The Stacks project

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 $C$?

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09U1. Beware of the difference between the letter 'O' and the digit '0'.



View Exercise 111.56.6 as pdf