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Tangent cones to generalised theta divisors and generic injectivity of the theta map

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George H. Hitching, Michael Hoff

Algebraic Geometry · Submitted 11 Oct 2016

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

The authors' main claim We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map $SU_C (r) -rightarrow |r \Theta|$ is generically injective for large values of $g$.

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arXiv ID1610.03456 (version 2)
Pith IDR52MUXYF, Pith's permanent record of this paper and its review
CitationGeorge H. Hitching, Michael Hoff. "Tangent cones to generalised theta divisors and generic injectivity of the theta map." arXiv:1610.03456 (2016). https://pith.science/paper/R52MUXYF
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,134 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.65 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.92 · compares with prior results 0.89 · mentions code or data availability 0.02.

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

Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^0 (C, E) = r+1$. For $g > (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the theta divisor $\Theta_E$ at ${\mathcal O}_C$. We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map $SU_C (r) -rightarrow |r \Theta|$ is generically injective for large values of $g$.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^0 (C, E) = r+1$. background 0.99 0.12 0.02 asserted 0.64
s1 For $g > (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the theta divisor $\Theta_E$ at ${\mathcal O}_C$. result 0.92 0.95 0.60 asserted 1.00
s2 We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map $SU_C (r) -rightarrow |r \Theta|$ is generically injective for large values of $g$. contribution 0.59 0.93 0.65 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 Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^0 (C, E) = r+1$. For $g > (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the theta divisor $\Theta_E$ at ${\mathcal O}_C$. We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map $SU_C (r) -rightarrow |r \Theta|$ is generically injective for large values of $g$.

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