Definition:Genus of Plane Algebraic Curve

Definition

Let $\CC$ be a plane algebraic curve with no singular points.

The genus of $\CC$ is defined as:

$\dbinom {d - 1} 2$

where $d$ denotes the degree of $\CC$.

Singular Points

Let $\CC$ be a plane algebraic curve which has $1$ or more singular points.

The genus of $\CC$ is defined as:

$\dbinom {d - 2} 2 - \sum \delta$

where:

$d$ denotes the degree of $\CC$

each term of the summation corresponds to one of the singular points of $\CC$

$\delta$ is $1$ for a double point, but for more complicated singular points is larger.

Also see

• Results about genera of plane algebraic curves can be found here.

Linguistic Note

The plural of genus is genera.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): genus: 3.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): genus: 3.