Gröbner-Cell Parametrization of Punctual Hilbert Schemes

Definitions and mathematical statement

Let $k$ be the coefficient field and let $R=k[[x,y]]$ be the ring of formal power series in two variables. The supplied record imposes no characteristic or algebraic-closure hypothesis on $k$. A monomial ideal is an ideal generated by monomials $x^a y^b$, with $a,b\in{\mathbb{Z}}_{\geq 0}$. Here a zero-dimensional monomial ideal $E\subset R$ is understood to have finite colength $n=\dim_k(R/E)$.

The record associates to $E$ a canonical Hilbert--Burch matrix $H$, a space $T'(E)$ of admissible perturbation matrices, and the Gröbner cell $V(E)$ of ideals with initial ideal $E$. For a matrix $M$, write $I_t(M)$ for the ideal generated by its $t\times t$ minors. In the intended Hilbert--Burch presentation, these are the maximal minors of $H+N$. The record does not specify the size of $H$, the value of $t$, the construction and normalization of $H$, the admissibility conditions defining $T'(E)$, or the order and initial-ideal convention defining $V(E)$. Thus these objects retain their source-dependent meanings; the supplied information does not give a self-contained definition of the proposed map.

Conjectural target, subject to those unresolved definitions. For every zero-dimensional monomial ideal $E\subset k[[x,y]]$ in the intended scope, the assignment \[ \Phi_E:T'(E)\longrightarrow V(E),\qquad N\longmapsto I_t(H+N), \] is a well-defined bijection. Equivalently, every admissible $N$ produces an ideal in $V(E)$, and \[ \forall J\in V(E)\quad \exists!\,N\in T'(E)\quad J=I_t(H+N). \] The scope over coefficient fields, and whether the unit ideal is excluded, require clarification from the source.

The supplied original excerpt reports that the conjecture holds for lex-segment ideals, an unspecified slightly more general class, and complete intersections. It also reports agreement, for $n\leq 30$, of the cell dimensions predicted by the conjecture for a cellular decomposition of $\operatorname{Hilb}^{n}(k[[x,y]])$. This dimension agreement is reported evidence, not a proof of bijectivity. The supplied dated assessment labels the general problem open and reports injectivity and surjectivity as unproved; these are supplied source assessments, not independently verified conclusions.

Short English statement

The proposed parametrization says that every ideal in the Gröbner cell of a finite-colength monomial ideal arises from exactly one admissible perturbation of its canonical Hilbert--Burch matrix. Precise matrix conventions, admissibility conditions, initial-ideal conventions, and field hypotheses are missing from the supplied record, so this remains a source-dependent, statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30005408$, title "Gröbner-Cell Parametrization of Punctual Hilbert Schemes," exact source problem code catalogue entry. The record gives proposed year 2023 and names no proposer.
• Resolutions in Local Algebra and Singularity Theory (2023), Oberwolfach Reports, pp. 327--395, DOI 10.4171/owr/2023/6: https://doi.org/10.4171/owr/2023/6. The original excerpt includes the report page marker 347 and supplies the special-case and dimension evidence described above.
• Supplied literature reference: "OWR-2023-6, Conjecture 6," https://oa.tib.eu/renate/server/api/core/bitstreams/11773ea5-b1e5-4e9e-b650-808e9d9dc944/content.
• Supplied literature reference: "Homs--Winz, canonical Hilbert--Burch matrices," https://doi.org/10.1016/j.jalgebra.2021.04.021.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports injectivity and surjectivity as unproved in general, and treats the checks for $n\leq 30$ as evidence only. This is supplied dated curation, not independently verified literature assessment. The record labels its statement corrected_verified and reports direct checking of the official report; that verification claim is also supplied metadata and has not been independently checked here.