Corollary 11.43.13.F57433433789 Put $a_ j=1-1/(j+2)$ and $r_ j=1+1/(j+2)$ for $j\geq 1$. On $\mathbb P^ n$, or on its blowup at a fixed point $q\ne o$, the references $r_ j\pi ^*\Phi _{P_{a_ j}}$ satisfy the hypotheses of theorem 11.43.6, with limit $\pi ^*\Phi _{P_1}$. Their ordinary types and masses vary with $j$, and
\[ \sup _ G|B_ j(G)-B(G)| \leq n|\log r_ j|+\frac{n\kappa _ n}{1-\kappa _ n}|1-a_ j|\longrightarrow 0. \]
The equation is solvable in the limiting ordinary type, whose mass is $1-\kappa _ n$; for $n=2$ this mass is $2/3$.
Proof.
If $p\in P_{a_ j}\setminus P_1$, then $u=(\sum _ i\sqrt{p_ i})^2\in [a_ j,1)$ and $p/u\in P_1$. Since $|p|\leq \sum _ ip_ i\leq u$, one has $|p/u-p|\leq 1-u\leq 1-a_ j$. Thus $d_ H(P_{a_ j},P_1)\leq 1-a_ j$, and the hypotheses of the theorem hold with the simplex found in the proof of proposition 11.43.10. The mass formula (11.43.11) and the derivative bound $|\mathrm{d}\log (1-\kappa _ n a^ n)/\mathrm{d}a|\leq n\kappa _ n/(1-\kappa _ n)$ give the estimate for the functionals. By (11.43.12), the directional coefficients of the $j$th reference are multiplied by $r_ ja_ j=1-(j+2)^{-2}$, which strictly increases with $j$, so the ordinary types are distinct, while the masses $r_ j^ n(1-\kappa _ n a_ j^ n)$ strictly decrease to $1-\kappa _ n>0$.
$\square$
Pome
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