On the G\"ottsche Threshold

Steven L. Kleiman; Vivek V. Shende; with an appendix by Ilya Tyomkin

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On the G\"ottsche Threshold

Steven L. Kleiman, Vivek V. Shende, with an appendix by Ilya Tyomkin

Algebraic Geometry · Submitted 27 Apr 2012

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

The authors' main claim For S rational, we relax the latter condition substantially: it suffices that three key loci be of codimension more than d.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.54 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.23 · compares with prior results 0.79 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

For a line bundle L on a smooth surface S, it is now known that the degree of the Severi variety of cogenus-d curves is given by a universal polynomial in the Chern classes of L and S if L is d-very ample. For S rational, we relax the latter condition substantially: it suffices that three key loci be of codimension more than d. As corollaries, we prove that the condition conjectured by G\"ottsche suffices if S is P^2 or S is any Hirzebruch surface, and that a similar condition suffices if S is any classical del Pezzo surface.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	For a line bundle L on a smooth surface S, it is now known that the degree of the Severi variety of cogenus-d curves is given by a universal polynomial in the Chern classes of L and S if L is d-very ample.	background 1.00	0.07	0.17	asserted 1.00
s1	For S rational, we relax the latter condition substantially: it suffices that three key loci be of codimension more than d.	result 0.63	0.85	0.37	asserted 1.00
s2	As corollaries, we prove that the condition conjectured by G\"ottsche suffices if S is P^2 or S is any Hirzebruch surface, and that a similar condition suffices if S is any classical del Pezzo surface.	result 0.94	0.91	0.45	asserted 1.00

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arXiv listing abstract v2

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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

For a line bundle L on a smooth surface S, it is now known that the degree of the Severi variety of cogenus-d curves is given by a universal polynomial in the Chern classes of L and S if L is d-very ample. For S rational, we relax the latter condition substantially: it suffices that three key loci be of codimension more than d. As corollaries, we prove that the condition conjectured by G\"ottsche suffices if S is P^2 or S is any Hirzebruch surface, and that a similar condition suffices if S is any classical del Pezzo surface.

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5 arXiv papers cite this in 9 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 · 8 sentences without a confident label

Cites it as support 1 paper

The equality at $y = 1$ follows from [ref] .

Refined curve counting on complex surfaces · Introduction · 0.96

What it cites · 7 sentences without a confident label

Cites as support 1 paper

Finally, as each $D ∈∩||^δ$ is immersed, $n_g(D) = 1$ by [ref] plus [ref] .

Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation · Proof of the main results · 0.99

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