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A rationality criterion for projective surfaces - partial solution to Kollar's conjecture

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JongHae Keum

Algebraic Geometry · Submitted 7 Oct 2005

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We confirm the conjecture under the additional condition that the exceptional divisor in a minimal resolution of $S$ has at most 3 components over each singular point of $S$.

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arXiv IDmath/0510137 (version 3)
Pith IDHYYV3T7T, Pith's permanent record of this paper and its review
CitationJongHae Keum. "A rationality criterion for projective surfaces - partial solution to Kollar's conjecture." arXiv:math/0510137 (2005). https://pith.science/paper/HYYV3T7T
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,369 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.94 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.83 · compares with prior results 0.15 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

Koll\'ar's conjecture states that a complex projective surface $S$ with quotient singularities and with $H^2(S,\bbQ)\cong \bbQ$ should be rational if its smooth part $S^0$ is simply connected. We confirm the conjecture under the additional condition that the exceptional divisor in a minimal resolution of $S$ has at most 3 components over each singular point of $S$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Koll\'ar's conjecture states that a complex projective surface $S$ with quotient singularities and with $H^2(S,\bbQ)\cong \bbQ$ should be rational if its smooth part $S^0$ is simply connected. background 1.00 0.03 0.05 asserted 0.62
s1 We confirm the conjecture under the additional condition that the exceptional divisor in a minimal resolution of $S$ has at most 3 components over each singular point of $S$. result 0.90 0.95 0.94 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Koll\'ar's conjecture states that a complex projective surface $S$ with quotient singularities and with $H^2(S,\bbQ)\cong \bbQ$ should be rational if its smooth part $S^0$ is simply connected. We confirm the conjecture under the additional condition that the exceptional divisor in a minimal resolution of $S$ has at most 3 components over each singular point of $S$.

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