The Stacks project

Exercise 111.47.1. Let $k$ be an algebraically closed field. Let $d \geq 1$ and $n \geq 1$. Let us say the moduli of hypersurfaces of degree $d$ in $P^ n$ is given by

  1. an object is a hypersurface $X \subset \mathbf{P}^ n_ k$ of degree $d$,

  2. an isomorphism between two objects $X \subset \mathbf{P}^ n_ k$ and $Y \subset \mathbf{P}^ n_ k$ is an element $g \in \text{PGL}_ n(k)$ such that $g(X) = Y$, and

  3. a family of hypersurfaces over a variety $V$ is a closed subscheme $X \subset \mathbf{P}^ n_ V$ such that for all $v \in V$ the scheme theoretic fibre $X_ v$ of $X \to V$ is a hypersurfaces in $\mathbf{P}^ n_ v$.

Compute (if you can – these get progressively harder)

  1. the number of moduli when $n = 1$ and $d$ arbitrary,

  2. the number of moduli when $n = 2$ and $d = 1$,

  3. the number of moduli when $n = 2$ and $d = 2$,

  4. the number of moduli when $n \geq 1$ and $d = 2$,

  5. the number of moduli when $n = 2$ and $d = 3$,

  6. the number of moduli when $n = 3$ and $d = 3$, and

  7. the number of moduli when $n = 2$ and $d = 4$.

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Your email address will not be published.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