Let C be a smooth projective (irreducible) curve of genus g over an algebraically closed field k. By Pic(C) (resp. Picd (C)) we denote the Picard group of C (resp. the degree d subset in it). In this chapter we introduce the structure of abelian variety on Pic0(C). More precisely, we construct an abelian variety J = J (C) called the Jacobian of C, such that the group of k-points of J is isomorphic to Pic0(C). The idea is to use the fact that every line bundle of degree g on C has a nonzero global section and that for generic line bundle L of degree g this section is unique (up to rescaling). Therefore, a big subset in Picg(C) can be described in terms of effective divisors on C. The set of effective divisors of degree d on C can be identified with the set of k-points of the symmetric power SymdC (the definition and the main properties of the varieties SymdC are given in Section 16.1). The subset in Picg(C) consisting of line bundles L with h0(L) = 1 corresponds to the set of k-points of an open subset in SymgC. Translating this subset by various line bundles of degree –g we obtain algebraic charts for Pic0(C). We define the Jacobian variety J by gluing these open charts.