Affine and Projective Varieties

In this book we will be dealing with varieties over a field K, which we will take to be algebraically closed throughout. Algebraic geometry can certainly be done over arbitrary fields (or even more generally over rings), but not in so straightforward a fashion as we will do here; indeed, to work with varieties over nonalgebraically closed fields the best language to use is that of scheme theory. Classically, much of algebraic geometry was done over the complex numbers ℂ, and this remains the source of much of our geometric intuition; but where possible we will avoid assuming K = ℂ.