The Stacks project

Exercise 111.47.2. Let $k$ be an algebraically closed field. Let $g \geq 2$. Let us say the moduli of hyperelliptic curves of genus $g$ is given by

  1. an object is a smooth projective hyperelliptic curve $X$ of genus $g$,

  2. an isomorphism between two objects $X$ and $Y$ is an isomorphism $X \to Y$ of schemes over $k$, and

  3. a family of hyperelliptic curves of genus $g$ over a variety $V$ is a proper flat1 morphism $X \to Y$ such that all scheme theoretic fibres of $X \to V$ are smooth projective hyperelliptic curves of genus $g$.

Show that the number of moduli is $2g - 1$.

[1] You can drop this assumption without changing the answer to the question.

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Your email address will not be published.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 0EGQ. Beware of the difference between the letter 'O' and the digit '0'.