The resolved four-ruling-lines degeneration and an explicit liaison audit

Definitions and mathematical statement

Write $H_{d,g}$ for the locally Cohen--Macaulay curve locus in the Hilbert scheme of ${\mathbb{P}}^3$ with Hilbert polynomial $dt+1-g$. A curve is locally Cohen--Macaulay if it has no embedded or isolated zero-dimensional components. Its Rao module is $M_C=\bigoplus_m H^1({\mathbb{P}}^3,\mathcal I_C(m))$. An extremal curve has the maximal Rao function allowed for its fixed degree and arithmetic genus, in the standard extremal-curve convention.

Problem 26. (i) If $C$ is a divisor of bidegree $(3,7)$ on a smooth quadric $Q\simeq{\mathbb{P}}^1\times{\mathbb{P}}^1$, can it be connected to an extremal curve in $H_{10,12}$? Its degree is $3+7=10$ and genus $(3-1)(7-1)=12$. (ii) Given four skew ruling lines on a smooth quadric $Q_1$, find a flat family of quadrics degenerating to a double plane $Q_0=2H$ and a flat family $C_t\subset Q_t$ whose special curve $C_0$ is locally Cohen--Macaulay. The original remark states (ii)$\Rightarrow$(i). The OCR arrow and footer damage are repaired according to the supplied clean text.

Supplied resolution. The aid attributes both affirmative answers to Lella--Schlesinger: for every $d\geq3$, $d$ disjoint ruling lines on a smooth quadric admit a flat degeneration to an extremal locally Cohen--Macaulay curve in a double plane, and every effective divisor on a smooth quadric belongs to the connected component containing extremal curves. In particular the $(3,7)$ curves are connected to the extremal locus in $H_{10,12}$. "Connected component" is essential; this does not say all such curves lie in the same irreducible component.

Reported characteristic-zero specialization. In coordinates $[x:y:z:w]$, for ruling parameters $\{0,1,3,7\}$ the aid gives the quadric family \[ Q_t:\ x^2+t^3xw-t^5yz=0, \] with central double plane $x^2=0$. Its reported weight degeneration has central curve ideal \[ I_E=(x^2,xy,y^4,xG-y^3F), \] \[ G=(w+z)(w+3z)(w+4z)(w+7z)(w+8z)(w+10z),\qquad F=z^3(2w+11z). \] This is the claimed extremal length-four-ruling-lines limit, of degree $4$ and genus $-3$. The full equations for $C_t$ and the ruling-parameter convention are not supplied, so the displayed endpoint and quadrics do not alone reconstruct the entire flat family.

Reported liaison audit. A proper direct complete-intersection link of type $(2,7)$ from this degree-four curve would have residual degree $10$ and genus $12$. The aid proves that its Rao-module length is $24$, whereas a degree-$10$, genus-$12$ extremal curve has Rao-module length $384$. Hence that direct residual is not itself extremal. The implication from the four-line degeneration to connectivity of $(3,7)$ divisors genuinely uses biliaison and double-plane connectedness rather than this naive single-link identification. The existence result, explicit specialization and numerical obstruction are supplied claims requiring source review.

Short English statement

The aid reports positive solutions to both the quadric-divisor connectivity and four-skew-lines degeneration questions. It gives an explicit double-plane limit with its ideal. A Rao-length calculation also prevents identifying a direct $(2,7)$ residual with the desired extremal curve, so connectedness must use the stated biliaison argument.

Sources

• Local UnsolvedMath registry, permanent ID $20000827$, catalogue entry; canonical aim-arithmetic-geometry-notes.json, notes[72].
• AIM workshop "Components of Hilbert schemes," PDF Problem 26 and remark (ii)$\Rightarrow$(i): https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf. The extracted footer says recorded on the whiteboard by Izzet Coskun and edited by Li Li; it is an editorial attribution, not part of the mathematical question. Registry statement status reconstructed_unverified.
• Lella--Schlesinger degeneration and connectedness results, as named in the attached summary; no full title, theorem number or bibliographic reference supplied.
• Research attempt 1, full_solution, run status solved, internal rigor proved, low novelty confidence; registry SOLVED-IN-LITERATURE. Explicitly machine-generated, not peer reviewed, and to be independently verified before citation. Original literature field empty.