4.5.1 Rational maps
We start with Hartshorne’s definition of a rational map [ 4 ] .
Suppose \(X\) and \(Y\) are schemes of finite type over \(k\) with \(X\) irreducible. A rational map \(\phi :X\dashrightarrow Y\) is an equivalence class of morphisms \(\phi _U: U\to Y\), where \(U\) runs over the nonempty open subsets of \(X\) and two morphisms \(\phi _{U_i}: U_i\to Y\) (\(i=1,2\)) are equivalent if they agree on the intersection \(U_1\cap U_2\). We say a rational map \(\phi :X\dashrightarrow Y\) is dominant if \(\phi (U)\) is dense.
Suppose \(\phi :X\dashrightarrow Y\) is a rational map represented by a morphism \(\phi _U:U\to Y\) with \(U\) nonempty open in \(X\). Then we say that \(\phi \) is defined on \(U\). If we take the union of all nonemtpy open subsets of \(X\) on which \(\phi \) is defined, we get an open set \(\operatorname{\mathrm{dom}}\phi \), which is the largest open subset on which \(\phi \) is defined.
The image \(\phi (X)\) of a rational map \(\phi :X\dashrightarrow Y\) is the closure of the image of the morphism \(\phi _U:U\to Y\) where \(U=\operatorname{\mathrm{dom}}\phi \).
Suppose \(X\) is normal, \(Y\) is proper and \(\phi :X\dashrightarrow Y\) is a rational map. Then the complement of \(\operatorname{\mathrm{dom}}\phi \) in \(X\) has codimension at least \(2\).
Suppose \(x\in X\setminus \operatorname{\mathrm{dom}}\phi \) is a point with \(\dim \overline{x} = \dim X -1\). Then, since \(X\) is normal \(\mathcal{O}_{X,x}\) is a discrete valuation ring with fraction field \(K(X)\). So the valuative criterion of properness for \(Y\) gives us a lifting \(\operatorname{\mathrm{Spec}}\mathcal{O}_{X,x}\to Y\) agreeing with \(\phi \) on the generic point of \(X\). But then this gives a definition of \(\phi \) on a Zariski open subset of \(V\) of \(X\) containing \(x\), which contradicts our assumption that \(x\not\in U\).
Suppose \(\phi :X\dashrightarrow Y\) and \(\psi : Y\dashrightarrow Z\) are two dominant rational maps with \(U=\operatorname{\mathrm{dom}}X\) and \(V = \operatorname{\mathrm{dom}}\psi \). Then \(U':= \phi _U^{-1}(V)\) is a nonemtpy open subset of \(X\). So we get a morphism \(\psi _V\circ \phi _{U'}:U'\to Z\) where \(\phi _{U'}\) is the restriction of \(\phi _U\) to \(U'\). Thus, we get a rational map \(\psi \circ \phi :X\dashrightarrow Z\).
We call this rational map the composition of \(\psi \) with \(\phi \). It makes varieties (\(=\) integral schemes of finite type over a field) with dominant rational maps into a category.