Positive tangent-bundle Seshadri constants characterize projective space

Abstract

Let X be a smooth integral projective variety over an algebraically closed field of arbitrary characteristic. We prove that positivity of the relative Seshadri constant of T_X at one closed point implies X ≃ P^(dim X), resolving the conjecture of Fulger and Murayama. An absolute minimum of ample degree among rational curves through the specified point gives a proper pointed family without boundary. Its very-free members yield surjective evaluation. A contracted marking section and line-bundle descent force Picard number one and ampleness of −K_X, reducing the problem to the established Fano case. No general-point or characteristic-zero assumption is used. This is an unrefereed preprint developed and checked extensively with AI tools.

The idea in plain English

Projective space is one of algebraic geometry’s basic spaces. A central question is how to recognize it from properties of a space that may initially look quite different.

The tangent bundle records the infinitesimal directions available at each point. Its relative Seshadri constant tests positivity using curves through a chosen point. Although the constant is attached to that point, computing it involves whole curves rather than only a small neighborhood.

The paper proves that a positive value at just one closed point already forces a smooth integral projective variety to be projective space. Earlier results required additional assumptions, such as being Fano, being a surface, or choosing a general point in characteristic zero. This argument retains the chosen point and works in every characteristic.

The precise result

Let X be a smooth integral projective variety of dimension n ≥ 1 over an algebraically closed field k of arbitrary characteristic. If the relative Fulger–Murayama Seshadri constant ε(T_X; x) is positive at one closed point x, then X is isomorphic over k to P^n.

For a vector bundle E, let ρ: P(E) → X parametrize one-dimensional quotients and let ξ = c₁(O_{P(E)}(1)). The relative constant is ε(E; x) = inf_Γ (ξ·Γ)/mult_x(ρ_*Γ), where Γ ranges over integral curves in P(E) whose images are curves containing x. If Γ maps to C with total generic degree q, then ρ_*Γ = q[C], including any inseparable degree.

The theorem establishes the statement of Fulger–Murayama Conjecture 4.9 without assuming that X is Fano, that its tangent bundle is globally nef, that the field has characteristic zero, or that x is a general point.

Why the result holds

Positivity forces the tangent bundle pulled back to the normalization of every rational curve through x to split into positive-degree line bundles. Among these curves, choose one with the smallest degree for a fixed very ample line bundle.

Absolute degree minimality rules out broken curves and multiple covers in the corresponding one-marked stable-map family. Deformation theory makes its evaluation map cover X while its marking section stays at x.

The contracted section lets line bundles descend and forces all numerical divisor classes to be proportional. The anticanonical divisor is then ample, so the established Fano case identifies X with projective space.

Scope and limitations

Smoothness, integrality, projectivity, an algebraically closed base field, and positive dimension are explicit hypotheses.

The constant is the relative Seshadri constant of Fulger and Murayama, with the quotient convention for the projectivized tangent bundle. The result should not be transferred silently to another positivity invariant.

The smoothness hypothesis cannot simply be dropped: Chang gives a singular projective toric threefold with a positive tangent-sheaf Seshadri constant that is not projective space.

The proof uses established rational-curve and moduli theory, and concludes by applying the previously proved Fano case.

Status and AI use

This is an unrefereed preprint. AI tools were used extensively in the research, proof development, computational checks, literature review, and writing. The supporting package contains the published source and verification materials.

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Collaborate on this result

I welcome substantive collaboration on an independent algebraic-geometric review, particularly of the pointed stable-map family in positive characteristic and the numerical descent argument, or on developing a journal submission.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). Positive tangent-bundle Seshadri constants characterize projective space (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23331395

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Background and prior work

• Mihai Fulger and Takumi Murayama, Seshadri constants for vector bundles (2021): Defines the relative invariant, states Conjecture 4.9, and establishes the Fano, surface, and characteristic-zero general-point cases. Its Fano case is an input to this proof.
• Chih-Wei Chang, The Seshadri constants of tangent sheaves on toric varieties (2025): Proves the smooth projective toric case in arbitrary characteristic and gives a singular counterexample to an unrestricted extension.
• Dan Abramovich and Frans Oort, Stable maps and Hurwitz schemes in mixed characteristic (1998): Supplies the stable-map framework used to construct the proper family through the prescribed point in arbitrary characteristic.
• Marco Andreatta and Gianluca Occhetta, Fano manifolds with long extremal rays (2005): Provides context for the established numerical-generation methods based on proper families of rational curves.