This article is based on a talk given by V. Srinivas at the MRI, Allahabad. We give an account of the theory of algebraic cycles where the stress is not on the spate of conjectures (Hodge, Tate, Grothendieck, Bloch-Beilinson, etc.) that define the picture of this theory today, but rather on the key examples that refined and delineated this picture. Some of the deepest aspects of the theory of algebraic cycles are related to number theory. However, because of our lack of expertise on this topic, and to limit the scope of the discussion, we concentrate on the geometric aspects of the theory.
The model case of the theory of divisors on an algebraic curve (or compact Riemann surface) is dealt with in Section 1. The attempt to generalise this theory in higher dimensions is what led to much of the later work. In Section 2 we introduce the Chow ring with its relation to Grothendieck's K-theory via the Grothendieck Riemann-Roch theorem. The theory of divisors on smooth projective varieties is the next best understood case and we describe its features in Section 3. A much studied case is that of zero-cycles which we discuss in Section 4. Various alternate equivalences were introduced on the group of algebraic cycles; we study these in Section 5. We carry on in this section with a survey of the examples that build up our picture of the relation between these equivalences. In Section 6 we see the results of the attempt to relate the Chow group to points on an Abelian variety. As mentioned above we do not survey the conjectures in the theory of algebraic cycles. We discuss some of these briefly in Section 7. More detailed accounts of these can be found in [12] and an excellent survey by U. Jannsen [20].
General references for the theory of algebraic cycles are the survey article [19], describing the status of the subject in the early 70's, and the book [4] giving subsequent developments and newer viewpoints. The material in Section 1 can be found in most books on curves, for example [17]. The book [14] is the most complete source for the construction of the Chow ring, Chern classes, the Grothendieck-Riemann-Roch theorem and other material in Section 2. The theory of divisors on surfaces is dealt with in detail in [25] and a similar treatment can also be given in higher dimensions. References for the remaining sections can be found within the text. Unfortunately no books exist which cover the developments in the theory of algebraic cycles after Bloch's monograph [4] on the subject in 1980.