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Picard Groups of Hypersurfaces in Toric Varieties

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Shi-shyr Roan

Algebraic Geometry · Submitted 8 Jun 2011

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

The authors' main claim By using the fan structure associated to the ambient toric variety, an explicit basis of the Picard group is described by certain combinatorial data.

how it is stated · asserted 1.00 read the record →

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arXiv ID1106.1615 (version 1)
Pith IDCKCM3M7A, Pith's permanent record of this paper and its review
CitationShi-shyr Roan. "Picard Groups of Hypersurfaces in Toric Varieties." arXiv:1106.1615 (2011). https://pith.science/paper/CKCM3M7A
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,843 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.04 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We study the structure of rational Picard groups of hypersurfaces of toric varieties. By using the fan structure associated to the ambient toric variety, an explicit basis of the Picard group is described by certain combinatorial data. We shall also discuss the application to Calabi-Yau spaces.

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 We study the structure of rational Picard groups of hypersurfaces of toric varieties. contribution 0.95 0.87 0.10 asserted 0.88
s1 By using the fan structure associated to the ambient toric variety, an explicit basis of the Picard group is described by certain combinatorial data. method 0.59 0.86 0.54 asserted 1.00
s2 We shall also discuss the application to Calabi-Yau spaces. future work 0.50 0.50 0.07 asserted 0.51

What kind of paper the classifier reports this is: theoretical result 0.97 · 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

We study the structure of rational Picard groups of hypersurfaces of toric varieties. By using the fan structure associated to the ambient toric variety, an explicit basis of the Picard group is described by certain combinatorial data. We shall also discuss the application to Calabi-Yau spaces.

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1 arXiv paper cite this in 25 sentences

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

Papers citing it · 17 sentences without a confident label

Contests it 1 paper
  • Equivalently, $_Σ_0$ is isomorphic to an abelian quotient of weighted projective space [ref] [Some misprints occur in RIMS journal paper [ref] page 832, where $d_j:= d/n_j$ and $dia.[ e^2πi/d_1, …, e^2πi/ d_5]$ (on line 14, 15) should be $d_j:= d/g.c.d(d, n_j)$, $dia.[ e^2πi n_1/d, …, e^2πi n_5/ d]$ respectively.]: $_Σ_0 ≃_(w_i)^n/G$, with weights $w_i ∈_>0 (1≤i ≤n+1)$, and $G$ a subgroup of the diagonal finite group $SD (:={dia[ t_1, …, t_n+1] | t_i^d_i = ∏_it_i=1 ( 1 ≤i ≤n+1)})$, containing the element $dia [ e^2πi w_1/d, …, e^2πiw_n+1/d] ∈G$, where $d= ∑_i=1^n+1 w_i$ and $d_i = d/g.c.d(d, w_i)$.

    Toric Representation and Positive Cone of Picard Group and Deformation Space in Mirror Symmetry of Calabi-Yau Hypersurfaces in Toric Varieties · Kahler cone of a toric variety · contest 0.82

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