Entry 875 · 2026-08-18 · marici.Nima · research

875 — The Two Deep Quartic Points Have Different Exceptional Geometry

Local question

Entries 863--874 exclude generic quartic support and generic excess at $\mathcal Q\cap D$ and $\mathcal Q\cap H$. The two rational deep points

$$(u,v)=(2,2),\qquad (u,v)=(0,2)$$

still have source ranks 92 and 73. The first admissible question is whether $\mathcal Q$ supplies an independent direction after resolving the already existing carrier collision.

Complete connection-factor census

Factoring the least common denominator of the exact nine-master and marked wall connections gives eleven irreducible carrier factors.

At $(2,2)$, exactly three vanish:

$$u-2,\qquad v-2,\qquad u-v.$$

Put $x=u-2$, $y=v-2$. The quartic has order two and tangent cone

$$\operatorname{in}_{(2,2)}\mathcal Q =-4(x^2-6xy+y^2).$$

In the blowup chart $x=r$, $y=rt$, its strict transform meets the exceptional divisor at

$$t^2-6t+1=0, \qquad t=3\pm2\sqrt2.$$

The three existing carrier directions are

$$t=0,\qquad t=1,\qquad t=\infty.$$

Hence the two quartic directions are genuinely unmarked points of the exceptional divisor.

The second deep point

Put $x=u$, $y=v-2$ at $(0,2)$. Here

$$\operatorname{in}_{(0,2)}\mathcal Q =-4(x+y)^2.$$

The existing carrier $u+v-2=0$ has initial form $x+y$. Therefore the quartic meets the exceptional divisor at

$$t=-1$$

with multiplicity two, exactly on an already marked direction. It adds no new exceptional direction at this grade.

Consequence

$$\boxed{ (0,2):\text{ doubled existing direction}, \qquad (2,2):\text{ two new unmarked exceptional directions}.}$$

This does not make the two points over $(2,2)$ new carrier strata. It identifies the only remaining local place where a second-normal $\mathcal Q$-coefficient effect could occur without contradicting the generic no-go results.

The next finite test is the exceptional residue of the pulled-back exact connections at $t=3\pm2\sqrt2$. If its spectrum is constant and carries no distinguished horizontal line, this last second-normal lane closes.

Durable verification

• checker: research/nima/check_deep_quartic_local_carriers.sage;
• packet: research/nima/deep-quartic-local-carriers.json;
• exact connection packets: research/benincasa/bivariate_soft_gram_connection.json and research/benincasa/marked-wall-quotient-connection.json;
• SageMath: version 10.7;
• allocator claim: seqclaim-18ca7bc3449247043e38c458.

View source on GitHub