Diameters and centers of algebraic hypersurfaces (Q1349446)

The authors generalize the notion of diameter and center of a quadric cone (Appolonius), resp. a plane algebraic curve (Newton) to a projective algebraic hyperplane. They discuss the properties of diameter and center as well as the existence of a unique center of an algebraic hyperplane. Let \(F\) denote the homogeneous polynomial of degree \(d\) defining a hypersurface in \(\mathbb P^n\) over the field of complex numbers. Then the diameter \(\Delta_F(P)\) of a point \(P\) of \(F\) is described by the equation \(\sum F_{x_i}(P) \cdot x_i = 0.\) Then \(M \in \mathbb P^n\) is called a center of \(F\) with respect to a hyperplane \(H\) if \(M\) belongs to any diameter \(\Delta_F(P)\) for any \(P \in H \setminus F.\) Then it is shown that a smooth hypersurface \(F\) of degree \(d \geq 2\) has - with respect to any hyperplane \(H\) - no or exactly one center. This is the point dual to \(P\) with respect to \(F.\) Finally a few examples are discussed. It follows that for a smooth cubic \(F \subset \mathbb P^2\) the set of all centers with respect to suitable lines coincides with the Hesse curve of \(F\) described by the vanishing of the determinant of the Hessean of \(F.\) The intersections of \(F\) with the Hesse curve are the 9 inflection points of \(F.\) In fact they are the centers with respect to the inflection line.