Plane curves whose tangent lines at collinear points are concurrent (Q1345126)

The author investigates the special geometry of a plane curve of degree \(d\) over an algebraically closed field of characteristic \(p>0\). Such curve is called strange, if all general tangent lines of the curve are concurrent. The author introduces a weaker condition (CC): for any general line and its \(d\) different (thus necessarily smooth) intersection points with the curve, all \(d\) tangent lines to the curve in these points are concurrent. The author proves that a curve which satisfies (CC) must either be strange, or be projectively equivalent to Fermat's curve \(X_0^{q+1}+ X_1^{q+1}+ X_2^{q+1} =0\) where \(q\) is a power of \(p\). The \textit{if} part is easy calculation. The \textit{only if} part is proved in two steps: the general case \(d\geq 4\) where the author uses some facts on non-reflexive curves, and the special case \(d=3\) dealed with by direct calculation.