110.47 Moduli

In this section we consider some naive approaches to moduli of algebraic geometric objects.

Let $k$ be an algebraically closed field. Suppose that $M$ is a moduli problem over $k$. We won't define exactly what this means here, but in each exercise it should be clear what we mean. To understand the following it suffices to know what the objects of $M$ over $k$ are, what the isomorphisms between objects of $M$ over $k$ are, and what the families of object of $M$ over a variety are. Then we say the number of moduli of $M$ is $d \geq 0$ if the following are true

1. there is a finite number of families $X_ i \to V_ i$, $i = 1, \ldots , n$ such that every object of $M$ over $k$ is isomorphic to a fibre of one of these and such that $\max \dim (V_ i) = d$, and

2. there is no way to do this with a smaller $d$.

This is really just a very rough approximation of better notions in the literature.

Exercise 110.47.1. 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 $d = 4$.

Exercise 110.47.2. 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.

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).