Smooth Varieties with Squarefree Gröbner Degenerations

Definitions and mathematical statement

Let $K$ be a field and let $S=K[x_0,\ldots,x_n]$, with $\deg x_i=1$, be a standard graded polynomial ring. Let $I\subseteq S$ be a homogeneous ideal, put $R=S/I$, and write $X=\operatorname{Proj}(R)$. For a monomial order $\prec$, the initial ideal $\operatorname{in}_{\prec}(I)$ is generated by the leading monomials of elements of $I$. It is square-free if it admits monomial generators in which every variable has exponent at most one.

Let $\mathfrak m=R_+$ and $d=\dim R$. The ring $R$ is Cohen--Macaulay if $\operatorname{depth}_{\mathfrak m}R=d$. Its graded $a$-invariant is \[ a(R)=\sup\{j\in{\mathbb{Z}}:H^d_{\mathfrak m}(R)_j\ne0\}, \] where $H^d_{\mathfrak m}(R)$ denotes graded local cohomology and the supremum of the empty set is $-\infty$.

Question. Suppose that $X$ is a nonsingular projective variety, as stipulated in the source, and that there exists a monomial order $\prec$ for which $\operatorname{in}_{\prec}(I)$ is square-free. Must \[ \operatorname{depth}_{\mathfrak m}R=\dim R \qquad\text{and}\qquad a(R)<0 \] hold for every such $K,S,I$? No restriction on the characteristic of $K$ or on the monomial order is stated in the supplied formulation.

Curve formulation. Under the same hypotheses, if $\dim X=1$, must $X$ have genus zero? For a smooth projective geometrically integral curve over $K$, genus means \[ g(X)=\dim_K H^1(X,\mathcal O_X). \] The record does not specify geometric integrality or the intended genus convention outside that setting. Its original statement calls the curve question equivalent to the ring-theoretic conclusion; this claimed equivalence is retained as a source qualification requiring review.

Unresolved hypotheses. The source does not specify whether nonsingularity means regularity over $K$ or smoothness over $K$, whether "variety" requires geometric integrality, or whether $I$ is intended to be prime or saturated. The expression $X=\operatorname{Proj}(S/I)$ records the scheme-theoretic reading of "defines"; these additional conventions have not been imposed.

Short English statement

The question asks whether a homogeneous coordinate ring defining a nonsingular projective variety and admitting a square-free initial ideal must be Cohen--Macaulay with negative $a$-invariant. It also asks whether every curve satisfying these hypotheses has genus zero. The supplied dated assessment reports results for degree-reverse-lexicographic order and for curves, while retaining an open label for the general assertion. Field, geometric, and equivalence conventions require review.

Sources

• Local UnsolvedMath registry, permanent ID $30004076$, source code catalogue entry, title Smooth Varieties with Squarefree Gröbner Degenerations, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2019; no individual proposer is named.
• Supplied report citation: Singularities and Homological Aspects of Commutative Algebra (2020), Oberwolfach Reports, pp. 407--464, DOI 10.4171/owr/2019/6. https://doi.org/10.4171/owr/2019/6. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The literature-source list labels this reference "Oberwolfach Report 6/2019."
• Supplied literature reference: Huang--Tarasova--Varbaro--Witt, https://arxiv.org/abs/2512.00584. No title or theorem identifier is supplied.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open and states that the conjecture is proved for degree-reverse-lexicographic order, that the curve case has since been settled, and that the full assertion for arbitrary square-free initial ideals remains open. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The background explicitly requests independent verification of the time-sensitive status and cited claims.