Regularization of a one-dimensional integral curve with finite normalization by point blowups

(Base of the recursion) Put $Y_0=Y$ and $f_0={\mathrm{id}}_Y$; then $Y_0$ is integral, Noetherian and one-dimensional, $\nu :Y^{\nu }\to Y_0$ is a finite normalization and $f_0$ is finite. We construct, as long as the current curve is not regular, a point blowup at a non-regular closed point and keep the data of [F1].

(Existence of a suitable center) Suppose $Y_{i-1}$ is not regular. By [F5] some closed point $p_{i-1}\in Y_{i-1}$ is not regular; choose it and let ${\beta }_i:Y_i={\mathrm{Bl}}_{p_{i-1}}{Y}_{i-1}\to Y_{i-1}$ be the blowup, with $f_i=f_{i-1}\circ {\beta }_i$. By [F1] the morphism ${\beta }_i$ is finite, is not an isomorphism (the center is non-regular), and $Y^{\nu }$ is a normalization of $Y_i$; by [F3] the scheme $Y_i$ is integral, Noetherian and one-dimensional, and $f_i$ is finite. By [F2] the inclusion $f_{i-1,\ast }{\mathcal{O}}_{Y_{i-1}}⊊f_{i,\ast }{\mathcal{O}}_{Y_i}$ is strict, both terms being coherent ${\mathcal{O}}_Y$-subalgebras of ${\nu }_{\ast }{\mathcal{O}}_{Y^{\nu }}$.

(Termination) Suppose, for contradiction, that $Y_{i-1}$ is non-regular for every $i\ge 1$. Then step 2.1 can be iterated for all $i$, and it produces the infinite strictly increasing sequence ${\mathcal{O}}_Y⊊f_{1,\ast }{\mathcal{O}}_{Y_1}⊊f_{2,\ast }{\mathcal{O}}_{Y_2}⊊\cdots \subseteq {\nu }_{\ast }{\mathcal{O}}_{Y^{\nu }}$ of coherent ${\mathcal{O}}_Y$-submodules of the coherent module ${\nu }_{\ast }{\mathcal{O}}_{Y^{\nu }}$ over the Noetherian scheme $Y$. This contradicts stabilization [F4]. Hence there is an integer $n\ge 0$ with $Y_n$ regular.

(Conclusion) The finite sequence $Y_n\to Y_{n-1}\to \cdots \to Y_1\to Y$ consists of blowups in closed points, each center non-regular in the preceding curve, and ends at the regular curve $Y_n$; no morphism in it is an isomorphism by [F1], and the associated sequence of coherent subalgebras strictly increases at every step by [F2]. Conversely, stabilization of this sequence is precisely what forces the termination: a stabilized non-regular stage would admit one further strict increase, contradicting stabilization. This proves all assertions, including the empty sequence when $Y$ is already regular. No assertion is made for non-finite normalizations or in dimension greater than one.