(Families of degree-zero line bundles)
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.