Projective-line projection dichotomy and endpoint coordinates

The closure of the irreducible set $\mathrm{\mid }W\mathrm{\mid }$ is irreducible; [F2] makes its reduced closed structure $T$ integral, and its restriction to $X\times {\mathbf{A}}^1$ is $W$ by uniqueness. By [F1], $T\to X$ is the composite of a proper closed immersion and the proper projective-line projection, hence proper and closed. Its image is a nonempty irreducible closed set, so its reduced structure $Y$ is integral. The factorization criterion of [F2] gives $T\to Y$, and then $T{\times }_{X}Y\cong T$ because $Y\to X$ is a closed immersion and $T\to X$ factors through it. Thus $T\to Y$ is a base change of $T\to X$, hence proper. It is surjective, in particular dominant. The same criterion factors $W$ as a closed subscheme of $Y\times {\mathbf{A}}^1$.

Put $K=k(Y)$. On nonempty affine charts over a nonempty affine open of $Y$, dominance gives injective maps of domains, so [F3] shows that the generic fiber is reduced. Every such chart contains the generic point of $T$, which survives localization; hence their generic-fiber charts share this point and their union is irreducible. Thus the generic fiber is nonempty and integral, has function field $k(T)=k(W)$ over $K$, and is closed in ${\mathbf{P}}_K^1$ by base change of its closed immersion. By [F4], a proper irreducible closed subset of the integral one-dimensional ${\mathbf{P}}_K^1$ is a closed point. Its reduced closed structure is $\mathrm{Spec}\kappa (p)$: on an affine chart the defining ideal is the corresponding maximal prime, by the quotient description in [F1]. The residue field is finite over $K$ by [F4]. The remaining possibility is the whole projective line, and uniqueness of reduced closed structures gives the asserted scheme equality.

In the two alternatives the function field over $K$ is respectively $K(t)$ or a finite extension, with transcendence degree one or zero. The proper-map formula in [F4] therefore gives $\dim T=\dim Y+1$ or $\dim T=\dim Y$, and $\dim T=\dim W$ because $W$ is a nonempty open of $T$. To check the first case, $Y\times {\mathbf{A}}^1$ is integral: its affine charts have domain rings $A[t]$ by [F3], and overlapping nonempty base charts give nonempty overlaps. Its function field is $K(t)$, so [F4] gives dimension $\dim Y+1$. The integral closed subscheme $W$ has that same dimension and hence codimension zero by [F4]; in an irreducible ambient space its support must be the whole space. Both closed structures are reduced, so [F2] gives $W=Y\times {\mathbf{A}}^1$. In the other case the finite generic field extension is exactly generic finiteness.

On the finite $u$-chart away from $u=1$ put $t=u\mathrm{/}(u-1)$; on the infinity chart with $v=u^{-1}$ put $t=1\mathrm{/}(1-v)$. These regular functions agree on their overlap and define a map to ${\mathbf{A}}^1$ by [F5]. Conversely, on $D(t-1)$ use $u=t\mathrm{/}(t-1)$, and on $D(t)$ use $v=(t-1)\mathrm{/}t$; these opens cover ${\mathbf{A}}^1$ and the formulas satisfy $uv=1$ on their overlap. They glue by [F1] and [F5], and substitution makes the composites identities. At $u=0$, $t=u\mathrm{/}(u-1)$ differs from $u$ by a unit; at infinity $t-1=v\mathrm{/}(1-v)$ differs from $v$ by a unit. Thus the endpoint fiber ideals, including their multiplicities after any base change, are those claimed.