TITLE: Why did Hartshorne bother on schemes in his textbook when he worked only over an algebraically closed field?
QUESTION [11 upvotes]: In his textbook Algebraic Geometry, he wrote in p. 58:

Now that we have seen a litle bit of what algebraic geometry is about,
  we should discuss the degree of generality in which to develop the foundations of the subject.
  In this chapter we have worked over an an algebraically closed field, because that is the simlest case.
  But there are good reasons for allowing fields which are not algebraically closed.

However, in Chapter 4 and 5 treating curves and surfaces, which are the "meat" of algebraic geometry, he worked only over an algebraically closed field.
Hence my title question.

REPLY [5 votes]: The arguments in chapters 4 and 5 use the scheme-theoretic machinery of chapters 2 and 3.  
See e.g. the proof of Castelnuovo's criterion, which among other things uses the theorem on formal functions.  
Basically, even when the ground field is algebraically closed, it is easy to 
encounter non-trivial schemes (e.g. as thickened neighborhoods of closed sub varieties), and these come up in the arguments and discussion of chapters 4 and 5.