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?

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

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