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Coherent systems and Brill-Noether theory

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Steven Bradlow, Oscar Garcia-Prada, Vicente Muñoz, Peter Newstead

Algebraic Geometry · Submitted 30 May 2002

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

The authors' main claim The study of coherent systems is applied to find the dimension, irreducibility, and in some cases, the Picard group, of the Brill-Noether loci with $k\leq 3$.

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arXiv IDmath/0205317 (version 1)
Pith IDRUDOG5V7, Pith's permanent record of this paper and its review
CitationOscar Garcia-Prada, Peter Newstead, Steven Bradlow, Vicente Mu\~noz. "Coherent systems and Brill-Noether theory." arXiv:math/0205317 (2002). https://pith.science/paper/RUDOG5V7
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,605 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.27 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.07 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Let $C$ be a curve of genus $g\geq 2$. A coherent system on $C$ consists of a pair $(E,V)$ where $E$ is an algebraic vector bundle of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of sections of $E$. The stability of the coherent systems depend on a parameter $\tau$. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyse the cases $k=1,2,3$ and $n=2$ explicitly. For small values of $\tau$, the moduli space of coherent systems is related to the Brill-Noether loci, the subspaces of the moduli space of stable bundles consisting of those bundles with a prescribed number of sections. The study of coherent systems is applied to find the dimension, irreducibility, and in some cases, the Picard group, of the Brill-Noether loci with $k\leq 3$.

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 $C$ be a curve of genus $g\geq 2$. background 1.00 0.05 0.01 asserted 0.89
s1 A coherent system on $C$ consists of a pair $(E,V)$ where $E$ is an algebraic vector bundle of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of sections of $E$. background 1.00 0.05 0.02 asserted 0.99
s2 The stability of the coherent systems depend on a parameter $\tau$. background 0.99 0.09 0.03 asserted 1.00
s3 We study the variation of the moduli space of coherent systems when we move the parameter. contribution 0.99 0.93 0.08 asserted 0.94
s4 As an application, we analyse the cases $k=1,2,3$ and $n=2$ explicitly. contribution 0.85 0.94 0.07 asserted 1.00
s5 For small values of $\tau$, the moduli space of coherent systems is related to the Brill-Noether loci, the subspaces of the moduli space of stable bundles consisting of those bundles with a prescribed number of sections. result 0.74 0.30 0.18 asserted 1.00
s6 The study of coherent systems is applied to find the dimension, irreducibility, and in some cases, the Picard group, of the Brill-Noether loci with $k\leq 3$. result 0.63 0.89 0.49 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

Let $C$ be a curve of genus $g\geq 2$. A coherent system on $C$ consists of a pair $(E,V)$ where $E$ is an algebraic vector bundle of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of sections of $E$. The stability of the coherent systems depend on a parameter $\tau$. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyse the cases $k=1,2,3$ and $n=2$ explicitly. For small values of $\tau$, the moduli space of coherent systems is related to the Brill-Noether loci, the subspaces of the moduli space of stable bundles consisting of those bundles with a prescribed number of sections. The study of coherent systems is applied to find the dimension, irreducibility, and in some cases, the Picard group, of the Brill-Noether loci with $k\leq 3$.

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