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Obrifold del Pezzo surfaces in $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$ format

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Muhammad Imran Qureshi

Algebraic Geometry · Submitted 28 Dec 2023

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

The authors' main claim We construct two types of wellformed and quasismooth biregular models (infinite series) of rigid orbifold del Pezzo surfaces having their (sub) anti-canonical embeddings in $\mathbb P^6(w_i) $.

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arXiv ID2312.17339 (version 2)
Pith IDWUEBIAON, Pith's permanent record of this paper and its review
CitationMuhammad Imran Qureshi. "Obrifold del Pezzo surfaces in $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$ format." arXiv:2312.17339 (2023). https://pith.science/paper/WUEBIAON
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,529 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.44 · compares with prior results 0.05 · mentions code or data availability 0.02.

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

We construct two types of wellformed and quasismooth biregular models (infinite series) of rigid orbifold del Pezzo surfaces having their (sub) anti-canonical embeddings in $\mathbb P^6(w_i) $. One type of model contains a family of rigid del Pezzo surfaces with a fixed Fano index and weights of ambient $\mathbb P^6(w_i)$ are parameterized by positive integers. In the other type of models, weights of $\mathbb P^6(w_i)$ and Fano index, both are parameterized by the positive integers. The equations describing their images under (sub) anti-canonical embeddings are given in terms of the equations of the Segre embedding of $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$, which has codimension 4 in $\mathbb P^7$. We also give a formula for the Hilbert series of a generic weighted $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$ variety, a key tool in these constructions.

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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 construct two types of wellformed and quasismooth biregular models (infinite series) of rigid orbifold del Pezzo surfaces having their (sub) anti-canonical embeddings in $\mathbb P^6(w_i) $. contribution 0.98 0.96 0.72 asserted 1.00
s1 One type of model contains a family of rigid del Pezzo surfaces with a fixed Fano index and weights of ambient $\mathbb P^6(w_i)$ are parameterized by positive integers. contribution 0.75 0.92 0.15 asserted 1.00
s2 In the other type of models, weights of $\mathbb P^6(w_i)$ and Fano index, both are parameterized by the positive integers. contribution 0.83 0.90 0.14 asserted 1.00
s3 The equations describing their images under (sub) anti-canonical embeddings are given in terms of the equations of the Segre embedding of $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$, which has codimension 4 in $\mathbb P^7$. method 0.59 0.86 0.16 asserted 1.00
s4 We also give a formula for the Hilbert series of a generic weighted $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$ variety, a key tool in these constructions. contribution 0.60 0.95 0.08 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · 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 construct two types of wellformed and quasismooth biregular models (infinite series) of rigid orbifold del Pezzo surfaces having their (sub) anti-canonical embeddings in $\mathbb P^6(w_i) $. One type of model contains a family of rigid del Pezzo surfaces with a fixed Fano index and weights of ambient $\mathbb P^6(w_i)$ are parameterized by positive integers. In the other type of models, weights of $\mathbb P^6(w_i)$ and Fano index, both are parameterized by the positive integers. The equations describing their images under (sub) anti-canonical embeddings are given in terms of the equations of the Segre embedding of $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$, which has codimension 4 in $\mathbb P^7$. We also give a formula for the Hilbert series of a generic weighted $\mathbb P^1 \times \mathbb P^1\times \mathbb P^1$ variety, a key tool in these constructions.

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