A detachment obstruction and three-generator threshold for generic embedded points

Definitions and mathematical statement

Use the full Hilbert scheme of subschemes of ${\mathbb{P}}^3$ with polynomial $dt+1-g$, allowing embedded points; it cannot be restricted to the locally Cohen--Macaulay open locus. An embedded point is a zero-dimensional associated point lying on the one-dimensional support, distinct from an isolated point off that support.

Problem 33. Find the best pair $(d,g)$ for which an irreducible component has a general member with an embedded point, with $d$ as small and $g$ as close to zero as possible. The source expressly calls this optimization vague and supplies no ordering of the competing goals. Its known example is $(d,g)=(4,-15)$, cited to Chen--Nollet; that is an achieved pair, not a proved optimum.

Reported detachment obstruction. In the projective-space/field setting of the cited detachment theorem, the aid states that if a general member is obtained from a codimension-two locally complete-intersection curve by embedded points whose local quotient lengths are all at most three, those points detach in the same irreducible component. Consequently this situation cannot produce an embedded-general component. Here locally complete intersection means that the curve ideal is locally generated by a regular sequence of length two, and local multiplicity refers to length of the added zero-dimensional quotient.

Thus a candidate embedded-general component must use either a non-LCI underlying curve or an embedded local quotient of length at least four. This is a necessary branch division, not a sufficiency theorem for either branch.

To make the genus bookkeeping explicit, an embedded extension is described by \[ 0\longrightarrow\mathcal I_Z\longrightarrow\mathcal I_C\longrightarrow Q\longrightarrow0, \qquad\ell=\operatorname{length}Q, \] so $\deg Z=\deg C=d$ and $p_a(Z)=p_a(C)-\ell$. In the LCI branch, $\ell\geq4$ and the usual degree-$d$ curve genus bound used by the report gives the necessary screen \[ g\leq\binom{d-1}{2}-4. \] This screen does not prove that an appropriate non-detaching extension or component exists.

Reported simple-extension dimension threshold. At a point $p$ where $\mathcal I_C$ has $r$ minimal generators, a simple quotient $\mathcal I_C\twoheadrightarrow k(p)$ is parametrized by ${\mathbb{P}}^{r-1}$. If such support points move in a one-dimensional stratum of $C$, the extension locus has relative dimension $1+(r-1)=r$, whereas one detached point moves in three dimensions in ${\mathbb{P}}^3$. Thus $r=3$ is the first dimensionally competitive case in this comparison. If the $r$-generator stratum is isolated, the extra one-dimensional term is absent; no unrestricted dimension formula is inferred.

The aid says the known thick-quartic-line example realizes the three-generator threshold. It supplies neither the explicit example ideal nor a complete component classification. The threshold is a search guide and the $(4,-15)$ example does not determine the source's optimum. The precise characteristic, detachment hypotheses and component-level argument require checking in Chen--Nollet.

Short English statement

The source seeks the smallest-degree, closest-to-zero-genus component whose general curve has an embedded point. The aid gives necessary obstructions: low-length embedded extensions of LCI curves detach, while a simple moving extension first competes dimensionally at three generators. The known $(4,-15)$ example is retained without claiming optimality.

Sources

• Local UnsolvedMath registry, permanent ID $20000834$, catalogue entry; canonical aim-arithmetic-geometry-notes.json, notes[79].
• AIM workshop "Components of Hilbert schemes," PDF Problem 33 and nested remark: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf.
• D. Chen, S. Nollet, "Detaching embedded points," https://arxiv.org/abs/0911.2221, cited in the original for $(d,g)=(4,-15)$.
• Research attempt 1, partial_result, run status open, internal rigor proved, low novelty confidence; current registry partially solved. Explicitly machine-generated, not peer reviewed, and to be independently verified before citation. Original literature field empty.