Question

Let |V| be a (incomplete) linear series on a nonsingular projective surface. Hironaka says that there is a resolution of the singularities of |V| along smooth centers. If the base locus of |V| is just a collection of points, does it mean I can acheive this resolution by a series of blow ups at points?

Answer 1

I may be misunderstanding something but this question does not seem to have anything to do with Hironaka's desingularization. You are asking if you can resolve the indeterminacy of a rational map, right? If this is the question, then you can do it with finitely many blow-ups at points.

Suppose you have a rational map $f$ from a smooth projective surface $S$ to $\mathbb{P}^{n}$ given by some linear system $V \subset |D|$. Without loss of generality you may assume that the image of $f$ is non-degenerate, i.e. is not contained in any hyperplane. In that case we can subtract the fixed component of $V$ to make the base locus of $V$ at most finite. Now take a base point of $V$ and blow it up $g : \widehat{S} \to S$. The pullback $g^{*}V$ will consist of divisors in $|g^{*}D|$ which contain a positive multiple $nE$ of the exceptional divisor $E$. So $g^{*}V$ has a fixed component and therefore is of the form $\widehat{V} + nE$, where $\widehat{V} \subset |g^{*}D - nE|$. The linear system $\widehat{V}$ gives a rational map $\hat{f}$ from $\widehat{S}$ to $\mathbb{P}^{n}$ which has no fixed component and coincides with the original $f$ on $\widehat{S} - E$. Now set $\widehat{D} = g^{*}D - nE$. Note that the fact that $\widehat{D}$ has no fixed component implies that $\widehat{D}^{2} = D^{2} - n^{2} < D^{2}$. If $\hat{f}$ has no base point, then we are done. If it does have base points, then we can repeat the process. Since the degree of the linear system decreases on each step the process will terminate.

Answer 2

Over $\mathbb C$ at least, If the surface is non-singular then a finite number of blow-ups at points suffices to resolve the linear system. Indeed any birational morphism between smooth surfaces is a finite composition of point blow-ups.

Answer 3

Not necessarily, I don't think. There are surface singularities (though I can't recall an example easily, but one whose resolution has exceptional divisor an elliptic curve) for which if you blow up at a point you get a singular curve. The example I'm thinking of is in Kollar's Exercises.

In the case I'm thinking of, you have a surface with a single point singularity, you blow it up, and you get a rational curve singularity, which if you blow up (or normalize, either one) gives you an elliptic curve over it.

EDIT: found it, it's exercise 68, do all three parts to see some of the things that can happen.