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On exceptional collections of line bundles and mirror symmetry for toric Del-Pezzo surfaces

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Yochay Jerby

Algebraic Geometry · Submitted 1 May 2016

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

The authors' main claim Our aim in this work is to describe a map $L : Crit(W) \rightarrow Fuk_{trop}(X^{\circ})$ whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles.

how it is stated · asserted 0.81 read the record →

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arXiv ID1605.00236 (version 2)
Pith ID6MIN6UO7, Pith's permanent record of this paper and its review
CitationYochay Jerby. "On exceptional collections of line bundles and mirror symmetry for toric Del-Pezzo surfaces." arXiv:1605.00236 (2016). https://pith.science/paper/6MIN6UO7
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,002 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Let $X$ be a toric Del-Pezzo surface and let $Crit(W) \subset (\mathbb{C}^{\ast})^n$ be the solution scheme of the Landau-Ginzburg system of equations. Denote by $X^{\circ}$ the polar variety of $X$. Our aim in this work is to describe a map $L : Crit(W) \rightarrow Fuk_{trop}(X^{\circ})$ whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles.

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 Let $X$ be a toric Del-Pezzo surface and let $Crit(W) \subset (\mathbb{C}^{\ast})^n$ be the solution scheme of the Landau-Ginzburg system of equations. background 1.00 0.10 0.02 asserted 0.70
s1 Denote by $X^{\circ}$ the polar variety of $X$. background 0.97 0.16 0.02 asserted 0.70
s2 Our aim in this work is to describe a map $L : Crit(W) \rightarrow Fuk_{trop}(X^{\circ})$ whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles. contribution 0.98 0.70 0.41 asserted 0.81

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 0.97.

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  • references and citing sentences
  • relations to other papers
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  • 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

Let $X$ be a toric Del-Pezzo surface and let $Crit(W) \subset (\mathbb{C}^{\ast})^n$ be the solution scheme of the Landau-Ginzburg system of equations. Denote by $X^{\circ}$ the polar variety of $X$. Our aim in this work is to describe a map $L : Crit(W) \rightarrow Fuk_{trop}(X^{\circ})$ whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles.

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1 arXiv paper cite this in 1 sentence

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 · 6 sentences without a confident label

Cites as background 2 papers
  • In [ref] Abouzaid established an equivalence between $𝒟^b(X)$, the derived category of coherent sheaves of $X$, and a certain full subcategory of $𝒟^π (Fuk( (ℂ^∗)^n, W))$, the derived Fukaya category of a specific mirror model $((ℂ^∗)^n, W)$, affirming Kontsevich's homological mirror symmetry conjecture for toric Fano manifolds, see [ref] .

    Mirror Symmetry · Introduction and Summary of Main Results · 0.81

  • We refer the reader to [ref] where the case of general nef toric stacks is studied.

    Quantum cohomology and toric minimal model programs · Introduction and Summary of Main Results · 0.93

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