Special divisors and the index of speciality (Tag D-CRVSPEC)
Let \(D\) be a divisor on a smooth projective curve \(C\) of genus \(g\). The index of speciality of \(D\) is \(\ell(K-D) = h^1(\OO_C(D))\). \(D\) is special if this is positive, and nonspecial if it vanishes.
If \(\deg D > 2g-2\) then \(D\) is nonspecial, and then Riemann–Roch reads \[ \ell(D) = \deg D + 1 - g . \]
Riemann–Roch expresses \(\ell(D)\) as \(\deg D+1-g+\ell(K-D)\). Once \(\deg D\) exceeds \(\deg K = 2g-2\), the divisor \(K-D\) has negative degree, so \(\ell(K-D)=0\). For every \(D\), Riemann–Roch gives the inequality \(\ell(D) \geq \deg D + 1 - g\).
The canonical divisor \(K\) is special, since \(\ell(K-K)=1\), and on a hyperelliptic curve of genus \(g\ge2\) every divisor of the \(g^1_2\) is special. For a special effective divisor \(D\), Clifford’s theorem gives \(\ell(D)-1\le\frac12\deg D\), with equality if and only if \(D=0\), \(D\sim K\), or \(C\) is hyperelliptic and \(D\) is a multiple of the \(g^1_2\) [@Har10a, Theorem IV.5.4].
Clifford's theorem (Tag T-CRVCLIFF)
Let \(D\) be an effective special divisor on a smooth projective curve \(C\). Then \[ \ell(D) \leq \tfrac{1}{2}\deg D + 1 , \] equivalently \(\dim \abs{D} \leq \tfrac{1}{2}\deg D\). Equality holds exactly when \(D = 0\), or \(D = K\), or \(C\) is hyperelliptic and \(D\) is a multiple of its unique \(g^1_2\).
Clifford is the tool for the range Riemann–Roch cannot reach. For \(\deg D > 2g-2\) the divisor is nonspecial and \(\ell(D)\) is known exactly; for \(0 \leq \deg D \leq 2g-2\) Riemann–Roch gives only a lower bound, and Clifford supplies the upper one.
The proof is the superadditivity \(\ell(D) + \ell(E) \leq \ell(D+E)\) for effective \(D, E\), applied to \(D\) and \(K - D\): it gives \(\ell(D) + \ell(K-D) \leq \ell(K) = g\), and Riemann–Roch turns that into the stated bound.
The equality cases are the content. They say that the only curves carrying unusually large special systems are the hyperelliptic ones, which is why Clifford is the standard first step in proving anything about non-hyperelliptic curves.
Counting maps to \(\PP^1\)
Linear series $g^r_d$, and the gonality of a curve (Tag D-CRVGON)
A \(g^r_d\) on a curve \(C\) is a linear system of degree \(d\) and projective dimension \(r\): a subspace \(V \subseteq H^0(\OO_C(D))\) with \(\deg D = d\) and \(\dim \PP V = r\). The gonality of \(C\) is the least \(d\) for which \(C\) carries a \(g^1_d\), equivalently the least degree of a nonconstant map \(C \to \PP^1\). The curve \(C\) is trigonal if its gonality is \(3\).
Gonality \(1\) means \(C \cong \PP^1\). For genus at least \(2\), gonality \(2\) means \(C\) is hyperelliptic; genus-\(1\) curves also have gonality \(2\). Every curve of genus \(g\) carries a \(g^1_d\) for \(d \geq \lfloor (g+3)/2 \rfloor\), and a general curve of genus \(g\) has no \(g^1_d\) below this bound.
A base-point-free \(g^r_d\) determines a morphism \(\phi\colon C\to\PP^r\) with \(\deg\phi^*\OO(1)=d\); for \(r=1\) this is a map \(C\to\PP^1\) of degree \(d\). The canonical system on a curve of genus \(g\ge2\) is a \(g^{g-1}_{2g-2}\).
For \(g=3\) and \(g=4\), \(\lfloor (g+3)/2\rfloor=3\), so every non-hyperelliptic curve of genus \(3\) or \(4\) is trigonal. A smooth plane quartic has a \(g^1_3\) for each of its points \(p\): the lines through \(p\) cut out \(p\) plus three further points.
In \(g^r_d\) notation, a \(g^1_2\) is a degree-two map to \(\PP^1\), a \(g^1_3\) makes the curve trigonal, and the canonical system on a non-hyperelliptic curve is a \(g^{g-1}_{2g-2}\).
Hyperelliptic curves and their canonical maps (Tag D-CRVHYP)
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 \(g=2\).
For a hyperelliptic curve \(C\) of genus \(g\ge2\) with double cover \(\pi\colon C\to\PP^1\), the canonical map is \(\pi\) followed by the degree-\((g-1)\) Veronese embedding \(\PP^1\to\PP^{g-1}\): it has degree two onto a rational normal curve.
Low-genus canonical models
The canonical models in genus $2$, $3$, and $4$ (Tag FE-CRVLOWG)
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.
A non-hyperelliptic curve of genus \(3\) is canonically a smooth plane quartic, and one of genus \(4\) is canonically the intersection of a quadric and a cubic in \(\PP^3\). Every curve of genus \(2\) is hyperelliptic, so none has a canonical embedding.
Complete intersections are not hyperelliptic
A hyperelliptic curve is never a complete intersection (Tag PR-CRVHYPCI)
Let \(C \subseteq \PP^n\) be a smooth curve of genus \(g \geq 2\) which is a complete intersection of hypersurfaces of degrees \(d_1,\ldots,d_{n-1}\). Then \(C\) is not hyperelliptic.
The reason is that a complete intersection has its canonical class forced to be a multiple of the hyperplane class, and a positive multiple of the hyperplane class is very ample.
Adjunction on the complete intersection gives \[ \omega_C \cong \OO_C(a), \qquad a = \sum_{i=1}^{n-1} d_i - n - 1 , \] and \(\deg \OO_C(1) = \prod d_i =: d > 0\). So \(2g-2 = ad\), and the hypothesis \(g \geq 2\) forces \(a \geq 1\).
Now \(\OO_{\PP^n}(a)\) is very ample for \(a \geq 1\), so the linear system it cuts on \(C\) separates points and tangent vectors. That cut system is a subsystem of the complete system \(\abs{\OO_C(a)} = \abs{K_C}\), and a linear system containing a very ample subsystem is itself very ample. Hence \(K_C\) is very ample, so the canonical map is an embedding, so \(C\) is not hyperelliptic. (Complete intersections are projectively normal, so in fact the cut system is all of \(\abs{K_C}\): the canonical embedding of \(C\) is the composite of \(C \subseteq \PP^n\) with the \(a\)-uple embedding. The weaker subsystem statement is already enough for the conclusion.)
The contrapositive is what the exercise asks for: a hyperelliptic curve embeds in projective space in many ways, but never as a complete intersection in any \(\PP^n\). Two checks against curves one can name. Genus \(2\) is entirely hyperelliptic, and such a curve embeds in \(\PP^3\) by a divisor of degree \(5\); it cannot be a complete intersection there, and indeed the only factorization \(5 = d_1 d_2\) is \(1 \cdot 5\), which would place the curve in a plane as a quintic of genus \(6\). Genus \(4\) runs the other way: the non-hyperelliptic canonical curve is the intersection of a quadric and a cubic in \(\PP^3\), with \(a = 2+3-3-1 = 1\) and \(\omega_C \cong \OO_C(1)\), exactly as the formula predicts for a canonically embedded complete intersection.
The canonical curve of genus \(4\) is the complete intersection of a quadric and a cubic in \(\PP^3\), with \(\omega_C\cong\OO_C(2+3-3-1)=\OO_C(1)\).
Brill–Noether theory
The Brill--Noether number, and Brill--Noether general curves (Tag D-CRVBRILLNOETHER)
Let \(C\) be a smooth projective curve of genus \(g\) over an algebraically closed field, and let \(r \geq 0\) and \(d\) satisfy \(g - d + r \geq 0\). The locus \(W^r_d(C) \subseteq \Pic^d(C)\) of line bundles \(L\) with \(h^0(C, L) \geq r + 1\) is a closed subscheme, cut out locally as a degeneracy locus of a map of vector bundles. Its expected dimension is the Brill–Noether number \[\rho(g, r, d) = g - (r+1)(g - d + r).\] The curve \(C\) is Brill–Noether general if for every such \(r\) and \(d\), \(W^r_d(C)\) is empty when \(\rho(g,r,d) < 0\) and has dimension \(\rho(g,r,d)\) when \(\rho(g,r,d) \geq 0\).
(Brill--Noether)
Let \(g\), \(r\), \(d\) be as above.
If \(\rho(g,r,d) \geq 0\), then every smooth projective curve of genus \(g\) has a \(g^r_d\), and every component of \(W^r_d(C)\) has dimension at least \(\rho(g,r,d)\) (Kempf, Kleiman–Laksov).
A general curve of genus \(g\) is Brill–Noether general (Griffiths–Harris).
For a general curve and \(\rho(g,r,d) \geq 0\), \(W^r_d(C) \setminus W^{r+1}_d(C)\) is smooth of dimension \(\rho(g,r,d)\) (Gieseker–Petri).
A \(g^1_d\) has \(\rho(g,1,d) = 2d - g - 2\). So every curve of genus \(g\) has a map of degree \(\lceil (g+2)/2 \rceil\) to \(\PP^1\), and the general curve has no map of smaller degree: its gonality is \(\lfloor (g+3)/2 \rfloor\). For \(d = 2\), \(\rho = 2 - g\), which is negative for \(g \geq 3\); this is why every curve of genus \(2\) is hyperelliptic while the general curve of genus \(g \geq 3\) is not.