Generalized pencils of conics derived from cubics

Abstract

Given a cubic K. Then for each point P there is a conic $C_P$ associated to P. The conic $C_P$ is called the polar conic of K with respect to the pole P. We investigate the situation when two conics $C_0$ and $C_1$ are polar conics of K with respect to some poles $P_0$ and $P_1$, respectively. First we show that for any point Q on the line $P_0P_1$, the polar conic $C_Q$ of K with respect to Q belongs to the linear pencil of $C_0$ and $C_1$, and vice versa. Then we show that two given conics $C_0$ and $C_1$ can always be considered as polar conics of some cubic K with respect to some poles $P_0$ and $P_1$. Moreover, we show that $P_1$ is determined by $P_0$, but neither the cubic nor the point $P_0$ is determined by the conics $C_0$ and $C_1$.