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

$f(x, y) = xy + h.o.t$

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

$v(x) = 1, v(y) > 1 \quad \text{and}\quad w(x) > 1, w(y) = 1$

1. 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.)

2. 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 $C$?

3. What is the genus of the curve $C$?

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).