Let \(X\) be a variety.
Let \(f\) and \(g\) be regular functions on open subsets \(U\) and \(V\) of \(X\), and suppose \(f = g\) on \(U \intersect V\). Show that the function which equals \(f\) on \(U\) and \(g\) on \(V\) is a regular function on \(U \union V\).
Conclude that if \(f\) is a rational function on \(X\), then there is a largest open subset \(U \subseteq X\) on which \(f\) is represented by a regular function. One says that \(f\) is defined at the points of \(U\).
Solution
1. The piecewise rule\[ h(x)= \begin{cases} f(x),&x\in U,\\ g(x),&x\in V \end{cases} \]defines a regular function on \(U\cup V\).
The rule is well-defined because \(f=g\) on \(U\cap V\). Fix \(x\in U\cup V\). If \(x\in U\), then on the open neighborhood \(U\) of \(x\) the function \(h\) equals the regular function \(f\). If \(x\in V\), then on the open neighborhood \(V\) it equals the regular function \(g\). Thus every point has an open neighborhood on which \(h\) is regular. Regularity is local, so \(h\) is regular on \(U\cup V\). This proves (a).
2. Let \(r\in K(X)\) be a rational function and let\[ D(r)=\bigcup\{U\subseteq X: U\text{ is open and }r\text{ is represented on }U\text{ by a regular function}\}. \]Then \(D(r)\) is open.
It is a union of open subsets of \(X\).
3. The local representatives of \(r\) glue to a regular function on \(D(r)\).
For every open \(U\) occurring in the union of step 2, let \(r_U\) be the regular representative of \(r\) on \(U\). If \(U\) and \(V\) are two such opens, then the two representatives define the same rational function. Hence they agree on \(U\cap V\). Therefore the rule \[ \widetilde r(x)=r_U(x)\qquad(x\in U) \] is independent of the choice of \(U\). At each \(x\in D(r)\), choose one such \(U\) containing \(x\); on that neighborhood \(\widetilde r=r_U\) is regular. Thus \(\widetilde r\) is regular on all of \(D(r)\) and represents \(r\) there.
4. The open subset \(D(r)\) is the largest open subset on which \(r\) is represented by a regular function.
By step 3, \(r\) is represented regularly on \(D(r)\). If \(W\subseteq X\) is any open set on which \(r\) has a regular representative, then \(W\) occurs in the union defining \(D(r)\), so \[ W\subseteq D(r). \] Thus \(D(r)\) is maximal by inclusion and is uniquely determined. This proves (b).
5. Q.E.D.
Step 1 proves (a), and steps <1>2–<1>4 prove (b).
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P-AGH43LINEARPROJDomain of $f = x_1/x_0$ on $\PP^2$ and of the induced map to $\PP^1$uses