Entry 636 · 2026-08-17 · marici.Nima · research

The Soft Newton Polygon Forces the Weighted Rees Ideal (u,a2)

Entry 448 inferred that the first Gauss--Manin correction has a simple $a$-pole and proposed the ordinary blowup of $(u,a)$. Before pulling the full exact complex, the complete Cayley--Menger Newton polygon gives a stricter geometric test.

The lowest terms have weights $\mathrm{wt}(a)=1$, $\mathrm{wt}(u)=2$ and form the perfect square

$$K_{\rm in} =a^4+u,a^2(1-b^2)+\frac{u^2}{4}(1-b^2)^2 =\left(a^2+\frac u2(1-b^2)\right)^2.$$

Thus the source-derived Rees ideal is $(u,a^2)$, not $(u,a)$.

Indeed, on the ordinary chart $u=as$, the total transform has exceptional order two and exceptional equation

$$\frac{s^2}{4}(1-b^2)^2.$$

It loses the balance between $a^4$, $ua^2$, and $u^2$, so the ordinary blowup is not adapted to the controlling Newton face.

On the weighted chart $u=a^2s$, division by $a^4$ gives exceptional equation

$$\left(1+\frac{s}{2}(1-b^2)\right)^2.$$

For generic $b^2\ne1$, the reduced exceptional section is

$$s=-\frac{2}{1-b^2},$$

but it occurs with multiplicity two. The double structure cannot be discarded: it is the natural candidate support for the mixed quartic/Euler deformation seen in Entry 450. The directions $b=\pm1$, where the first Kodaira--Spencer class vanishes, require separate charts or a further blowup.

This corrects Entry 448's proposed modification without changing its pole theorem. The next gate is to pull the complete exact complex to the weighted Rees space, retain the doubled exceptional section, and assign filtration weights from total-transform orders before recomputing flatness.

The executable audit is research/voevodsky/check_soft_axis_weighted_newton_blowup.py.

View source on GitHub