Complex Curves

Abstract

In all this welter of original work on the geometry of curves, one matter has remained stubbornly unclear, although to a modern mathematician the need for it is painfully apparent. Is the subject the algebraic geometry of real curves in the real projective plane, or has everything migrated to the complex projective plane? In a sense, the question is anachronistic. It does not arise for Poncelet because, as we saw, he had his own way of talking about imaginary points and resisted very strongly Cauchy’s suggestion of making everything algebraic. But Plücker and Hesse were avowedly algebraic, yet they do not seem to have openly confronted the issue. At one stage, Plücker even outlined a way in which complex coordinates could be written out of the theory in favour of certain symmetry consideration (a topic there is not room to explore here). Generally speaking, he and Hesse seem to have pursued a policy of quiet acceptance: intersection points of one curve with another, tangents to a curve, and all manner of objects may be complex if the algebra forces them to be so — but one will not explore too closely. This is a curious position. A cubic curve, let us say, was thought of as a real object in the real plane with nine inflection points, six of which at least were necessarily imaginary. A cubic and a quartic curve meet in 12 points, of which in any given case quite a number might be complex. But a curve was not made up of complex points — points with complex coordinates — or if it was one did not enquire too closely how this could be so.