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 $d = 4$.

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