An exact ACM certificate for reverse-lexicographic gins of locally Cohen-Macaulay space curves

Definitions and mathematical statement

The source asks for necessary or sufficient conditions on Borel-fixed monomial ideals to occur as generic initial ideals of locally Cohen--Macaulay curves. It does not state the field, term order or ambient projective space. The supplied aid treats the following precise space-curve setting: $k$ is infinite of characteristic zero, $R=k[x_0,x_1,x_2,x_3]$, and initial ideals use degree reverse lexicographic order with $x_0>x_1>x_2>x_3$.

An ideal is strongly stable if replacing a factor $x_j$ of any monomial by $x_i$, $i<j$, keeps the monomial in the ideal; in this characteristic this is the relevant Borel-fixed condition. For a homogeneous ideal $I$, $\operatorname{gin}(I)$ is the initial ideal after a general linear coordinate change. Suppose the candidate monomial ideal $J$ is saturated and strongly stable and has a curve Hilbert polynomial $dt+1-g$, $d>0$.

The aid reports the exact ACM criterion: \[ \begin{aligned} J\text{ is the gin of an arithmetically Cohen--Macaulay curve in }{\mathbb{P}}^3_k \quad\Longleftrightarrow\quad &\text{no minimal generator of }J\text{ involves }x_2\text{ or }x_3\\ \quad\Longleftrightarrow\quad &J=B R\text{ for a finite-colength strongly stable }B\subset k[x_0,x_1]. \end{aligned} \] Here ACM means that the homogeneous coordinate ring is Cohen--Macaulay, which is stronger than the local Cohen--Macaulay condition on the projective curve. Under the baseline saturation and strong-stability assumptions, no minimal generator involves $x_3$ already, so the remaining finite check is absence of $x_2$.

When the condition holds, $J$ itself supplies a realization: its projective scheme is an ACM, hence locally Cohen--Macaulay, multiple line supported on $x_0=x_1=0$, and $\operatorname{gin}(J)=J$. For the finite Hilbert function \[ h_i=\dim_k(k[x_0,x_1]/B)_i, \] the degree and arithmetic genus are \[ d=\sum_i h_i,\qquad g=1-d+\sum_i i h_i. \] Conversely, if a minimal generator involves $x_2$, every locally Cohen--Macaulay curve realizing $J$ as gin must be non-ACM and have nonzero Rao module \[ M_C=\bigoplus_{t\in{\mathbb{Z}}}H^1({\mathbb{P}}^3_k,\mathcal I_C(t)). \] This is a conditional obstruction on any realization, not a proof that such a non-ACM realization exists or cannot exist. Thus the reported criterion settles the entire ACM region of the stated reverse-lexicographic problem but leaves the broader locally Cohen--Macaulay realizability problem open in the supplied assessment. No characteristic-$p$ replacement of Borel fixation by strong stability is asserted.

Short English statement

In the aid's characteristic-zero reverse-lexicographic setting, a candidate gin has an ACM curve realization exactly when its minimal generators use only the first two variables. The ideal itself then gives a multiple-line witness and explicit degree and genus. A generator in the third variable forces any locally Cohen--Macaulay witness to be non-ACM, without resolving its existence.

Sources

• Local UnsolvedMath registry, permanent ID $20000828$, source code catalogue entry, supplied complete source record and research summary.
• American Institute of Mathematics, Components of Hilbert Schemes, Problem 27, https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf; canonical local record aim-arithmetic-geometry-notes.json, notes[73]. No PDF was retrieved.
• The original literature field is empty. No full authoritative bibliographic citation for the generic-initial-ideal criterion or Rao-module implication is supplied.
• Research attempt 1: partial result; problem open at the recorded run, registry status partially solved; supplied rigor label "proved", novelty confidence "low". The machine-generated aid is not peer reviewed and requires independent verification before citation.