Agnes Scott College

Elizabeth Morgan Cooper

Perspective Elliptic Curves
American Journal of Mathematics, Vol. 53, No. 3 (July 1931), 555-572


Two curves in a plane are said to be perspective if the points of one and the tangents of the other can be put into one-to-one correspondence in such a way that each point of the one lies on the corresponding tangent of the other. The property is not confined to curves in the plane, (for a discussion of many possible cases of perspective space curves and surfaces, see Segre), but this paper will deal with perspective plane curves only.

A curve perspective to a curve C is a birational transform of C, for if an envelope E is perspective to C a line r of E, incident with the point t of C, will have, in general, a unique point of contact t' with E. Therefore the points of contact of E will be in one-to-one correspondence with the points of C, and their locus C', the dual form of E, will be a birational transform of C. Hence two perspective curves have, necessarily, the same genus. If, on the other hand, C' is a birational transform of C, the joins of corresponding points of C and C' envelope a curve E perspective to both C and C'. Two birational transforms, C' and C", of C will, however, lead to the same envelope perspective to C if it happens that corresponding points of C, C' and C" are collinear.

Some, at least, of the curves perspective to C are obtained as transforms of C by quadratic null systems, i. e., quadratic correlations in which corresponding point and line are incident.

A conic generated by, two perspective points, a cubic generated by a line and conic, a quartic by two conics, or, if it is " singular," by a line and cubic are simple examples of a curve generated by two of its perspective curves. The fact that a curve can be so generated leads to a classification of rational curves by means of their perspective curves. Take, for example, the rational sextic. A rational curve. has ∞2-n+1 perspective m-ics, hence the general rational sextic has ∞1 perspective cubics and is the locus of points of intersection of corresponding lines of any two of these cubics. A condition on a sextic gives it a perspective conic. It can then be generated by that conic and by one of its ∞3 perspective quartics, while a sextic with a five-fold point can be generated by that point and a perspective quintic. There are, then, three types of rational sextics, the classification depending on the curve of lowest class perspective to the sextic. From this point of view perspective curves are important, and it is desirable to know the distribution of curves of given class (or order) perspective to a curve of given order (or class) and given genus. In this paper the number and arrangement of curves perspective to a curve of genus 1 is found.

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