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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.

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View Exercise 111.47.2 as pdf