## IV.5: The Canonical Embedding

### IV.5.1.  #to-work

Show that a hyperelliptic curve can never be a complete intersection in any projective space. Cf. (Ex. 3.3).

### IV.5.2.  #to-work

If $X$ is a curve of genus $\geqslant 2$ over a field of characteristic 0 , show that the group $\Aut X$ of automorphisms of $X$ is finite.[^hint.4.5.2]

[^hint.4.5.2]: 
Hint: If $X$ is hyperelliptic, use the unique $g_2^1$ and show that $\Aut X$ permutes the ramification points of the 2 -fold covering $X \rightarrow \mathbf{P}^1$. If $X$ is not hyperelliptic, show that $\Aut X$ permutes the hyperosculation points (Ex. 4.6) of the canonical embedding. Cf. (Ex. 2.5).

### IV.5.3.  Moduli of Curves of Genus 4. #to-work

The hyperelliptic curves of genus 4 form an irreducible family of dimension 7 . The nonhyperelliptic ones form an irreducible family of dimension 9. The subset of those having only one $g_3^1$ is an irreducible family of dimension 8.[^hint.4.5.3.moduli]

[^hint.4.5.3.moduli]: 
Hint: Use (5.2.2) to count how many complete intersections $Q \cap F_3$ there are.

### IV.5.4.  #to-work

Another way of distinguishing curves of genus $g$ is to ask, what is the least degree of a birational plane model with only nodes as singularities (3.11)? Let $X$ be nonhyperelliptic of genus 4 . Then:

a. if $X$ has two $g_3^1$,s, it can be represented as a plane quintic with two nodes, and conversely;

b. if $X$ has one $g_3^1$, then it can be represented as a plane quintic with a tacnode (I, Ex. 5.14d), but the least degree of a plane representation with only nodes is 6 .

### IV.5.5. Curves of Genus 5. #to-work

Assume $X$ is not hyperelliptic.

a. The curves of genus 5 whose canonical model in $\mathbf{P}^4$ is a complete intersection $F_2 . F_2 . F_2$ form a family of dimension 12 .

b. $X$ has a $g_3^1$ if and only if it can be represented as a plane quintic with one node. These form an irreducible family of dimension 11.[^4.5.5.hint]

c. \* In that case, the conics through the node cut out the canonical system (not counting the fixed points at the node). Mapping $\mathbf{P}^2 \rightarrow \mathbf{P}^4$ by this linear system of conics, show that the canonical curve $X$ is contained in a cubic surface $V \subseteq \mathbf{P}^4$, with $V$ isomorphic to $\mathbf{P}^2$ with one point blown up (II, Ex. 7.7). 

    Furthermore, $V$ is the union of all the trisecants of $X$ corresponding to the $g_3^1(5.5 .3)$, so $V$ is contained in the intersection of all the quadric hypersurfaces containing $X$. Thus $V$ and the $g_3^1$ are unique.[^rmk.4.5.5.c]

[^rmk.4.5.5.c]: 
Note. Conversely, if $X$ does not have a $g_3^1$, then its canonical embedding is a complete intersection, as in (a). More generally, a classical theorem of Enriques and Petri shows that for any nonhyperelliptic curve of genus $g \geqslant 3$, the canonical model is projectively normal, and it is an intersection of quadric hypersurfaces unless $X$ has a $g_3^1$ or $g=6$ and $X$ has a $g_5^2$. See Saint-Donat $[1]$.

[^4.5.5.hint]: 
Hint: If $D \in g_3^1$, use $K-D$ to $\operatorname{map} X \rightarrow \mathbf{P}^2$.

### IV.5.6.  #to-work

Show that a nonsingular plane curve of degree 5 has no $g_3^1$. Show that there are nonhyperelliptic curves of genus 6 which cannot be represented as a nonsingular plane quintic curve.

### IV.5.7.  #to-work

a. Any automorphism of a curve of genus 3 is induced by an automorphism of $\mathbf{P}^2$ via the canonical embedding.

b. \* Assume char $k \neq 3$. If $X$ is the curve given by
\[
x^3 y+y^3 z+z^3 x=0
\]
the group $\Aut X$ is the simple group of order 168 , whose order is the maximum $84(g-1)$ allowed by (Ex. 2.5).[^4.5.7.reference]

c. \* Most curves of genus 3 have no automorphisms except the identity.[^4.5.7.c.hint][^4.5.7.note]

[^4.5.7.reference]: 
See Burnside $[1, \S  232]$ or Klein $[1]$.

[^4.5.7.note]: 
More generally it is true (at least over $\mathbf{C}$ ) that for any $g \geqslant 3$, a "sufficiently general" curve of genus $g$ has no automorphisms except the identity-see Baily [1].

[^4.5.7.c.hint]: 
Hint: For each $n$, count the dimension of the family of curves with an automorphism $T$ of order $n$. For example, if $n=2$, then for suitable choice of coordinates, $T$ can be written as $x \rightarrow-x, y \rightarrow y, z \rightarrow z$. Then there is an 8-dimensional family of curves fixed by $T$; changing coordinates there is a 4-dimensional family of such $T$, so the curves having an automorphism of degree 2 form a family of dimensional 12 inside the 14-dimensional family of all plane curves of degree 4.