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Geometry And Topology

By Hopf H.

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We consider then a complex projective plane P (V ) with a conic C ⊂ P (V ) defined by a non-degenerate symmetric bilinear form B. A tangent to C is a line which meets C at one point. Proposition 12 Let C be a nonsingular conic in a complex projective plane, then 32 • each line in the plane meets the conic in one or two points • if P ∈ C, its polar line is the unique tangent to C passing through P • if P ∈ C, the polar line of P meets C in two points, and the tangents to C at these points intersect at P .

Uip , vj1 , . . , vjq ).

Similarly for [v2 ]. ✷ 33 The picture to bear in mind is the following real one, but even that does not tell the full story, since if P is inside the circle, its polar line intersects it in two complex conjugate points, so although we can draw the point and its polar, we can’t see the two tangents. Quadrics are nonlinear subsets of P (V ) but they nevertheless contain many linear subspaces. For example if Q ⊂ P (V ) is a nonsingular quadric, then P (U ) ⊂ Q if and only if B(u, u) = 0 for all u ∈ U .

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