Notation in Hartshorne book about monoidal transformations (blow-ups of surfaces along points)

Question

I'm starting the process of learning the concept of blow-up for surfaces along a point.

At page 386 of Hartshorne's book, the author defines the monoidal transformation of a surface $X$. But at the beginning of the chapter I find the following sentence:

Throughtout this chapter a surface will mean a non-singular projective surface over an algebraically closed field.

Now, if the "aim" of the blow-up is the resolution of singularities, what is the sense of defining it for non-singular surfaces?

edit: Why for example in Beauville's book there is no mention to the non-singularity?

Answer 1

You say

if the "aim" of the blow-up is the resolution of singularities, what is the sense of defining it for non-singular surfaces?

But that is a false premise. Blow-ups arise unavoidably in solving all kinds of problems in algebraic geometry, even when you start from a nonsingular variety. The basic reason is that rational maps can be resolved by blowups.

As an example of how this arises in practice, look no further than the exact page of Hartshorne you mention (p. 386): he says there

We will see later (5.5) that any birational transformation of surfaces can be factored into monoidal transformations and their inverses.