TITLE: How to Compute Genus
QUESTION [8 upvotes]: How to compute the genus of  $ \{X^4+Y^4+Z^4=0\} \cap \{X^3+Y^3+(Z-tW)^3=0\}  \subset \mathbb{P}^3$?
We know that the genus of  $ \{X^4+Y^4+Z^4=0\} \subset \mathbb{P}^3$ is 3 because the degree is 4.
Now, I want to know the genus of the intersection as a curve. For that I have to use the adjunction formula and the fact that $K_{\mathbb{P}^3}=O(-4)$.

REPLY [6 votes]: If $X$ is the the intersection in $\mathbb{P}^3$ of two hypersurfaces of degrees $d_1$, $d_2$, respectively, and $\mathrm{dim} \ X = 1$, then the genus of $X$ is:
$$
g = \frac{d_1^2d_2 + d_1 d_2^2}{2} - 2 d_1 d_2 + 1.
$$
So, in your case $d_1 = 4$ and $d_2 = 3$, therefore $g = 19$.