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Computing node polynomials for plane curves
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Block, Florian (2011) Computing node polynomials for plane curves. Mathematical Research Letters, Vol.18 (No.4). pp. 621-643. ISSN 1073-2780.
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Abstract
According to the Göttsche conjecture (now a theorem), the degree Nd,δ of the Severi variety of plane curves of degree d with δ nodes is given by a polynomial in d, provided d is large enough. These “node polynomials” Nδ(d) were determined by Vainsencher and Kleiman–Piene for δ≤6 and δ≤8, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute Nδ(d) for δ≤14. Furthermore, we improve the threshold of polynomiality and verify Göttsche’s conjecture on the optimal threshold up to δ≤14. We also determine the first nine coefficients of Nδ(d), for general δ, settling and extending a 1994 conjecture of Di Francesco and Itzykson.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Mathematical Research Letters | ||||
Publisher: | Mathematical Publishing | ||||
ISSN: | 1073-2780 | ||||
Official Date: | 2011 | ||||
Dates: |
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Volume: | Vol.18 | ||||
Number: | No.4 | ||||
Page Range: | pp. 621-643 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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