A support-deficit criterion and the fold-singularity test family

Definitions and mathematical statement

Let $C$ be a curve over an algebraically closed field $k$, and let $\operatorname{Hilb}^d(C)$, $d\geq1$, parametrize zero-dimensional closed subschemes of length $d$. Dimension means the Krull dimension of an irreducible component. The intended nontrivial question is whether a pure one-dimensional curve can have an irreducible Hilbert component $W$ with \[ \dim W<d. \] The source does not specify purity, reducedness, irreducibility, or characteristic. Without a pure-curve convention, isolated zero-dimensional components would change the question. For an integral curve the open locus of $d$ distinct smooth points has dimension $d$; its closure is the smoothable component. A subscheme is smoothable here if it is a flat limit, on the same curve, of such reduced smooth-supported configurations.

The source remark reports a nonsmoothable component of dimension exactly $d>1$. This does not answer whether a component of smaller dimension exists. The remark gives neither its curve nor a reference, and nonsmoothable must not be equated with smaller-dimensional.

The attached aid supplies a support-separated reduction for integral curves with isolated singularities $p_1,\ldots,p_s$. Fix local lengths $a_j\geq0$, $\sum_j a_j\leq d$, and chosen punctual families $Q_j$ of subschemes supported at $p_j$, with dimensions $q_j$. Let the remaining length $d-\sum_j a_j$ vary over the smooth locus. The reported stratum dimension is \[ d-\sum_j a_j+\sum_j q_j. \] For its closure to be a counterexample component, both conditions are required: a genuine punctual deficit $\sum_jq_j<\sum_ja_j$, and global maximality of that closure among irreducible closed subsets of $\operatorname{Hilb}^d(C)$. A small punctual family can lie inside the closure of a larger deforming family and hence need not be a component.

For irreducible curves with one rational $r$-fold singularity, the aid invokes a 2025 classification and reports that the nonsmoothable components indexed by an admissible $a$ have dimension \[ D_{r,a}(d)=d+(a-1)(r-a)-1. \] It says every admitted component has dimension at least $d$, with equality only at $(r,a)=(3,2)$. The local singularity convention and complete admissible $a,d$ range are not given; this formula must not be evaluated at arbitrary indices and treated as a component-existence assertion. In particular this test family supplies no smaller-dimensional component. The aid describes the unrestricted pure-curve question as still open in the literature it checked. Its reduction and classification application are supplied unreviewed partial claims.

Short English statement

Can the length-$d$ Hilbert scheme of a pure curve have an irreducible component smaller than $d$? The supplied aid separates small support strata from actual components and reports that one rational-fold-singularity family has no such deficit, despite possible nonsmoothable components of dimension $d$.

Sources

• AIM Problem 20, Components of Hilbert schemes, including the dimension-$d$ nonsmoothable-component remark: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf.
• Local registry data/unsolvedmath/problems.json, permanent ID $20000821$, exact source code catalogue entry; canonical aim-arithmetic-geometry-notes.json, notes[66].
• The attached aid invokes a 2025 classification of curves with a rational $r$-fold singularity, without authors, title, identifier, or full statement. Attempt 1 is explicitly machine-generated, not peer reviewed, and requires independent verification before citation.