The study of algebraic cycles in algebraic and arithmetic geometry in terms of the Chow group $\operatorname{CH}^(X)$, the grothendieck group $\operatorname{K}0(X)$ and the Chern character $ch:\operatorname{K}_0(X){\mathbb{Q}}\rightarrow \operatorname{CH}^(X)_{\mathbb{Q}}$
An algebraic cycle on an algebraic variety $V$ is a formal linear combination of subvarieties of $V$. These are the part of the algebraic topology of $V$ that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety.
The most trivial case is co-dimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of co-dimension one subvarieties, called divisors. The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann surface, and to extrinsic properties, such as embeddings of the curve into projective space.