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If two algebraic plane curves with only ordinary singular points and Cusps are related such that the coordinates of a point on either are Rational Functions of a corresponding point on the other, then the curves have the same Genus (Curve). This can be stated equivalently as the Genus of a curve is unaltered by a Birational Transformation.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 120, 1959.