If two projective Pencils of curves of orders and have no common curve, the Locus of the intersections of corresponding curves of the two is a curve of order through all the centers of either Pencil. Conversely, if a curve of order contains all centers of a Pencil of order to the multiplicity demanded by Noether's Fundamental Theorem, then it is the Locus of the intersections of corresponding curves of this Pencil and one of order projective therewith.
See also Noether's Fundamental Theorem, Pencil
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 33, 1959.