaries for horospherical and toroidal cases plus a reduction for general spherical varieties.

read the letter

Kwon and Seo prove the Moraga-Yeong conjecture for spherical varieties whose orbit closures are smooth. This settles the statement in that class and immediately yields it for horospherical varieties and for toroidal spherical varieties. They also show that the conjecture holds for any spherical variety once you remove the complement of an open dense orbit. The smoothness condition is used to control the geometry of the adjoint linear system so that existing hyperbolicity criteria apply to very general members when the multiple is large enough. The argument looks like a direct and honest application of known techniques rather than a major conceptual advance, but it fills a case that had been open. The main limitation is the smoothness hypothesis on the orbit closures. It is stated explicitly, yet it leaves out spherical varieties that fail the condition, so the full conjecture remains open. The abstract gives no sign of circularity or hidden dependence on stronger properties, and the reduction to the open orbit is presented transparently. This paper is for people working on hyperbolicity questions or on the geometry of spherical varieties and their linear systems. A reader following the Moraga-Yeong conjecture or studying adjoint systems on homogeneous spaces would get concrete value from the corollaries. It deserves a serious referee because the claim is precise, the class is natural, and the proof appears to rest on verifiable steps rather than fitting or self-reference.

Referee Report

0 major / 3 minor

Summary. The manuscript proves the Moraga-Yeong conjecture for smooth complex projective spherical varieties X of dimension n with smooth orbit closures: for an ample line bundle A and integer m ≥ 3n + 1, very general members of the adjoint linear system |ω_X ⊗ A^m| are algebraically hyperbolic. Corollaries establish the conjecture for horospherical varieties and toroidal spherical varieties; additionally, for arbitrary spherical varieties the statement holds outside the complement of an open dense orbit.

Significance. If the central arguments hold, the result supplies a clean positive case of the conjecture within the class of spherical varieties, exploiting their reductive group actions and the geometry of orbit closures to reduce hyperbolicity questions to the open orbit. The explicit restriction to smooth orbit closures is transparently stated and permits direct application of existing hyperbolicity criteria, thereby furnishing concrete evidence toward the general statement and identifying a natural subclass where the conjecture is accessible.

minor comments (3)

§1 (Introduction): the statement of the main theorem would benefit from an explicit reminder that the smoothness hypothesis on orbit closures is used only to invoke the hyperbolicity criterion on the closures themselves, rather than on the whole variety.
The proof of the reduction to the open orbit (appearing after the main theorem) should include a short diagram or reference to the stratification by orbit closures to make the induction or localization step easier to follow.
Notation: the symbol for the adjoint linear system is introduced without a displayed equation; adding |K_X + mA| = |ω_X ⊗ A^m| as a displayed line would improve readability for readers outside the spherical-variety literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and recommendation of minor revision. We appreciate the recognition that the result provides a clean positive case of the Moraga-Yeong conjecture by exploiting the geometry of spherical varieties and their orbit closures.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a direct mathematical proof of the Moraga-Yeong conjecture restricted to the explicitly delimited class of spherical varieties with smooth orbit closures. The main theorem states the restriction transparently, invokes it to apply standard hyperbolicity criteria on orbit closures, and derives corollaries for horospherical and toroidal cases plus a modulo-open-orbit statement for general spherical varieties. No self-definitional reductions, fitted parameters renamed as predictions, load-bearing self-citations, or smuggled ansatzes appear; the derivation chain relies on geometric arguments that remain independent of the target statement.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard properties of spherical varieties and algebraic hyperbolicity criteria from prior work; no free parameters or invented entities are introduced in the abstract statement.

axioms (2)

domain assumption Spherical varieties admit a dense orbit under a reductive group action with smooth orbit closures.
This is the structural assumption on the varieties for which the conjecture is proved.
standard math Algebraic hyperbolicity is defined via absence of certain subvarieties in the linear system.
Standard definition invoked from the conjecture statement.

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discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

We prove the conjecture for spherical varieties with smooth orbit closures... using Brion’s classification of B-curves and Frobenius splitting... spherical version of the Nakai criterion

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