## IV.5: The Canonical Embedding

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Idea: $K_C$ yields an effective basepoint-free linear system and thus determines a rational map $C\rational \PP^N$, and for $g\geq 3$ this is the **canonical embedding**.

- If $g\geq 2$ then $\abs{K_C}$ is bpf.
  - Proof: STS $h^0(K\sm P) = h^0(K) - 1$ which is $g-2$ in this case since $h^0(K) = h^0(\omega_C) - 1 = g-1$
- $C$ is hyperelliptic iff $C$ admits a linear system of dimension 1 and degree 2.
- $\abs{K_C}$ is very ample iff $X$ is not hyperelliptic.
- Define a **canonical curve** to be the image of $X$ non-hyperelliptic, $g\geq 3$, under the canonical embedding $X\injects \PP^{g-1}$.
  This is a curve of degree $2g-2$.
- Examples showing there exist non-hyperelliptic curves of low genus:
  - For $g = 3$ one has $X\embeds \PP^2$ as a quartic. 
  Conversely, any smooth quartic in $\PP^2$ has $\omega_X\cong \OO_X(1)$ and is thus canonical.
  To show this, take the ideal sheaf LES and show $h^0(\PP^2, \mci(2)) \geq 1$.
  - For $g=4$, $X\embeds \PP^3$ as a degree 6 curve.
  It is contained in a unique irreducible quadric surface $Q_X$ and $X$ is the complete intersection of $Q_X$ with an irreducible cubic surface $F$.
  Conversely, any smooth curve in $\PP^3$ which is a complete intersection of a quadric and cubic has $\deg X = 6$ and $\omega_X \cong \OO_X(1)$ and is thus a canonical curve of genus $4$.
  To show this, take the ideal sheaf LES and show $h^0(\PP^2, \mci(3)) \geq 5$.
- Theorem: $X$ hyperelliptic with $g\geq 2$ has a unique $g_2^1$.
  - The corresponding $X\to \PP^1$ is degree 2 and factors the canonical embedding, and the image in $\PP^{g-1}$ is a rational normal curve of degree $g-1$.
  - One can also write $\abs{K_X} = \sum_{i=1}^{g-1} g_2^1$ since every effective canonical divisor is a sum of $g-1$ divisors in $g_2^1$.
- Clifford's theorem: RR can be used to estimate $h^0(D)$ in terms of degree for nonspecial divisors.
  For *special* divisors, we can at least get a bound:
  - If $D\in \Div(X)$ is effective and special, then $h^0(D) \leq {1\over 2}\deg D$ with equality iff $D= 0$, $D=K_X$, or $X$ is hyperelliptic with $D$ a multiple of the unique $g_2^1$ on $X$.
- Lemma: $h^0(D) + h^0(E) \leq h^0(D+E)$ for $D, E$ effective divisors on a curve.
- Classification of curves: 
  - $g=0\implies X\cong \PP^1$
  - $g=1 \implies X$ is classified by its $j\dash$invariant.
  - $g\geq 2$: difficult! 
- Idea: stratify $\mcm_g$ according to admitting linear systems of specified degrees and dimensions.
  - There are hyperelliptic curves of every $g\geq 2$, and non-hyperelliptic for at least $g=3, 4$ (and in fact for every $g\geq 3$).
  - Can subdivide according to admitting a $g_d^1$ for various $d$.
- A **trigonal curve** is any admitting a $g_3^1$.
- Facts: 
  - $d\geq {1\over 2}g +1 \implies$ any $X$ of genus $g$ has a $g_d^1$.
  - $d\lt {1\over 2}g + 1 \implies$ there exist curves $X$ of genus $g$ have no $g_d^1$.
  - For $g=3,4$, this means there exist nonhyperelliptic curves and they are all trigonal.
  In fact there are infinitely many: the canonical embedding is a plane quartic, projecting away from any point to $\PP^1$ yields a distinct $g_3^1$.
- The $j\dash$invariant gives a coarse space for $\mcm_g$, but no fine space can exist since that are nontrivial families of curves whose fibers are all isomorphic.
- A **coarse moduli space** satisfies: 
  - $\abs{\mcm_g} \mapstofrom\ts{\text{Curves of genus } g}$ is a bijection of sets.
  - For any flat family $X\to T$ of genus $g$ curves, there is a morphism $h: T\to \mcm_g$ where for each $t\in \abs{T}$, the isomorphism class of curves is determined by $h(t)$.
- Deligne-Mumford show $\mcm_g$ for $g\geq 2$ is irreducible quasiprojective of dimension $3g-3$ over any fixed $k=\kbar$.
- Hyperelliptic curves with $g\geq 2$ are determined by 2-fold covers of $\PP^1$ branched along $0,1,\infty$, and $2g-1$ other points up to a finite group quotient.
  This gives an irreducible subvariety of $\mcm_g$ of dimension $2g-1$, which for $g=2$ is the whole space and forces $\mcm_2$ irreducible of dimension 3.
- $\mcm_1 \cong \AA_1$ since the $j\dash$invariant is a rational function in the coefficients of a plane embedding of a curve.
- $\dim \mcm_3 = 6$: non-hyperelliptic curves of genus 3 are nonsingular plane quartic curves.
  Under the canonical embedding, any two such curves are isomorphic iff they differ by the $\PGL_2$ action.
  All such curves are parameterized by $U \subseteq \PP^{14}$ since a degree 4 form has 15 coefficients.
  This yields a morphism $F\injects U \to \mcm_3$ where $\dim F = 8 \implies \dim \im U = 14-8 = 6$.
- Some remarks from the exercises: 
  - Hyperelliptic curves are never complete intersections.
  - $\size \Aut X < \infty$ for $g\geq 2$.
  - $\PGL_2 \surjects \Aut(X)$ for $g = 3$, i.e. every automorphism is induced by an automorphism of $\PP^2$ under the canonical embedding.
  - A general curve of genus $g\geq 3$ has no nontrivial automorphisms.
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