A tangent-weight criterion for rational Hilbert components

Definitions and mathematical statement

The original asks whether a Hilbert-scheme component containing a smooth Borel-fixed point must be rational. In an explicit projective setting, let $k$ be algebraically closed of characteristic zero, $S=k[x_0,\ldots,x_n]$, $H=\operatorname{Hilb}^{P}({\mathbb{P}}_k^n)$, and let $[J]\in H$ be a Borel-fixed saturated homogeneous ideal at which the whole Hilbert scheme is smooth. There is then a unique component $C$ through $[J]$. Rational means birational over $k$ to ${\mathbb{P}}^{\dim C}$. The original does not specify the ambient Hilbert scheme or characteristic; this is the setting of its attached aid.

The diagonal torus acts on the tangent space \[ T_{[J]}H=\operatorname{Hom}_S(J,S/J)_0=\bigoplus_\chi V_\chi. \] List its occurring characters, with repetitions allowed, as $\chi_1,\ldots,\chi_m$. The aid's sufficient criterion is the existence of an integral cocharacter $w$ such that \[ \langle\chi_j,w\rangle>0\quad\text{for every }j. \] Then the one-parameter subgroup $\lambda_w(t)$ makes $[J]$ an isolated attracting fixed point in the appropriate direction, and the aid identifies its full-dimensional Białynicki--Birula attracting cell with affine space. Consequently the cell is dense in $C$ and $C$ is rational.

The finite alternative attributed to Gordan's theorem is that exactly one of the following holds: such an integral strict separator exists; or there are $c_j\in{\mathbb{Q}}_{\ge0}$, not all zero, with \[ \sum_jc_j\chi_j=0. \] The latter is an exact certificate that no diagonal one-parameter subgroup makes all tangent weights strictly positive or all strictly negative, since reversing a cocharacter reverses every sign. It does not prove that $C$ is nonrational, or rule out other rationality arguments.

The original remark gives a known special case for segment ideals and cites Lella--Roggero. It describes a segment ideal as a monomial ideal generated in some degree $s$, with $s$ at most its regularity, by the maximal monomials for a term ordering. It states that such ideals are Borel fixed, while the converse fails, and that the corresponding component is known to be rational. The precise segment/truncation and ordering convention should be checked against the cited work. The general smooth-Borel-fixed question remains open as the aid's literature assessment; the half-space criterion alone does not settle it.

Short English statement

Does a smooth Borel-fixed Hilbert point force its component to be rational? The supplied aid proves rationality under a strict tangent-weight separation criterion and gives a finite certificate when that particular source-or-sink method fails. It does not turn failure of the criterion into nonrationality.

Sources

• Local UnsolvedMath registry, permanent ID $20000809$, source code catalogue entry; canonical aim-arithmetic-geometry-notes.json, notes[54].
• American Institute of Mathematics, 2010 workshop "Components of Hilbert schemes," problem 8: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf. No PDF inspection is claimed here.
• Original remark: P. Lella and M. Roggero, Rational components of Hilbert schemes, arXiv:0903.1029. This is the supplied reference for the segment-ideal special case.
• Attached research attempt 1: partial result, recorded run status open, rigor proved, novelty confidence low; registry status partially solved, classification PARTIAL-PROGRESS. It invokes Białynicki--Birula cells and Gordan's theorem without further bibliography. This machine-generated, non-peer-reviewed aid requires independent verification before citation.