A fixed-locus criterion and an obstruction in the standard nonreduced toric chart

Definitions and mathematical statement

For a polynomial ring $S$ with an abelian-group grading and an admissible Hilbert function $h$, the multigraded Hilbert scheme parametrizes homogeneous ideals whose quotient graded pieces have the prescribed dimensions, with the corresponding flat-family condition. A fat point here means $\operatorname{Spec}A$ for a finite-dimensional local algebra $A$ with nonzero nilradical, hence length greater than one and a single reduced support point.

Problem 21. Does some multigraded Hilbert scheme have a connected component isomorphic, as a scheme, to a fat point? The source does not fix the grading, Hilbert function, ring, field or characteristic. Merely having a nonreduced component of positive dimension does not answer the question.

Reported fixed-locus criterion. A grading refinement can realize a refined multigraded Hilbert scheme inside the scheme-theoretic fixed locus of the additional grading torus. If the fixed local scheme at an isolated point is Artin local and nonreduced, and lies in the corresponding refined-Hilbert-function locus, it gives the requested fat connected component. Isolation and the fixed scheme structure are both required.

For a torus-graded affine ring $R=\bigoplus_\chi R_\chi$, the fixed subscheme uses the quotient \[ R\big/\langle R_\chi:\chi\ne0\rangle, \] not the invariant subring $R^T$, which instead defines an affine quotient. The report applies this distinction to a particular nonreduced toric Hilbert chart.

Reported chart obstruction. Over ${\mathbb{C}}$ let \[ R={\mathbb{C}}[z_5,z_{10},z_{11},z_{13}] \big/\big((z_5z_{10}^2-z_{13})z_{10}z_{11}^2\big), \] and consider the origin, corresponding to the designated monomial point. The $z_{11}^2$ factor gives the chart's doubled irreducible component. For an additional grading torus diagonal in these minimal coordinates, with character weights $w_i$, preservation of the displayed relation imposes \[ w_{13}=w_5+2w_{10}. \] The scheme-theoretic fixed quotient is obtained by setting to zero every coordinate of nonzero character. The aid proves that every such compatible diagonal action has fixed locus through the origin either a reduced zero-dimensional point or a positive-dimensional scheme. It is never an Artin nonreduced local scheme there.

Thus refining the grading diagonally in these four displayed coordinates cannot isolate the doubled component as a fat point. The conclusion is limited to that chart and those diagonal refinements; it does not exclude other grading actions, other charts or other multigraded Hilbert schemes. The aid reports a failed candidate construction and a criterion, not an answer to the general existence problem.

Short English statement

The source seeks a connected component that is entirely a nonreduced point. The aid gives a scheme-theoretic fixed-locus route, then shows that every compatible diagonal refinement of a standard four-parameter toric Hilbert chart fails: its fixed locus is either a reduced point or remains positive-dimensional. Other constructions remain unaddressed.

Sources

• Local UnsolvedMath registry, permanent ID $20000822$, catalogue entry; canonical aim-arithmetic-geometry-notes.json, notes[67].
• AIM workshop "Components of Hilbert schemes," PDF Problem 21: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf.
• Stillman--Sturmfels--Thomas chart, as identified in the supplied summary by its explicit four-variable ring. No full title, location or bibliographic reference is supplied.
• Research attempt 1, reduction, run status open, internal rigor proved, low novelty confidence; current registry partially solved. Explicitly a machine-generated research aid, not peer reviewed; independently verify before citation. Original literature field empty.