Primer: Automorphisms of algebraic curves and moduli space of curves

Here we will introduce some well-known results about the automorphisms of algebraic curves over an algebraic closed field $k$ and their relations between the moduli space of curves.

For $g=0$

If $X$ be a proper smooth curve of genus $0$ over an algebraic closed field $k$, then one can easy to see $X\cong\mathbb{P}^1$ as any two points in $X$ are linear equivalent (consider Jacobian variety). Then we can easy to see that $\mathrm{Aut}(X)=\mathrm{PGL}_2$. Moreover, if we consider the subgroup fixed $n$ points in $X$, then this subgroup is finite if and only if $n\geq 3$ by easy linear algebra.

For $g=1$

Let $X$ be a proper smooth curve of genus $1$ over an algebraic closed field $k$. As $X$ is a group variety, then any closed point in $X$ can act on $X$ which forms an automorphism. Hence $\mathrm{Aut}(X)$ is also an infinity group! But if we consider the subgroup fixed $1$ points in $X$, that is, the automorphism group of elliptic curves, then we have:

Theorem 1. Let $X$ be a proper smooth elliptic curve over an algebraic closed field $k$ for $\mathrm{char}(k)\neq 2$. Pick $p\in X$, then $\mathrm{Aut}(X;p)$ is finite of order: $2$ if $j\neq 0,1728$; $4$ if $j=1728$ and $\mathrm{char}(k)\neq 3$; $6$ if $j=0$ and $\mathrm{char}(k)\neq 3$; $12$ if $j=0=1728$ and $\mathrm{char}(k)=3$.

Proof. Omitted, see [Har77]¹ Corollary IV.4.7. $\blacksquare$

For $g\geq 2$

Let $X$ be a proper smooth curve of genus $g\geq 2$ over an algebraic closed field $k$. We have the following well-known result:

Theorem 2. The group $\mathrm{Aut}(X)$ is finite.

Proof 1. Deformation theory. We need two following lemmas:

Lemma A. If $X$ be a proper smooth curve, then $\underline{\mathrm{Aut}}(X)\to\mathrm{Spec}k$ is unramified if and only if $\mathrm{Der}_k(\mathscr{O}_X,\mathscr{O}_X)=0$.

Proof of Lemma A. Actually $\underline{\mathrm{Aut}}(X)\to\mathrm{Spec}k$ is unramified if and only if $\Omega_{\underline{\mathrm{Aut}}(X)/k}=0$ if and only if $T_{\underline{\mathrm{Aut}}(X)/k}=0$ if and only if any automorphism $\alpha:X\times_k\mathrm{Spec}k[\varepsilon]\to X\times_k\mathrm{Spec}k[\varepsilon]$ over $\mathrm{Spec}k[\varepsilon]$ whose restriction to $\mathrm{Spec}k$ is $id$. By St 0DY9 this if and only if $\mathrm{Der}_k(\mathscr{O}_X,\mathscr{O}_X)=0$. $\blacksquare$

Lemma B. If $X$ be a proper smooth curve of genus $g\geq 2$ over an algebraic closed field $k$, then $\mathrm{Der}_k(\mathscr{O}_X,\mathscr{O}_X)=0$.

Proof of Lemma B. As $\deg (T_{X/k})=2-2g<0$, we get

\[\mathrm{Der}_k(\mathscr{O}_X,\mathscr{O}_X)=\hom_{\mathscr{O}_X}(\Omega_{X/k},\mathscr{O}_X)=\Gamma(X,T_{X/k})=0.\]

Well done. $\blacksquare$

Then the proof finish by Lemma A and B. $\blacksquare$

Proof 2. Using ramification and Weierstrass points (for charactristic zero).

Proof 3. Using graph trick and numerical equivalence.

Moreover, for $\mathrm{char}(k)=0$, we have:

Theorem 3. If $X$ be a proper smooth curve of genus $g\geq 2$ over an algebraic closed field $k$ with $\mathrm{char}(k)=0$, then $\sharp(\mathrm{Aut}(X))\leq 84g − 84$.

Proof.

Moduli space of curves