Genus of a Curve
a number characterizing an algebraic curve. The genus of the nth degree curve f(x, y)= 0 is
where r is the number of double points. When more complex singular points are present, they are counted as the corresponding number of double points; for example, a cusp is counted as one double point, and a triple point is counted as two.
Second-degree curves are of genus 0. Third-degree curves can be of genus 0 or 1. For example, y – x3= 0 is of genus 1. On the other hand, the semicubical parabola y2 – x3 = 0, which has one cusp, is of genus 0, as is the folium of Descartes x3 + y3 – 3axy = 0, which has one double point. Curves of genus 0 are called unicursal curves.
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