Fixed-resolution tie-breaking to one log canonical place

Write $K_Y-{\mu }^{\ast }{K}_X=\sum$ $k_j$ $E_j$ for the exceptional primes. In the smooth branch each $k_j$ is a nonnegative integer: a smooth-centre blowup of codimension c has Jacobian factor of order c−1, and the composition adds pullbacks of these effective exceptional divisors. In the normal terminal branch each $k_j$ is a positive integer by the stated Cartier-terminal discrepancy hypothesis.

There is a positive rational exceptional divisor $T_0$, supported on every exceptional prime, with $m{\mu }^{\ast }A-T_0$ ample for some m. In the smooth branch the finite blowup construction is explicit. A single blowup has its tautological relatively ample $O(-E_{new})$. If a positive exceptional T has relatively ample negative for a previous stage, at the next blowup take a sufficiently large integral multiple of its pullback plus the new exceptional divisor. Its negative is relatively ample over the original base: the negative new exceptional is ample over the previous stage, and tensoring it with a sufficiently high pullback of the previously relatively ample negative gives a projective embedding over the original base. This is the ordinary relative Segre/Serre construction for two projective embeddings. All coefficients remain positive. Induct over the finite blowup sequence.

For some positive integer m, $m{\mu }^{\ast }A-T_0$ is ample on Y: a relatively ample line bundle tensored with a sufficiently high pullback of an ample line bundle is ample, by a relative projective embedding followed by Segre and Serre twisting. In the normal terminal branch first apply this to the SUPPLIED $T_{\ast }$; if any exceptional coefficients vanish, add sufficiently small positive rational coefficients on those primes, preserving the resulting absolute ample divisor by the finite cone argument below. This gives the claimed positive $T_0$ on the actual model without assuming a smooth-base blowup factorization there.

The coefficients of $T_0$ can be varied independently in a rational neighbourhood while keeping positivity and mμ^*A−T ample. Here is a finite algebraic ample-cone argument. Put $H=m{\mu }^{\ast }A-T_0$. For each of the finitely many Cartier primes $E_j$, choose $q_j$ such that $q_{j}H+E_j$ and $q_{j}H-E_j$ are ample. Such $q_j$ exist by ample twisting. Then H+Σ δ_j $E_j$ is a positive rational combination of H and those finitely many ample divisors whenever Σ $q_j\mathrm{\mid }{\delta }_j\mathrm{\mid }<1$; hence is ample. Shrink further to retain positive coefficients of $T=T_0-\sum {\delta }_j{E}_j$.

Let $E_1,\mathrm{.}\mathrm{.}\mathrm{.},E_s$ be the coefficient-one prime divisors of $B_0$. They are all exceptional and map to W: a nonexceptional coefficient-one divisor would give a divisorial non-klt centre, contrary to codim W≥2 and uniqueness of W. At such a divisor the order of ${\mu }^{\ast }{G}_0$ is $k_i+1$.

Choose a rational T inside the preceding positive ample neighbourhood so that the numbers ${\rho }_i=T_i\mathrm{/}(k_i+1)$, $i=1,\mathrm{.}\mathrm{.}\mathrm{.},s$,

are pairwise distinct. There are finitely many forbidden affine hyperplanes $T_i\mathrm{/}(k_i+1)=T_j\mathrm{/}(k_j+1)$; none contains the open coefficient neighbourhood because those coefficients vary independently. Rational points are dense, so such a positive rational choice exists. Let $i_0$ be the unique index with maximal ρ_i.

Take a sufficiently divisible N so that NT and N(mμ^*A−T) are Cartier and the latter is very ample. Choose a general smooth member $H_Y$ of |N(mμ^*A−T)| meeting the existing SNC divisor normally and containing none of its components. Bertini applied to that complete very ample system gives exactly this finite collection of transversality conditions.

Multiplying its defining section by the canonical section of NT gives a section of ${\mu }^{\ast }{O}_X(NmA)$. Since ${\mu }_{\ast }{O}_Y=O_X$ by normality of X, this descends uniquely to a section of $O_X(NmA)$. Let D be its effective divisor. Its ACTUAL pullback is ${\mu }^{\ast }D=NT+H_Y$.

Thus on the fixed resolution the orders of D at the original $E_i$ are exactly $N{T}_i$, and no unrecorded resolution or new original coefficient-one divisors are introduced.

For rational τ>0 put $c_{\tau }=1\mathrm{/}(1+\tau N{\rho }_{i_0})$, $G_{\tau }=c_{\tau }(G_0+\tau D)$.

The coefficient of $E_i$ in its log pullback is $b_i(\tau )=c_{\tau }(k_i+1+\tau N{T}_i)-k_i$.

It equals one exactly for $i=i_0$, and is strictly below one for the other original coefficient-one divisors, because their ρ_i are smaller. Every original coefficient below one remains below one for all sufficiently small τ by continuity and the finite number of components. The new free member $H_Y$ has coefficient $c_{\tau }\tau <1$ for sufficiently small τ. Its intersections with the existing boundary are normal crossing by construction. Every coefficient on the fixed SNC model is therefore at most one, with equality on exactly $E=E_{i_0}$. Negative coefficients can occur only on exceptional divisors, since the strict transform of the effective downstairs $G_{\tau }$ is effective.

The numerical coefficient is ${\gamma }_{\tau }=c_{\tau }(\gamma +\tau Nm)$,

and tends to γ as τ tends to zero. Choose a positive rational τ sufficiently small for all preceding strict inequalities and |γ_τ−γ|<ε. Set $G_1=G_{\tau }$ and let F be its log boundary with E removed. This gives exactly the Statement form, with E mapping to the original W. No one-sided inequality γ_1≤γ is asserted.

There is a direct differential calculation for every divisorial valuation v, avoiding an assumption about how it is extracted by boundary-compatible blowups. At the generic point of its centre on smooth Y, take regular parameters $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_c$ for that centre, adapted to the SNC boundary components containing it, and complete them by parameters along the centre. On a smooth model carrying v, write $x_i=t^{a_i}{u}_i$ with $a_i=v(x_i)>0$ and $u_i$ a unit along the valuation divisor. Expanding the pulled-back top differential shows its order is at least $\sum a_i-1$: at most one differentiated factor can supply the dt term reducing the total order by one. Therefore the log discrepancy satisfies $A_Y(v)\ge \sum a_i$.

Assign coefficient $b_i$ to a parameter defining a boundary component and coefficient zero to the remaining centre parameters. The log discrepancy for the pair is at least $\sum (1-b_i)a_i$. If v is exceptional over Y, its centre has codimension at least two. At most one $b_i$ is one, and every other $b_i$ is strictly below one, so this sum is strictly positive. For a prime divisor already on Y, only E has coefficient one. Thus E is the sole lc place of $(X,G_1)$, with no hidden additional place above this model.