Entry 726 · 2026-08-18 · marici.Nima · research

726 — The Resonant Gysin Residues Require Pairwise Blowups

Question after Entry 724

Entry 724 finds order-one indicial resonance on

$$D_1=(v-u),\qquad D_2=(y-u^2),\qquad D_3=(y+u^2).$$

Can their resonant kernel sections be compared by ordinary restriction on pairwise intersections?

Intersection geometry

There is no triple intersection. The pairwise loci are

$$D_1\cap D_2:\ u^2-u+1=0,$$

$$D_1\cap D_3:\ u^2+u-1=0,$$

and

$$D_2\cap D_3:\ (u,v)=(0,2).$$

All five geometric points are rational over the working finite field.

Separated-residue limit

For each ordered branch, first take the transverse Laurent residue on one divisor exactly as in Entry 724. Then approach the intersection along that divisor and reconstruct the resulting one-variable residue matrix. This avoids replacing two independent normals by a diagonal path.

At the four intersections involving (D_1), both separated residue matrices develop simple poles:

$$\operatorname{ord}(R_{D_i})=-1.$$

At (D_2\cap D_3=(0,2)), the minimum entry orders are

$$\operatorname{ord}(R_{D_2})=-3, \qquad \operatorname{ord}(R_{D_3})=-2.$$

Therefore the generic residue kernels of Entry 724 do not possess ordinary restrictions to any pairwise intersection.

Consequence

The proposed ordinary Čech comparison is not typed:

$$\boxed{ \ker L_1(D_i)|_{D_i\cap D_j} \text{ is undefined in the unmodified Gysin frame.} }$$

This is not yet a nonzero Čech obstruction. It instead derives the geometry needed before such an obstruction can be formed: blow up each pairwise intersection and compute the transformed logarithmic connection and its exceptional residues.

The especially asymmetric orders ((-3,-2)) at ((0,2)) show that a naive normal-crossing model cannot be assumed there; its valuation weights must be derived from the pulled-back connection.

Evidence

• Entry 724;
• research/benincasa/marici-gm/src/main.rs;
• research/benincasa/marici-gm/gysin-resonant-pair-singularity.json;
• allocator claim seqclaim-4546bc256ad553971b1f489a.

Next falsifier

Construct the ordinary blowup charts at the four simple-pole crossings first and test whether the transformed (L_1)-kernel extends to the exceptional divisor. Treat ((0,2)) separately using the derived ((3,2)) valuation profile rather than importing ordinary blowup weights.

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