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arxiv: 2606.05881 · v1 · pith:LYAORHBInew · submitted 2026-06-04 · 🧮 math.AG

Non-projective complete log canonical surfaces

Osamu Fujino , Nao Moriyama , Hiroshi Sato This is my paper

Pith reviewed 2026-06-27 23:43 UTC · model grok-4.3

classification 🧮 math.AG
keywords log canonical surfacesnon-projective varietiessemi-ample canonical divisorKodaira dimensioncomplete algebraic surfacesalgebraic geometry
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The pith

Non-projective complete log canonical surfaces with semi-ample canonical divisors exist for every non-negative Kodaira dimension.

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

The paper constructs complete log canonical algebraic surfaces that fail to be projective yet have semi-ample canonical divisors, working over algebraically closed fields in any characteristic except the algebraic closure of a finite field. It supplies one framework that produces examples for Kodaira dimension zero, one, or two. It also proves that any complete log canonical surface with Kodaira dimension minus infinity must be projective. Together these results show that non-projective examples occur precisely when the Kodaira dimension is non-negative.

Core claim

We construct non-projective complete log canonical algebraic surfaces whose canonical divisors are semi-ample over an algebraically closed field of any characteristic other than the algebraic closure of a finite field. We provide a unified framework to construct such surfaces for any given non-negative Kodaira dimension, namely, zero, one, or two. Furthermore, we show that any complete log canonical algebraic surface with Kodaira dimension minus infinity is automatically projective.

What carries the argument

A unified framework that produces complete log canonical surfaces with semi-ample canonical divisor while deliberately avoiding projectivity for each non-negative Kodaira dimension.

If this is right

  • Non-projective examples exist for Kodaira dimensions zero, one, and two in all allowed characteristics.
  • Projectivity is forced exactly when Kodaira dimension equals minus infinity.
  • The constructions cover every possible Kodaira dimension value for non-projective complete log canonical surfaces.
  • The same surfaces have semi-ample canonical divisors.

Where Pith is reading between the lines

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

  • The projectivity theorem may extend to related classes of surfaces with milder singularities.
  • Similar non-projective examples could appear in higher dimensions once analogous frameworks are found.
  • The characteristic restriction suggests that finite-field phenomena may force projectivity in special cases.

Load-bearing premise

Such surfaces can be built without the characteristic or field condition forcing projectivity when the Kodaira dimension is non-negative.

What would settle it

Either a complete log canonical algebraic surface with Kodaira dimension minus infinity that is non-projective, or the inability to produce the required non-projective examples for one of the non-negative Kodaira dimensions in an allowed characteristic.

read the original abstract

We construct non-projective complete log canonical algebraic surfaces whose canonical divisors are semi-ample over an algebraically closed field of any characteristic other than the algebraic closure of a finite field. We provide a unified framework to construct such surfaces for any given non-negative Kodaira dimension, namely, zero, one, or two. Furthermore, we show that any complete log canonical algebraic surface with Kodaira dimension minus infinity is automatically projective. This projectivity result confirms that our construction covers all possible values for the Kodaira dimension of non-projective complete log canonical surfaces.

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

0 major / 3 minor

Summary. The paper constructs non-projective complete log canonical algebraic surfaces with semi-ample canonical divisors over algebraically closed fields of characteristic not equal to the algebraic closure of a finite field, for each Kodaira dimension κ=0,1,2 via a unified framework. It additionally proves that every complete log canonical surface with κ=-∞ is projective, thereby showing that the constructions cover all possible Kodaira dimensions for non-projective examples.

Significance. If the constructions and projectivity theorem hold, the work supplies explicit examples realizing non-projective lc surfaces in every non-negative Kodaira dimension and establishes a clean projectivity criterion for the remaining case. This completes the Kodaira-dimension picture for non-projective lc surfaces and supplies a unified method that may be useful for further classification questions in the minimal model program.

minor comments (3)
  1. The abstract states the characteristic restriction but does not indicate whether the constructions rely on any specific vanishing theorems or base-change arguments that might fail in positive characteristic; a brief remark in the introduction clarifying the characteristic dependence would help readers.
  2. Notation for the unified framework (e.g., the parameters or gluing data used to produce the surfaces for each κ) should be introduced consistently in §2 or §3 so that the three cases can be compared directly.
  3. The projectivity theorem for κ=-∞ is stated without an explicit reference to the relevant vanishing or contraction criterion used in its proof; adding a pointer to the standard result (e.g., Kawamata-Viehweg vanishing or the contraction theorem) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. The recommendation for minor revision is noted. No specific major comments were provided in the report, so we have no points requiring detailed rebuttal at this stage. We will incorporate any minor suggestions during revision.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results consist of explicit constructions of non-projective log canonical surfaces with semi-ample canonical divisors for each non-negative Kodaira dimension, together with a separate projectivity theorem when κ = −∞. These are presented as new geometric constructions resting on standard log canonical surface theory over algebraically closed fields (excluding algebraic closures of finite fields). No equations, definitions, or self-citations are shown to reduce the claimed existence or projectivity statements to quantities defined in terms of themselves or to prior fitted parameters from the same authors. The derivation chain therefore remains self-contained against external benchmarks in algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger records the explicit domain assumptions stated there plus standard background results in algebraic geometry that any such paper must invoke. No numerical free parameters or new postulated entities appear in the abstract.

axioms (2)
  • domain assumption The base field is algebraically closed of characteristic other than the algebraic closure of a finite field.
    Explicitly stated in the abstract as the setting in which the constructions and projectivity theorem hold.
  • standard math Standard results on log canonical pairs and the minimal model program for surfaces remain valid in the given characteristic range.
    Implicit background required for any statement about log canonical surfaces and semi-ampleness.

pith-pipeline@v0.9.1-grok · 5609 in / 1578 out tokens · 34040 ms · 2026-06-27T23:43:55.278328+00:00 · methodology

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Reference graph

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11 extracted references · 7 canonical work pages

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