On the differential geometry of holomorphic plane curves

Source abstract

We consider the geometry of regular holomorphic curves in C 2 \mathbb C^2 viewed as surfaces in the affine space R 4 \mathbb R^4 . We study the A \mathcal A -singularities of parallel projections of generic such surfaces along planes to transverse planes. We show that at any point on the surface which is not an inflection point of the curve there are two tangent directions determining two planes along which the projection has singularities of type butterfly or worse. The integral curves of these directions form a pair of foliations on the surface defined by a binary differential equation (BDE). The singularities of this BDE are the inflection points of the curve together with other points that we call butterfly umbilic points . We determine the configurations of the solution curves of the BDE at its singularities. Finally, we prove that an affine view of an algebraic curve of degree d ≥ 2 d\ge 2 in C P 2 \mathbb CP^2 has 8 d ( d − 2 ) 8d(d-2) butterfly umbilic points.