Abstract: These varieties play an important role in the theory of algebraic curves, even though — formally speaking — they are outside its one-dimensional scope. In fact, their geometry is no more complicated than that of curves. For instance, the Jacobian of a complex algebraic curve C can be thought of as a complex torus. The lattice is given by the period matrix of regular differentials on C (cf. Example 2 of Sect. 1.3). This torus is algebraic, since it is associated with a (non-singular) algebraic subvariety of ℂℙ n (cf. Chap. 2, Sect. 1.9). In view of Abel’s theorem, the points on the torus can be identified with the linear equivalence classes of divisors of degree 0 on C. This presentation of Jacobians is adapted for applications and holds over any ground field k. It is developed in §2. As we see, a Jacobian has two algebraic structures at once: it is both a variety and a group. This brings us to the subject of algebraic groups and abelian varieties.
Publication Year: 1994
Publication Date: 1994-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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