A curve \(C\) of genus \(g \geq 2\) is hyperelliptic if it admits a finite morphism \(C \to \PP^1\) of degree \(2\), that is, a \(g^1_2\).
A hyperelliptic curve has exactly one \(g^1_2\). The canonical system \(\abs{K}\) is base-point free for every \(g \geq 2\), so it always defines a morphism \(C \to \PP^{g-1}\); on a hyperelliptic curve that morphism factors as \[ C \to \PP^1 \to \PP^{g-1} , \] the degree-two map followed by the \((g-1)\)-uple embedding, whose image is a rational normal curve of degree \(g-1\). Equivalently \(\abs{K} = (g-1)\, g^1_2\).
Base-point freeness of \(\abs{K}\) follows from Riemann–Roch: \(\ell(K)=g\), and \(\ell(K-p)=g-1\) because \(\ell(p)=1\) for a non-rational curve. By Riemann–Roch, \(\ell(K - p - q) = \ell(K) - 2\) for points \(p,q\) unless \(\ell(p+q) = 2\), that is, unless \(p+q\) is a divisor of a \(g^1_2\). Hence \(\abs{K}\) is very ample if and only if \(C\) is not hyperelliptic [@Har10a, Proposition IV.5.2].
Every curve of genus \(2\) is hyperelliptic, since \(\abs{K}\) is then a \(g^1_2\). Hyperelliptic curves of genus \(g\) are the double covers of \(\PP^1\) branched at \(2g+2\) points, so they form a locus of dimension \(2g-1\) in \(\mathcal{M}_g\), which has dimension \(3g-3\); the two agree only for \(g=2\).
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