A Minimal-Surface Counterexample to a Seshadri Lower Bound

Abstract

We give an explicit family of counterexamples to the proposed lower bound ε(S,L;x) ≥ (2+|K_S²|^(1/4))⁻¹ for minimal complex projective surfaces. For every integer d ≥ 7, a classical plane-pencil construction followed by a double base change gives a smooth surface with ample canonical bundle, K_S² = 6(d−1)(d−3), and an ample integral line bundle L with L² = 6 and ε(S,L;x) ≤ 1/(d−1). Already d = 7 contradicts the bound. The pencil and covering ingredients are classical; historical priority has not been fully cleared. This unrefereed note used AI tools extensively and has not received human peer review.

The idea in plain English

A Seshadri constant measures how positive a line bundle is near one point of an algebraic surface. One way it becomes small is when a curve has little intersection with the line bundle but many branches through that point.

A proposed lower bound tied this local quantity to the surface’s canonical self-intersection. This paper gives a family that falls below that bound, while keeping the line bundle’s self-intersection equal to 6.

The surfaces are minimal in the algebraic-geometric sense: they contain no smooth rational curve of self-intersection −1. The small value occurs at a deliberately chosen singular point of a curve on the smooth surface.

The precise result

For every integer d ≥ 7, the paper constructs a smooth integral complex projective surface S with ample canonical bundle K_S, an ample integral line bundle L and a point x such that K_S² = 6(d−1)(d−3), L² = 6 and ε(S,L;x) ≤ 1/(d−1) < 1/(2+(K_S²)^(1/4)).

At d = 7, the displayed curve has L-degree 1 and multiplicity 6 at x. Thus ε(S,L;x) ≤ 1/6, while K_S² = 144 and the proposed lower bound is larger than 1/6.

The target is Szemberg’s arbitrary-point bound for minimal surfaces, restated as Conjecture 2 in Oberwolfach Report 21/2009 and Question 6.1.6 of the cited primer. The additive 2 is outside the fourth root.

Why the result holds

The proof starts from Miranda’s classical pencil construction. A finite-degree argument chooses a plane pencil whose fibres are all integral and reduced. Blowing up its simple base points gives a surface Y with fibre F and a section E₁; the divisor A = 2F+E₁ is ample and has degree 1 on the chosen singular fibre.

A double base change, branched at four smooth fibres and away from the chosen singular fibre, produces a smooth surface with ample canonical bundle. The chosen fibre splits into two isomorphic copies, so its degree 1 and multiplicity d−1 survive on either copy.

Pulling back A gives L² = 6. The curve then supplies the stated upper bound directly from the definition of the Seshadri constant, and the strict comparison holds for every d ≥ 7.

Scope and limitations

Only an upper bound for ε(S,L;x) is asserted; the paper does not determine its exact value.

The surfaces vary with d and x is a special point. This does not establish vanishing of the infimum on one fixed complex surface, or contradict very-general-point, Picard-rank-one, canonical or adjoint polarization bounds.

The second pencil member and branch values are chosen through nonempty-open existence arguments; the paper does not list explicit coefficients for every choice.

The pencil, polarization and covering methods are classical. Historical priority and the problem’s status immediately before this note have not been fully established.

This is an unrefereed preprint made with extensive AI assistance. Internal AI checks and finite symbolic controls do not constitute human peer review or a proof-assistant certificate.

Status and AI use

AI tools were used extensively in solving, research, writing, source comparison and internal verification. This work has not received human peer review or proof-assistant certification.

Zenodo labels this deposit version 1.0. The manuscript is dated 14 October 2026 and does not print a numbered version.

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Collaborate on this result

I welcome independent scrutiny from researchers working on algebraic surfaces and local positivity, especially of the construction and its relation to earlier literature. Substantive verification, corrections or extensions could provide a basis for collaboration and a jointly prepared journal submission.

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

Cite this paper

Alec Kriebel. (2026). A Minimal-Surface Counterexample to a Seshadri Lower Bound (Zenodo version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23387814

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Background and prior work

• Tomasz Szemberg, An effective and sharp lower bound on Seshadri constants on surfaces with Picard number 1: The final question on p. 7 asks for a minimal polarized surface below the proposed bound; it is broader than the paper’s Picard-rank-one theorem.
• Tomasz Szemberg, Seshadri constants and geometry of surfaces (2009): Oberwolfach Report 21/2009, p. 1124, Conjecture 2 states the minimal-surface bound for arbitrary points and arbitrary Picard number.
• Ein and Lazarsfeld, Seshadri constants on smooth surfaces (1993): Example 3.1 records Miranda’s classical pencil construction and its small local Seshadri quotient.
• Bauer and Szemberg, On the Seshadri constants of adjoint line bundles: Proposition 2.1 uses the same section-plus-two-fibres polarization on a plane blowup to realize arbitrary positive rational Seshadri values.
• Fuentes García, Cyclic coverings and Seshadri constants on smooth surfaces: Section 2 recalls the classical cyclic-cover canonical-bundle formula used here.