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Relative node polynomials for plane curves
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Block, Florian (2012) Relative node polynomials for plane curves. Journal of Algebraic Combinatorics, Vol.36 (No.2). pp. 279-308. doi:10.1007/s10801-011-0337-x ISSN 0925-9899.
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Official URL: http://dx.doi.org/10.1007/s10801-011-0337-x
Abstract
The degree of the Severi variety of plane curves of degree d and ¿ nodes is given by a polynomial in d, provided ¿ is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves that, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined "relative node polynomial" in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary ¿, and use it to present explicit formulas for ¿¿6. We also give a threshold for polynomiality, and compute the first few leading terms for any ¿.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Journal of Algebraic Combinatorics | ||||
Publisher: | Springer New York LLC | ||||
ISSN: | 0925-9899 | ||||
Official Date: | 2012 | ||||
Dates: |
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Volume: | Vol.36 | ||||
Number: | No.2 | ||||
Page Range: | pp. 279-308 | ||||
DOI: | 10.1007/s10801-011-0337-x | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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