Maximal number of Skew lines on Fermat Surfaces

Jacqueline Rojas; Sally Andria; W\'allace Mangueira

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Maximal number of Skew lines on Fermat Surfaces

Sally Andria, Jacqueline Rojas, Wállace Mangueira

Algebraic Geometry · Submitted 23 Mar 2024

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

The authors' main claim In this article we show that the maximal number of skew lines on the Fermat surface of degree $d\geq 4$ is $3d$, either $d$ even or $d$ odd distinct of 5, otherwise ($d=5$) it contains no more than 13 pairwise disjoint lines.

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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, 3,045 input tokens.

What the abstract itself states, as probabilities: claims first 0.19 · states limitation 0.34 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.52 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

It is well-known that the Fermat surface of degree $d\geq 3$ has $3d^2$ lines. However, it has not yet been established what is the maximal number of pairwise disjoint lines that it can have if $d\geq 4$. In this article we show that the maximal number of skew lines on the Fermat surface of degree $d\geq 4$ is $3d$, either $d$ even or $d$ odd distinct of 5, otherwise ($d=5$) it contains no more than 13 pairwise disjoint lines.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	It is well-known that the Fermat surface of degree $d\geq 3$ has $3d^2$ lines.	background 1.00	0.03	0.01	asserted 1.00
s1	However, it has not yet been established what is the maximal number of pairwise disjoint lines that it can have if $d\geq 4$.	gap 1.00	0.04	0.02	asserted 0.90
s2	In this article we show that the maximal number of skew lines on the Fermat surface of degree $d\geq 4$ is $3d$, either $d$ even or $d$ odd distinct of 5, otherwise ($d=5$) it contains no more than 13 pairwise disjoint lines.	result 0.97	0.96	0.98	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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arXiv listing abstract v2

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full text
figures and tables
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P abstract

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S sections
T body sentences
C citations
R cross-paper relations

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

It is well-known that the Fermat surface of degree $d\geq 3$ has $3d^2$ lines. However, it has not yet been established what is the maximal number of pairwise disjoint lines that it can have if $d\geq 4$. In this article we show that the maximal number of skew lines on the Fermat surface of degree $d\geq 4$ is $3d$, either $d$ even or $d$ odd distinct of 5, otherwise ($d=5$) it contains no more than 13 pairwise disjoint lines.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 2 sentences without a confident label

Cites as background 1 paper

For Fermat surfaces over fields with characteristic $p ≠0$, [ref] provides certain bounds for $p=2,3$.

Maximal skew sets of lines on a Hermitian surface and a modified Bron-Kerbosch algorithm · Introduction · 0.83

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