Let \(f\) be a rational function on the surface \(X\). Show that it is possible to “resolve the singularities of \(f\)” in the following sense: there is a birational morphism \(g: X^{\prime} \rightarrow X\) so that \(f\) induces a morphism of \(X^{\prime}\) to \(\PP^1\).
Hints: Write the divisor of \(f\) as \((f)=\sum n_i C_i\). Then apply embedded resolution (3.9) to the curve \(Y=\bigcup C_i\). Then blow up further as necessary whenever a curve of zeros meets a curve of poles until the zeros and poles of \(f\) are disjoint.
Solution
If \(f\) is constant, it already defines a constant morphism \[ X\longrightarrow\PP^1, \] so take \(g=\operatorname{id}_X\). Assume from now on that \[ f\in K(X)^\times \] is nonconstant.
Write its principal divisor as \[ (f)=\sum_i n_iC_i=Z-P, \] where \[ Z=\sum_{n_i>0}n_iC_i, \qquad P=\sum_{n_i<0}(-n_i)C_i \] are effective divisors with no common irreducible component.
1. After a birational morphism\[ g_1:X_1\longrightarrow X \]which is a finite composition of point blowups, the support of the divisorof \(f\) on \(X_1\) is a simple normal-crossing divisor.
Let \[ Y=\operatorname{Supp}(Z)\cup\operatorname{Supp}(P). \] Apply embedded resolution V.3.9 to the reduced curve \(Y\) on the nonsingular surface \(X\). This gives a finite composition of point blowups \[ g_1:X_1\longrightarrow X \] such that the reduced total transform of \(Y\) is simple normal crossings.
The birational map identifies the function fields, \[ K(X_1)=K(X), \] so we regard the same rational function \(f\) on \(X_1\). Its divisor is the pullback of \((f)\): its components are the strict transforms of the \(C_i\) and some exceptional curves, with integer coefficients. Hence its support is a union of components of the resolved total transform of \(Y\). A union of components of a simple normal-crossing divisor is again simple normal crossings.
On any subsequent surface write \[ (f)=Z'-P' \] for the positive and negative parts.
2. Suppose a zero component of multiplicity \(a>0\) and a pole componentof multiplicity \(b>0\) meet at a point \(Q\). After blowing up \(Q\), theirstrict transforms are disjoint and the exceptional curve \(E\) occurs in\((f)\) with coefficient\[ \boxed{a-b.} \]
Because the support is simple normal crossings, there are regular parameters \[ x,y\in\OO_{X,Q} \] such that the zero component is \(x=0\), the pole component is \(y=0\), and no other component of \((f)\) passes through \(Q\). Thus locally \[ f=u\,x^a y^{-b} \] for a unit \(u\).
Let \[ \pi:\widetilde X\longrightarrow X \] be the blowup at \(Q\). The total transforms of the two local curves are \[ \pi^*(x=0)=C_x'+E, \qquad \pi^*(y=0)=C_y'+E. \] Therefore \[ \operatorname{div}_{\widetilde X}(f) = aC_x'-bC_y'+(a-b)E \] near the exceptional fibre.
The strict transforms \(C_x'\) and \(C_y'\) meet \(E\) at the two distinct points corresponding to the tangent directions \(x=0\) and \(y=0\), so they no longer meet one another.
3. The effect of step 2 on zero–pole crossings is the Euclideanalgorithm:\[ \boxed{ (a,b)\longmapsto \begin{cases} (a-b,b),&a>b,\\ \text{no crossing},&a=b,\\ (a,b-a),&b>a. \end{cases}} \]
If \(a=b\), step 2 gives coefficient zero on \(E\). Since the two strict transforms are disjoint, no zero component meets a pole component over \(Q\).
If \(a>b\), then \(E\) is a zero component of multiplicity \(a-b\). It meets the strict transform of the pole component, of multiplicity \(b\), at one point. Its other intersection is with the strict transform of the original zero component, so that intersection is zero–zero and is irrelevant. Thus the only new zero–pole pair over \(Q\) has orders \[ (a-b,b). \]
The case \(b>a\) is symmetric: \(E\) is a pole component of multiplicity \(b-a\) and the unique new zero–pole crossing has orders \[ (a,b-a). \]
4. A finite sequence of further point blowups makes\[ \boxed{\operatorname{Supp}(Z')\cap\operatorname{Supp}(P')=\varnothing.} \]
At every stage the support of \((f)\) remains simple normal crossings: blowing up a transverse crossing of two components replaces it by an exceptional curve meeting the two strict transforms transversely at distinct points.
Consequently every zero–pole intersection point \(Q\) has exactly one zero component and one pole component through it. Let their positive multiplicities be \(a_Q,b_Q\), and define the positive integer \[ M = \sum_{Q\in\operatorname{Supp}(Z')\cap\operatorname{Supp}(P')} (a_Q+b_Q). \] There are only finitely many terms because the two effective divisors have finitely many components, and two distinct irreducible curves on the noetherian surface have zero-dimensional, hence finite, intersection unless they share a component. The positive and negative parts share none.
Blow up one zero–pole crossing of orders \((a,b)\). By step 3:
- if \(a=b\), its contribution \(a+b\) disappears;
- if \(a>b\), it is replaced by a contribution \[(a-b)+b=a<a+b;\]
- if \(b>a\), it is replaced by a contribution \[a+(b-a)=b<a+b.\]
No other zero–pole crossing is changed, since the blowup is an isomorphism away from its centre. Hence \(M\) strictly decreases after every such blowup. Because \(M\) is a nonnegative integer, the process terminates after finitely many steps. At termination \(M=0\), which is exactly the displayed disjointness.
4. On \(X'\) write\[ (f)=Z'-P', \qquad \operatorname{Supp}(Z')\cap\operatorname{Supp}(P')=\varnothing. \]
5. The rational functions\[ 1,\qquad f \]are global sections of the line bundle\[ \OO_{X'}(P'), \]and their zero divisors as sections are respectively\[ P', \qquad Z'. \]
For a divisor \(D\) on a nonsingular variety, \[ H^0(X',\OO_{X'}(D)) = \{\,h\in K(X'):\operatorname{div}(h)+D\ge0\,\}\cup\{0\}. \] For \(h=1\), \[ \operatorname{div}(1)+P'=P'\ge0. \] For \(h=f\), \[ \operatorname{div}(f)+P' = (Z'-P')+P' = Z'\ge0. \] Thus both are global sections. Under this description the zero divisor of a section \(h\) of \(\OO(P')\) is \[ \operatorname{div}(h)+P', \] which gives \(P'\) for \(1\) and \(Z'\) for \(f\).
6. The two-dimensional subspace\[ V=\langle f,1\rangle \subseteq H^0(X',\OO_{X'}(P')) \]is base-point free.
Since \(f\) is nonconstant, the sections \(f\) and \(1\) are linearly independent. By step 5, their common zero locus is \[ \operatorname{Supp}(Z')\cap\operatorname{Supp}(P'). \] Step 4 makes this intersection empty. Hence at every point of \(X'\) at least one of the two sections is nonzero, which is precisely base-point freeness of \(V\).
7. The base-point-free pencil \(V\) defines a morphism\[ \boxed{\varphi_f:X'\longrightarrow\PP^1} \]whose induced rational function is \(f\).
The standard construction of a morphism from a base-point-free linear system T-DIVMAPPN applied to the ordered basis \((f,1)\) gives regular local coordinate pairs which are never simultaneously zero. On the open set \[ X'\setminus\operatorname{Supp}(P'), \] the section \(1\) is nonvanishing and therefore trivializes \(\OO_{X'}(P')\); in this trivialization the two sections are represented by \(f\) and \(1\). Hence on this open set \[ \varphi_f(x)=[f(x):1] \] and the ratio of the two homogeneous coordinates is \(f\).
Therefore in the common function field the rational function induced by \(\varphi_f\) is \[ \frac{f}{1}=f. \] Therefore the morphism \(\varphi_f\) represents the pullback of the original rational map defined by \(f\).
8. The morphism \(g:X'\to X\) is birational and resolves the rationalfunction \(f\) in the required sense.
Every morphism used to construct \(g\) is the blowup of a point on a nonsingular surface. Each is birational, and a finite composition of birational morphisms is birational. Hence \(g\) is a birational morphism.
Step 7 gives a genuine morphism \[ \varphi_f:X'\to\PP^1 \] which agrees in the common function field with the rational function \(f\). This is exactly the required resolution.
9. Q.E.D.
Step 1 makes the divisor of \(f\) simple normal crossings. Steps <1>2–<1>4 separate its zero and pole divisors by finitely many additional blowups. Steps <1>5–<1>7 turn the resulting disjoint positive and negative parts into a base-point-free pencil and hence a morphism to \(\PP^1\). Step 8 verifies that the composite modification is birational.