If \(X\) is a curve of genus 2, show that a divisor \(D\) is very ample \(\iff \deg D \geq 5\). This strengthens (3.3.4).
Solution
Write \[ \ell(D)=h^0\bigl(X,\mcl(D)\bigr). \] Since \(g(X)=2\), a canonical divisor \(K\) has degree \[ \deg K=2g-2=2. \]
1. If \(\deg D\geq5\), then \(D\) is very ample.
Corollary IV.3.2(b) states that on a curve of genus \(g\), every divisor of degree at least \(2g+1\) is very ample. Here \[ 2g+1=5, \] so every divisor of degree at least \(5\) is very ample.
2. If \(D\) is very ample, then \(\deg D>0\) and \(\ell(D)\geq3\).
A very ample divisor is ample, so Corollary IV.3.3 gives \[ \deg D>0. \]
The complete linear system \(|D|\) gives a closed immersion \[ X\longrightarrow\PP^{\ell(D)-1}. \] If \(\ell(D)=1\), there is no nonconstant morphism. If \(\ell(D)=2\), the target is \(\PP^1\); a closed irreducible one-dimensional subscheme of \(\PP^1\) is \(\PP^1\) itself, which has genus \(0\). Since \(g(X)=2\), neither case can occur. Thus \[ \ell(D)\geq3. \]
3. A very ample divisor on \(X\) cannot have degree \(1\).
A very ample divisor has a nonzero global section, so after replacing \(D\) by a linearly equivalent effective divisor we may write \[ D=P \] for some point \(P\in X\). The exact sequence \[ 0\longrightarrow\OO_X \longrightarrow\mcl(P) \longrightarrow\mcl(P)|_P \longrightarrow0 \] has a one-dimensional quotient on global sections at most. Since \(h^0(X,\OO_X)=1\), it follows that \[ \ell(D)\leq2, \] contradicting step 2.
4. A very ample divisor on \(X\) cannot have degree \(2\).
Riemann-Roch gives \[ \ell(D)-\ell(K-D)=\deg D+1-g=1. \] Now \[ \deg(K-D)=0. \] A degree-\(0\) line bundle has at most one nonzero global section: if it has one, the corresponding effective divisor has degree \(0\) and is therefore zero, so the line bundle is trivial. Hence \[ \ell(K-D)\leq1 \] and therefore \[ \ell(D)\leq2, \] again contradicting step 2.
5. A very ample divisor on \(X\) cannot have degree \(3\).
Here \[ \deg(K-D)=-1, \] so \[ \ell(K-D)=0. \] Riemann-Roch yields \[ \ell(D)=\deg D+1-g=3+1-2=2, \] contradicting step 2.
6. A very ample divisor on \(X\) cannot have degree \(4\).
Again \[ \deg(K-D)=-2, \] so \(\ell(K-D)=0\). Riemann-Roch gives \[ \ell(D)=4+1-2=3. \] Thus, if \(D\) were very ample, its complete linear system would give a closed immersion \[ X\hookrightarrow\PP^2. \] For this embedding, \[ \mcl(D)\cong\iota^*\OO_{\PP^2}(1), \] so the degree of the image is \(\deg D=4\). Hence \(X\) would be isomorphic to a nonsingular plane quartic.
By Exercise I.7.2, the genus formula for a nonsingular plane curve of degree \(4\) gives \[ g=\frac{(4-1)(4-2)}2=3, \] contradicting \(g(X)=2\).
7. If \(D\) is very ample, then\[ \boxed{\deg D\geq5}. \]
Step 2 gives \(\deg D>0\), and steps <1>3–<1>6 exclude the four possible positive degrees below \(5\).
8. Therefore\[ \boxed{D\text{ is very ample}\iff\deg D\geq5}. \]
Step 1 proves the reverse implication, and step 7 proves the forward implication.
9. Q.E.D.
Step 8 is exactly the required equivalence.