Exercise 111.56.6. In Situation 111.56.5 assume $d \geq 3$ and that the curve $D$ has exactly one singular point, namely $P = (1 : 0 : 0)$. Assume further that we have the expansion

around $P = (0, 0)$. Then $C$ has two points $v$ and $w$ lying over $P$ characterized by

Show that the element $\omega = \text{d}x/f_ y = - \text{d}y/f_ x$ of $\Omega _{K/k}$ has a first order pole at both $v$ and $w$. (The behaviour of $\omega $ at nonsingular points is as discussed in the lectures.)

In the lectures we have shown that $\omega $ vanishes to order $d - 3$ at the divisor $X_0 = 0$ pulled back to $C$ under the map $C \to D$. Combined with the information of (1) what is the degree of the divisor of zeros and poles of $\omega $ on $C$?

What is the genus of the curve $C$?

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