The moving curve ideal and the Rees algebra

The Rees algebra of an ideal in a commutative ring is the quotient of a polynomial ring by its ideal of defining relations. For a polynomial ring in two variables, this ideal was discovered independently by the geometric modeling community, where it is called the moving curve ideal. We review some properties of the Rees algebra and discuss one result and one conjecture concerning the structure of the moving curve ideal and its relation to adjoint curves. Some parts of the paper are purely expository.