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Observations similar to these can be made at each point of Aj.

Hence **zg must have either a regulus of Af-fold secants or a regulus of Af-fold secants.

Moreover, if Af, no two of the multiple secants can intersect.

For if such were the case, either the plane of the two lines would meet **zg in more than K points or, alternatively, the order of the image regulus of the pencil determined by the two lines would be too high.

But if no two lines of the regulus of multiple secants of **zg can intersect, then the regulus must be quadratic, or in other words, **zg must be either a Af or a Af curve on a nonsingular quadric surface.