I.4: Rational Maps
4.1. #to_work
If \(f\) and \(g\) are regular functions on open subsets \(U\) and \(V\) of a variety \(X\), and if \(f=g\) on \(U \cap V\). show that the function which is \(f\) on \(U\) and \(g\) on \(V\) is a regular function on \(U \cup V\). Conclude that if \(f\) is a rational function on \(X\), then there is a largest open subset \(U\) of \(X\) on which \(f\) is represented by a regular function. We say that \(f\) is defined at the points of \(U\).
4.2. Same problem for rational maps. #to_work
If \(\varphi\) is a rational map of \(X\) to \(Y\), show there is a largest open set on which \(\varphi\) is represented by a morphism. We say the rational map is defined at the points of that open set.
4.3. #to_work
- Let \(f\) be the rational function on \({\mathbb{P}}^2\) given by \(f=x_1 / x_0\). Find the set of points where \(f\) is defined and describe the corresponding regular function.
- Now think of this function as a rational map from \({\mathbb{P}}^2\) to \(\mathbf{A}^1\). Embed \(\mathbf{A}^1\) in \({\mathbb{P}}^1\), and let \(\varphi: {\mathbb{P}}^2 \rightarrow {\mathbb{P}}^1\) be the resulting rational map. Find the set of points where \(\varphi\) is defined, and describe the corresponding morphism.
4.4. #to_work
A variety \(Y\) is rational if it is birationally equivalent to \({\mathbb{P}}^n\) for some \(n\) (or, equivalently by (4.5), if \(K(Y)\) is a pure transcendental extension of \(k\) ).
- Any conic in \({\mathbb{P}}^2\) is a rational curve.
- The cuspidal cubic \(y^2=x^3\) is a rational curve.
- Let \(Y\) be the nodal cubic curve \(y^2 z=x^2(x+z)\) in \({\mathbb{P}}^2\). Show that the projection \(\varphi\) from the point \(P=(0,0,1)\) to the line \(z=0\) (Ex. 3.14) induces a birational map from \(Y\) to \({\mathbb{P}}^1\). Thus \(Y\) is a rational curve.
4.5. #to_work
Show that the quadric surface \(Q: x y=z w\) in \({\mathbb{P}}^3\) is birational to \({\mathbb{P}}^2\), but not isomorphic to \({\mathbb{P}}^2\) (cf. Ex. 2.15).
4.6. Plane Cremona Transformations. #to_work
A birational map of \({\mathbb{P}}^2\) into itself is called a plane Cremona transformation. We give an example, called a quadratic transformation. It is the rational map \(\varphi: {\mathbb{P}}^2 \rightarrow {\mathbb{P}}^2\) given by \(\left(a_0, a_1, a_2\right) \rightarrow\left(a_1 a_2, a_0 a_2, a_0 a_1\right)\) when no two of \(a_0, a_1, a_2\) are \(0\).
- Show that \(\varphi\) is birational, and is its own inverse.
- Find open sets \(U, V \subseteq {\mathbb{P}}^2\) such that \(\varphi: U \rightarrow V\) is an isomorphism.
- Find the open sets where \(\varphi\) and \(\varphi^{-1}\) are defined. and describe the corresponding morphisms. See also (Chapter V, 4.2.3).
4.7. #to_work
Let \(X\) and \(Y\) be two varieties. Suppose there are points \(P \in X\) and \(Q \in Y\) such that the local rings \({\mathcal{O}}_{P, X}\) and \({\mathcal{O}}_{Q, Y}\) are isomorphic as \(k{\hbox{-}}\)algebras. Then show that there are open sets \(P \in U \subseteq X\) and \(Q \in V \subseteq Y\) and an isomorphism of \(U\) to \(V\) which sends \(P\) to \(Q\).
4.8. #to_work
- Show that any variety of positive dimension over \(k\) has the same cardinality as \(k\). 1
- Deduce that any two curves over \(k\) are homeomorphic (cf. Ex. 3.1).
4.9. #to_work
Let \(X\) be a projective variety of dimension \(r\) in \(P^n\). with \(n \geqslant r+2\). Show that for suitable choice of \(P \notin X\). and a linear \({\mathbb{P}}^{n-1} \subseteq {\mathbb{P}}^n\). the projection from \(P\) to \({\mathbb{P}}^{n-1}\) (Ex. 3.14) induces a birational morphism of \(X\) onto its image \(X' \subseteq {\mathbb{P}}^{n-1}\). You will need to use (4.6A). (4.7A). and (4.8A). This shows in particular that the birational map of (4.9) can be obtained by a finite number of such projections.
4.10. #to_work
Let \(Y\) be the cuspidal cubic curve \(y^2=x^{3}\) in \(\mathbf{A}^2\). Blow up the point \(O=(0.0)\). Let \(E\) be the exceptional curve. and let \(\tilde{Y}\) be the strict transform of \(Y\). Show that \(E\) meets \(\tilde{Y}\) in one point. and that \(\tilde{Y} \cong \mathbf{A}^1\). In this case the morphism \(\rho: \tilde{Y} \rightarrow Y\) is bijective and bicontinuous. but it is not an isomorphism.