Macaulay 2 and the geometry of schemes
  This tutorial illustrates how to use Grayson and Stillman's computer algebra system, Macaulay2, to study schemes. The examples are taken from the homework for an algebraic geometry class given at the University of California, Berkeley in the fall of 1999. This graduate course was taught by the second author with assistance for the first author. Our choice of problems follows the material in Eisenbud and Harris' textbook, The Geometry of Schemes. 
Gregory G. Smith and Bernd Sturmfels