Rational maps of integral finite-type schemes

Let $k$ be a field and let $X$ be an integral $k$-scheme of finite type and $Y$ a $k$-scheme of finite type (Integral schemes, Locally finite type and finite type morphisms) with $Y$ separated over $k$ (Separated morphism of schemes).

A rational map $\phi :X⇢Y$ is an equivalence class of pairs $(U,{\phi }_U)$, where $U\subseteq X$ is a nonempty open subscheme (Open immersions of schemes) and ${\phi }_U:U\to Y$ is a $k$-morphism (Morphisms of schemes). Two pairs $(U,{\phi }_U)$ and $(V,{\psi }_V)$ are equivalent when the two morphisms agree on a nonempty open subscheme of $U\cap V$, that is, when there is a nonempty open $W\subseteq U\cap V$ with ${\phi }_U{\mathrm{\mid }}_W={\psi }_V{\mathrm{\mid }}_W$.

The relation is an equivalence relation. Reflexivity and symmetry are immediate. For transitivity let $(U,{\phi }_U)\sim (V,{\psi }_V)$ through $W\subseteq U\cap V$ and $(V,{\psi }_V)\sim (T,{\chi }_T)$ through $Z\subseteq V\cap T$. Since $X$ is integral, hence irreducible, any two nonempty open subschemes meet, so $W\cap Z$ is a nonempty open subscheme of $U\cap T$, and on $W\cap Z$ the morphisms ${\phi }_U$ and ${\chi }_T$ agree with ${\psi }_V$, hence with one another. Thus $(U,{\phi }_U)\sim (T,{\chi }_T)$. No integrality or reducedness of the target is needed.

A rational map is dominant when some representative ${\phi }_U$ has dense image. This is independent of the representative: if $(U,{\phi }_U)$ and $(V,{\psi }_V)$ are equivalent through a nonempty open $W\subseteq U\cap V$, then $W$ is dense in the irreducible scheme $U$, so continuity gives ${\phi }_U(U)\subseteq \overline{{\phi }_U(W)}=\overline{{\psi }_V(W)}\subseteq \overline{{\psi }_V(V)}$; hence ${\phi }_U$ dominant implies ${\psi }_V$ dominant, and the converse is symmetric. A point of $X$ at which no representative of $\phi$ is defined is a point of indeterminacy of $\phi$.

When $X$ is integral and of finite type over $k$, its function field $k(X)={\mathcal{O}}_{X,{\eta }_X}$ is the stalk at the generic point, and the function field of every nonempty affine open is the fraction field of its coordinate ring (Function field of an integral finite-type scheme); this is the description used whenever a rational map of curves is converted into a map of function fields below.