4_5x

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 \(\mathop{\mathrm{Aut}}X\) of automorphisms of \(X\) is finite. 1

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. 2

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:

  • if \(X\) has two \(g_3^1\),s, it can be represented as a plane quintic with two nodes, and conversely;

  • 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.

  • 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 .

  • \(X\) 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. 3

  • * 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. 4

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

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

  • * Assume char \(k \neq 3\). If \(X\) is the curve given by \begin{align*} x^3 y+y^3 z+z^3 x=0 \end{align*} the group \(\mathop{\mathrm{Aut}}X\) is the simple group of order 168 , whose order is the maximum \(84(g-1)\) allowed by (Ex. 2.5). 5

  • * Most curves of genus 3 have no automorphisms except the identity. 6 7
Footnotes
1.
Hint: If \(X\) is hyperelliptic, use the unique \(g_2^1\) and show that \(\mathop{\mathrm{Aut}}X\) permutes the ramification points of the 2 -fold covering \(X \rightarrow \mathbf{P}^1\). If \(X\) is not hyperelliptic, show that \(\mathop{\mathrm{Aut}}X\) permutes the hyperosculation points (Ex. 4.6) of the canonical embedding. Cf. (Ex. 2.5).
2.
Hint: Use (5.2.2) to count how many complete intersections \(Q \cap F_3\) there are.
3.
Hint: If \(D \in g_3^1\), use \(K-D\) to \(\operatorname{map} X \rightarrow \mathbf{P}^2\).
4.
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 \([1]\).
5.
See Burnside \([1, \S 232]\) or Klein \([1]\).
6.
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.
7.
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].
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