Let \(C\) be a smooth projective curve of genus \(g\) over \(k = \bar{k}\). The canonical divisor \(K\) is very ample exactly when \(g \geq 3\) and \(C\) is not hyperelliptic.
(The criterion it comes from)
For every divisor \(D\) and closed point \(p\) on \(C\), \(\ell(D) - 1 \leq \ell(D - p) \leq \ell(D)\). A divisor \(D\) on \(C\) is very ample if and only if \[ \ell(D - p - q) = \ell(D) - 2 \quad \text{for all points } p, q , \] including \(p = q\). It is base-point free if and only if \(\ell(D - p) = \ell(D) - 1\) for all \(p\). Any \(D\) with \(\deg D \geq 2g+1\) is very ample. [@Har10a, Proposition IV.3.1, Corollary IV.3.2]
The criterion says the linear system separates points and tangent vectors, which is exactly what an embedding must do. The inequalities come from the exact sequence \(0 \to \mcl(D - p) \to \mcl(D) \to k(p) \to 0\), with \(k(p)\) the skyscraper sheaf at \(p\), whose global sections drop by at most \(\dim_k k(p) = 1\). Applied to \(K\) and using duality, \(\ell(K-p-q) = \ell(K) - 2\) fails precisely when \(\ell(p+q) = 2\), that is, when there is a degree-two map to \(\PP^1\) — the definition of hyperelliptic.
The excluded cases are genuinely excluded: for \(g = 0\), \(\deg K < 0\); for \(g = 1\), \(K = 0\); for \(g = 2\), every curve is hyperelliptic and \(K\) gives the degree-two map rather than an embedding. For \(g \geq 3\) non-hyperelliptic, \(K\) embeds \(C\) as a curve of degree \(2g-2\) in \(\PP^{g-1}\), the canonical curve, which for \(g = 3\) is a smooth plane quartic and for \(g = 4\) a curve of degree \(6\) in \(\PP^3\).
Dependencies
T-MWDVLRiemann--Roch and Serre duality on a curveuses
Backlinks
PR-DIVAMPCURVEAmpleness of a divisor on a curverelated-toT-DIVMAPPNThe morphism to projective space attached to a linear systemrelated-toD-CRVHYPHyperelliptic curves and their canonical mapsusesFE-CRVLOWGThe canonical models in genus $2$, $3$, and $4$uses