This book came out of a course I taught, twice, at the Australian National University. I taught it first in the Fall of 2004, and then again, because of interest from some students and colleagues, in the Fall of 2005. The course was a one-semester affair, and the students were fourth-year undergraduates.

Given that these were undergraduate students in their final year, this could be one of the last few mathematics courses they would ever see. They might well decide to pursue interests having nothing to do with mathematics; they could, for all I know, choose to become doctors, or lawyers, or bankers, or politicians. My task was to present to them an overview of algebraic geometry. It would be premature to give them a thorough grounding in the field; a broad, panoramic picture seemed far more appropriate, and if possible the panorama should include glimpses into a wide assortment of pretty vistas, into more specialized areas, each of which is beautiful in its own right. I tried to cover interesting topics, without delving into too much detail on any one of them.

The first order of business was to choose the subject matter for the course. I had the option of teaching classical algebraic geometry; there are several excellent textbooks to choose from, written specifically for students at this level.