# Irreducibility of the Jacobians of a curves.

I'm studying Jacobian varieties.I assume that the existence of the Jacobian variety for a curve and attempt to show irreducibility of the Jacobian for a curve according to Remark:IV.4.10.9 of Hartshorne's book.

Let $$C$$ be a complete smooth curve of genus $$g$$ over an algebraic field $$k$$, let $$P_0 \in C$$ and let $$J(C)$$ is the Jacobian of C. By the universal property of the Jacobian, we can obtain the morphism $$\varphi : C^g \rightarrow J(C)$$ such that $$\varphi(k): C(k) \ni(P_1 ,\dots,P_n) \mapsto \mathcal{O}_C(P_0+\dots+P_n-nP_0)\in J(C)(k)=\mathrm{Pic}^\circ(C)$$. Then there exists an open subset $$U$$ of $$J(C)$$ such that the fibers of $$\varphi$$ at any point of U are finite sets. My book conclude that this means $$J(C)$$ is irreducible. But I cannot understand this. Could someone please tell me this reason?

• Searching for "jacobian hyperelliptic curve" shows many texts about the construction of the abelian group $Pic^0(C)$ and its algebraized variety – reuns Dec 14 '18 at 15:55