Bravo-Villamayor strengthening of embedded desingularization

Construct and check the modified algorithm. Split the smooth ambient into its disjoint open-and-closed components $X_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}$ contained in $Y$ and $X_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}$; use the identity on the former. On the latter start with $(I_Y,\mathrm{\varnothing },1)$ and use the original Steps 1–2, with the following extra branch at every recursive mark-one input. After higher residual-order strata have been handled, where the residual order is at most one and $M$ is nonunit, resolve $(M,1)$ before the residual ideal, assigning source invariant $(3\mathrm{/}2,0,\dots )$ and the monomial $\nu ,\rho$; where $M$ is a unit use the original residual branch. At mark one each minimal threshold subset is a single positive-exponent boundary divisor, so a maximal-$\rho$ center is a union of disjoint regular components of that divisor. It lies in the marked support, is SNC with the full boundary, and its labelled Cartier blowup reduces its exponent by one. The residual factor $N$ is unchanged, and no new non-monomial singularity is introduced. There are finitely many positive exponents, so these passes terminate; in higher recursive marks use the unchanged finite monomial procedure of [F1]. The rational $3\mathrm{/}2$ lies strictly between residual orders one and two, so it gives the intended priority on these closed boundary pieces without changing the higher-order passes. Successive assignment on untouched strata, as in [F1], preserves the primary invariant comparisons; the closed monomial strata give the chosen centers. No standalone upper semicontinuity of the auxiliary functions is used here. All centers are defined by ordered boundary equations, exponents and lower-dimensional invariant maxima; these data commute with smooth pullback and semilinear isomorphisms, and the maximal-contact gluing is unchanged. Under an ambient embedding the extra tangent parameters give the same constant prefixes before the induced recursive problem, so this modification commutes with that comparison as well. Thus the dimension induction proving canonicity and SNC centers for the original algorithm applies with this checked finite extra branch. The stopping convention of [F2] preserves the original smooth locus; any exceptional divisor encountered there would require an earlier center meeting that locus. Over nonclosed fields these intrinsic centers are Galois-stable and descend as in [F1].

Prove the source's local ideal claim by induction on the codimension $c$ of an irreducible component $Z$ of the support, whose ideal agrees generically with the active mark-one ideal $J$. At the stage with maximal source invariant $(1,0;\dots ;1,0;\mathrm{\infty };0,\dots )$, with $c$ copies of $(1,0)$, the added monomial branch has already removed residual exceptional factors, and the active input is $(J,1)$ of maximal order. Since $H(J,1)=C(J,1)=(J,1)$, the boundary reduction and maximal-contact reductions act on this same ideal. For $c=1$, Codimension-one components of a maximal-order support gives $J=I_Z$ along the smooth isolated component. For $c>1$, choose a tangent parameter $u\in J$ and let $H=V(u)$. The coefficient-restriction lemma identifies the induced mark-one ideal and its sequence on $H$, where $Z$ has codimension $c-1$. The induction gives $J{\mathcal{O}}_H=I_{Z,H}$ near the final strict transform. Since $u\in J$ and $Z\subseteq H$, equality modulo $(u)$ lifts to $J=I_Z$ in the ambient ring: both are inverse images of the same ideal under its quotient by $(u)$. The smoothness and SNC position of $Z$ also lift from the successive compatible parameter reductions. Thus when such a component would be the next center it is already smooth, isolated in the active support, and its active ideal is exactly its strict-transform ideal. Retain it and omit that blowup; repeat on the remaining components.

The remaining components. After ${\tilde{Y}}_1$ is separated, all strict transforms of components of codimension $r_1$ are isolated and are ignored in the further resolution; the process continues with the next codimension $r_2>r_1$ and the same argument shows that at the moment the strict transform of a codimension-$r_2$ component becomes a center, the controlled transform near it is its ideal. Iterating over the finitely many codimensions and components separates the strict transforms ${\tilde{Y}}_i$, which are smooth and pairwise disjoint. Any remaining active support is disjoint from these completed strict transforms and is principalized by the canonical process of [F1].

The full transform. On $X_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}$, the controlled-transform rule [F5] separates the exceptional monomial factors from the residual ideal. By steps 1.2 and 2.1, that residual ideal is the ideal of each completed strict transform near it and is the unit ideal away from their union after the remaining principalization; these local identities glue. Thus the full transform factors as $M((\tilde{\sigma }{)}^{\ast }{\mathcal{I}}_Y)\cdot {\mathcal{I}}_{\tilde{Y}}$, where the monomial part collects exceptional-divisor factors and the remaining factor is the ideal of the disjoint strict transforms. On $X_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}$ the sequence is the identity, ${\mathcal{I}}_Y=0$, and by convention the factorization is $0={\mathcal{O}}_{X_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}}\cdot 0$. This proves the componentwise formula and the theorem.