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On parameterizations of plane rational curves and their syzygies
Author(s) -
Bernardi A.,
Gimigliano A.,
Idà M.
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201500264
Subject(s) - mathematics , normalization (sociology) , rational surface , plane curve , projection (relational algebra) , plane (geometry) , pure mathematics , ideal (ethics) , quartic plane curve , mathematical analysis , geometry , algorithm , physics , philosophy , plasma , epistemology , quantum mechanics , sociology , anthropology
Let C be a plane rational curve of degree d and p : C ̃ → C be its normalization. We are interested in the splitting type( a , b ) of C , whereO P 1( − a − d ) ⊕ O P 1( − b − d )gives the syzigies of the ideal( f 0 , f 1 , f 2 ) ⊂ K [ s , t ] , and ( f 0 , f 1 , f 2 ) is a parameterization of C . We want to describe in which cases ( a , b ) = ( k , d − k ) ( 2 k ≤ d ) , via a geometric description; namely we show that ( a , b ) = ( k , d − k ) if and only if C is the projection of a rational curve on a rational normal surface in P k + 1 .