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Generalized pencils of conics derived from cubics. (English) Zbl 1448.51002

Summary: 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 \(P_0 P_1\), 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 \(C_1\).

MSC:

51A05 General theory of linear incidence geometry and projective geometries
51A20 Configuration theorems in linear incidence geometry
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