Polar curves (Q1913963)

Let \(C\) be a plane curve of degree \(d\). For every integer \(t>0\) and every \(Q\in \mathbb{P}^2\) it is defined the polar curve of \(C\) of order \(t\) with respect to \(Q\). The characteristic 0 case was classical. Here the authors extend the general definitions to the positive characteristic case using Hasse derivations. They make a very detailed analysis of the pathologies arising in characteristic \(p\), giving very nice classifications of the examples with ``extremal'' behaviour and the analysis (in terms of \(d\), \(t\) and \(p)\) of the case of a general \(C\) and of a curve \(C\) with ``mild'' singularities. Then they introduce the polar morphisms of plane curves, which generalize the Gauss map, and prove a formula that generalizes Plücker's formula relating the degree of a curve, the degree of its Gauss map, and the degree of its dual curve.