(Genus 2)
Every curve of genus \(2\) is hyperelliptic: \(\deg K = 2\) and \(\ell(K) = 2\), so \(\abs{K}\) is itself the \(g^1_2\). The canonical map is the double cover \(C \to \PP^1\), branched at \(6\) points. \(C\) embeds in \(\PP^3\) by any divisor of degree \(5\), since \(5 \geq 2g+1\).
(Genus 3)
\(\deg K = 4\) and \(\abs{K}\) maps \(C \to \PP^2\). If \(C\) is not hyperelliptic this is an embedding as a smooth plane quartic, and conversely every smooth plane quartic has \(\omega_C \cong \OO_C(1)\) and so is canonical. The hyperelliptic curves of genus \(3\) are the remaining case, a divisor in \(\mathcal{M}_3\) of dimension \(5\) inside \(\dim \mathcal{M}_3 = 6\).
(Genus 4)
\(\deg K = 6\) and \(\abs{K}\) embeds a non-hyperelliptic \(C\) in \(\PP^3\) as a sextic. That curve lies on a unique quadric surface \(Q\) and is the complete intersection of \(Q\) with a cubic surface. Conversely a smooth complete intersection of a quadric and a cubic in \(\PP^3\) has degree \(6\) and \(\omega \cong \OO(1)\), so it is the canonical curve of a genus-\(4\) curve. The quadric \(Q\) is smooth or a cone: on a smooth \(Q\) the curve is of type \((3,3)\) and carries exactly two \(g^1_3\)’s, cut out by the two rulings; on the cone it carries exactly one, cut out by the lines through the vertex.
For a non-hyperelliptic curve, \(\abs{K}\) embeds \(C\) as a curve of degree \(2g-2\) in \(\PP^{g-1}\), and for \(g=3,4\) that image is the plane quartic or the quadric-cubic complete intersection of the genus-3 and genus-4 examples. Counting equations gives the unique quadric in genus \(4\): \(h^0(\PP^3, \OO(2)) = 10\) while \(h^0(C, \OO_C(2)) = 2 \cdot 6 + 1 - 4 = 9\), so the ideal of \(C\) contains a quadric, and it contains only one.
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T-CRVMG$\mathcal{M}_g$ is irreducible of dimension $3g-3$related-toPR-CRVHYPCIA hyperelliptic curve is never a complete intersectionrelated-to