Jim Pettigrew and John A G Roberts 2008 J. Phys. A: Math. Theor. 41 115203 doi:10.1088/1751-8113/41/11/115203
Characterizing singular curves in parametrized families of biquadratics
Jim Pettigrew and John A G Roberts
Show affiliationsWe consider families of biquadratic curves B = 0 on 
, defined with respect to arbitrarily many complex parameters. Due to the fact that these families include curve intersections across different parameter combinations, they represent a generalization of the non-intersecting foliations of one-parameter invariant curves associated with the QRT mapping. We use algebraic methods involving discriminants to provide a complete classification of the singular curves in these families. In developing this classification, we exploit the special symmetric nature of B; namely, that it is a quadratic in x and y whose reflection in the line y = x is given by a simple change of parameters. We also define a range of conditions in the biquadratic's parameters and demonstrate the manner in which they correspond to different geometric realizations of the singular curves.
- Dates
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Issue 11 (21 March 2008)
Received 8 November 2007, in final form 27 November 2007
Published 4 March 2008
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Characterizing singular curves in parametrized families of biquadratics
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