When the canonical divisor is very ample (Tag T-D8TUX)
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 \(\PP^3\).
In the criterion, \(p\neq q\) is separation of points and \(p=q\) is separation of tangent vectors at \(p\). Every \(D\) with \(\deg D\ge2g+1\) is very ample; the canonical divisor, of degree \(2g-2\), is very ample exactly when \(g\ge3\) and \(C\) is not hyperelliptic.
If \(g\ge2\), \(\pi\colon C\to\PP^1\) has degree two, and \(p+q\) is a fibre, then \(K\sim(g-1)(p+q)\) and \(\ell(K-p-q)=g-1=\ell(K)-1\), so \(\abs{K}\) does not separate \(p\) and \(q\), and the canonical map factors through \(\pi\).
The Jacobian
The Jacobian, Abel's theorem, and the Abel--Jacobi map (Tag T-U5QSY)
For \(X\) a smooth projective curve of genus \(g\) over \(\CC\), \[ \Jac(X) = \dualof{H^0(X,\Omega^1)} / H_1(X,\ZZ) , \] a complex torus of dimension \(g\), and an abelian variety. The Abel–Jacobi map sends a degree-zero divisor \(\sum (p_i - q_i)\) to \(\sum \int_{q_i}^{p_i}\), a linear functional on \(H^0(\Omega^1)\) taken modulo periods.
(Abel)
A divisor of degree zero is principal exactly when its image under the Abel–Jacobi map is zero. Hence \(\Pic^0(X) \cong \Jac(X)\).
(Jacobi inversion)
The Abel–Jacobi map is surjective, so \(\Jac(X)\) is exactly the group of degree-zero divisor classes.
The significance is the sentence the two theorems combine to: the Jacobian is a projective variety whose points are the degree-zero line bundles on \(X\), so a discrete-looking classification problem becomes a geometric object of dimension \(g\).
Fixing a base point \(p_0\) gives \(X \to \Jac(X)\), \(p \mapsto [p - p_0]\), which is an embedding for \(g \geq 1\). In genus \(1\) it is an isomorphism: an elliptic curve is its own Jacobian, and the group law on \(E\) is the statement that \([p]+[q]=[r]+[0]\) exactly when \(p+q=r\). Everything special about elliptic curves is this coincidence of \(X\) with \(\Jac X\).
By Abel’s theorem and Jacobi inversion, the Abel–Jacobi map induces an isomorphism of groups \(\Pic^0(X)\cong\Jac(X)=H^0(X,\Omega^1)^\vee/H_1(X,\ZZ)\), a \(g\)-dimensional complex torus that is an abelian variety.
In genus one, \(p\mapsto[p-p_0]\) is an isomorphism \(E\to\Jac(E)\), and it carries the group law of \((E,p_0)\) to addition in \(\Jac(E)\).
The period construction is analytic and applies to curves over \(\CC\).
The Jacobian represents $\Pic^0(X/-)$, and is a smooth proper group scheme of dimension $g$ (Tag T-CRVJACFUN)
Let \(X/k\) be a smooth projective curve of genus \(g\) with a point \(p_0\), and let \(T \in \Sch^{\mathrm{ft}}_{/k}\) with second projection \(p \colon X \times T \to T\). Put \[ \Pic^0(X \times T) \da \ts{ \mcf \in \Pic(X \times T) \st \deg \ro{\mcf}{X_t} = 0 \ \forall t \in T }, \qquad \Pic^0(X/T) \da \Pic^0(X \times T) \, / \, p^* \Pic(T) , \] the families of degree-zero line bundles on \(X\) parameterised by \(T\).
(Representability)
There is a \(k\)-scheme \(\Jac(X)\) of finite type together with a universal \(\mcl \in \Pic^0\big(X / \Jac(X)\big)\) such that for every \(T\) and every \(\mcm \in \Pic^0(X/T)\) there is a unique \(f \colon T \to \Jac(X)\) with \(f^* \mcl = \mcm\). That is, \(\Jac(X)\) represents \(\Pic^0(X/{-})\). Moreover:
On \(k\)-points, \(\Jac(X)(k) = \Pic^0(X)\).
\(\Jac(X)\) is a group scheme over \(k\): the identity is the class of \(\OO_X\), inversion is \(\mcl \mapsto \mcl^{-1}\), and addition is \(p_1^*\mcl \tensor p_2^*\mcl\).
\(\T_0 \Jac(X) \iso H^1(X; \OO_X)\), so \(\Jac(X)\) is smooth of dimension \(g\).
\(\Jac(X)\) is proper over \(k\), hence an abelian variety.
For \(X = E\) elliptic, \(E \iso \Jac(E)\).
(The symmetric product)
For each \(n\) the assignment \[ \phi^n \colon X^{\times n} \to \Jac(X), \qquad (p_1, \ldots, p_n) \mapsto \mcl\big( \textstyle\sum p_i - n p_0 \big) \] is a morphism, symmetric in its arguments, so it factors through \(\Sym^n X\). It is surjective for \(n \geq g\), and the fibre of \(\Sym^n X \to \Jac(X)\) over the class of \(D\) is the complete linear system \(\abs{D} \iso \PP^{\ell(D)-1}\).
(Representability and rigidification)
Two features of the functor control representability.
The quotient by \(p^*\Pic(T)\) is not a technicality: without it the functor is not representable, because a line bundle on \(X \times T\) can be twisted by one pulled back from the base without changing any fibre, so the assignment \(T \mapsto \Pic^0(X\times T)\) has automorphisms and cannot have a universal object. Quotienting kills exactly that ambiguity. The same issue reappears as the reason a universal family needs a rigidification, and it is why \(\Jac\) is a fine moduli space while \(\AA^1\) for elliptic curves is only coarse.
That \(k\)-points give \(\Pic^0(X)\) is then formal: a \(k\)-point is a map \(\spec k \to \Jac(X)\), which by the universal property is an element of \(\Pic^0(X/k) = \Pic^0(X)\). Thus \(\Jac(X)\) upgrades the set \(\Pic^0(X)\) of degree-zero line-bundle classes to a smooth proper group scheme representing families of such classes.
(Smooth, of dimension $g$, and proper)
Dual numbers identify the tangent space directly. A tangent vector at \(0\) is a map \(T = \spec k[\eps]/\eps^2 \to \Jac(X)\) sending the closed point to \(0\), hence a class in \(\Pic^0(X/T)\) restricting to \(0\) over \(\spec k\). The exponential sequence for dual numbers gives \[ 0 \to H^1(X; \OO_X) \to \Pic\big(X[\eps]\big) \to \Pic(X) \to 0 , \] so those classes are exactly \(H^1(X;\OO_X)\), of dimension \(g\). Translation by group elements propagates smoothness from the identity to every point.
Properness is the valuative criterion applied to a DVR \(R\) with fraction field \(K\): one must extend a line bundle on \(X \times \spec K\) over \(X \times \spec R\). But \(X \times \spec R\) is regular, so a Weil divisor extends and is automatically Cartier, and the extension is unique. Irreducibility comes from the other direction, via \(\phi^n\): for \(n \geq g\) Riemann–Roch makes every degree-\(n\) class effective, so \(\phi^n\) is surjective from an irreducible variety, and the generic fibre is finite because a general divisor of degree \(g\) is nonspecial with \(\ell(D) = 1\). Hence \(\dim \Jac(X) = g\) by a second, independent route.
(Comparison with the analytic construction)
Over \(\CC\) one can write \(\Jac(X) = \dualof{H^0(\Omega^1)} / H_1(X,\ZZ)\) and prove Abel and Jacobi inversion by integration; that construction produces a complex torus and needs GAGA to become algebraic. The functorial construction above needs no ground-field hypothesis and therefore applies in characteristic \(p\) and to families, where the analytic construction is unavailable. The two agree over \(\CC\), and the agreement is Abel’s theorem.
In genus \(1\) the two theories coincide: \(\phi^1 \colon X \to \Jac(X)\) is an isomorphism, so an elliptic curve carries its group law directly and statements about \(\Pic^0\) become statements about points of the curve.
The functorial construction applies over any field and in families. In \(\Pic^0(X/T)=\Pic^0(X\times T)/p^*\Pic(T)\), the quotient identifies two families that differ by a line bundle pulled back from \(T\); such families have isomorphic restrictions to every fibre \(X_t\).
The tangent space at the origin, computed from \(T=\Spec k[\varepsilon]/(\varepsilon^2)\), is \(H^1(X,\OO_X)\), of dimension \(g=\dim\Jac(X)\), so \(\Jac(X)\) is smooth at the origin; translations act transitively on its points, so it is smooth everywhere. Properness follows from the valuative criterion: for a discrete valuation ring \(R\), \(X\times\Spec R\) is regular, so a line bundle on the generic fibre extends by taking the closure of a divisor.
The map \(\Sym^nX\to\Pic^n(X)\), \(D\mapsto\OO_X(D)\), has fibre \(\abs{L}\cong\PP^{\ell(L)-1}\) over \(L\). For \(n\ge g\) it is surjective by Riemann–Roch, and for \(n=g\) its general fibre is a point, so \(\dim\Jac(X)=g\).