Plücker Formulas

Abstract

The plane curves break into pairs: with each plane curve one can associate a dual curve, the dual of the dual coinciding with the original curve. Usually, the dual of a smooth curve turns out to be singular, hence, studying duality, we cannot content ourselves with considering only smooth curves. Moreover, it does not suffice to consider only curves with singularities of simplest form, points of transversal self-intersection. However, the pairs of dual curves having only points of transversal self-intersection and cusps form an open subset in the space of pairs of dual curves of given degrees, which makes it natural to study such pairs. Plücker formulas are relations on the number of singularities of various types for a pair of dual curves of given degrees.