Characterising singular curves in parametrised families of biquadratics: a classification and bifurcation analysis of singular biquadratic curves using discriminants and resultants

摘要

We consider families of biquadratic curves B = 0 on C2, defined with respect to arbitrarilymany complex parameters. We use algebraic methods involving discriminantsto provide a complete classification of the singular curves in these families. This isa key result of the thesis. In assembling the various propositions and corollariesthat underpin the classification, we define a range of conditions in the curve family’sparameters and demonstrate the manner in which they correspond to differentgeometric realisations of the singular curves.We develop the classification by: (1) providing normal form representations ofthe various types of singular curves that fall within it; and (2) considering, in aprojective geometric context, the effect of relaxing an assumption on the degree ofthe discriminants, discry(B) and discrx(B), on which the classification is built.We apply Gröbner basis techniques to various systems of polynomial equationsthat arise in our singular curve manipulations. In addition to strengthening thecomputational aspects of our investigations, our work in this area highlights thecentrality of the role of the so-called double discriminant of B in determining theparametric combinations associated with the singular curves of B = 0.We deepen our engagement with the “atypical” singular curves of our classificationby investigating their bifurcation behaviour. Specifically, we prove that theparameter combinations associated with these curves are themselves singular pointsof the 0-contour of the above-mentioned double discriminant.Finally, we perform a dynamical investigation of a class of discrete dynamicalmappings acting on the non-singular curves in specialised families of biquadratics.