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 curve genus. This can be stated equivalently as the genus of a curve is unaltered by a birational transformation.

# Riemann Curve Theorem

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## References

Coolidge, J. L.*A Treatise on Algebraic Plane Curves.*New York: Dover, p. 120, 1959.

## Referenced on Wolfram|Alpha

Riemann Curve Theorem## Cite this as:

Weisstein, Eric W. "Riemann Curve Theorem."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannCurveTheorem.html