Algebraic geometry is an old subject. There are many introductory books about it, at various levels. There are even some introductory texts which, like the present one, are addressed primarily to advanced undergraduates or beginning graduate students. The idea of these elementary introductions is to sell the subject. We do not yet attempt to train people in the field, only to convince them that it is fascinating and well worth the effort required to learn it.

There is no doubt that learning algebraic geometry entails substantial effort. The modern way of approaching the subject makes use of several technical machines, and a well-trained algebraic geometer needs to master at least one of these machines, preferably more than one. The very elementary introductions to the field try to avoid the machinery. They are generally very classical, using mathematics from the nineteenth century and the first half of the twentieth, before algebraic geometry underwent the Industrial Revolution and became so mechanized. The classical introductory books talk a great deal about curves, using the Riemann-Roch theorem to study them. They also might deal a little with simple singularities and their resolutions.

There are also many excellent books which do a thorough job teaching the foundations. These are for the serious graduate student, who already knows that this is the subject in which she wants to write her PhD.