A rank-three flat family where the restricted Gröbner fan jumps

Definitions and mathematical statement

The source asks whether the Gröbner fan is a discrete invariant distinguishing irreducible components of $\operatorname{Hilb}^d({\mathbb{A}}^n)$. It does not specify weight domain, coordinates, labels or whether "invariant" means pointwise constancy or a generic value. The aid refutes the literal pointwise-constancy interpretation and keeps a separate generic question.

Over any field $k$, consider \[ J=(x^2-ty,xy,y^2)\subset k[t,x,y]. \] The quotient is free over $k[t]$ with basis $1,x,y$, so it defines a flat length-three family and an irreducible curve in $\operatorname{Hilb}^3({\mathbb{A}}^2_k)$. Restrict weights to the positive orthant $w_x,w_y>0$ and use highest-weight initial forms. At $t=0$, the fiber ideal is $(x,y)^2$, whose restricted fan has one maximal cone. At every nonzero fiber $t=c$, the aid reports exactly two maximal cones: \[ \begin{array}{c|c} 2w_x>w_y&\operatorname{in}_w(J_c)=(x,y)^2\\ 2w_x<w_y&\operatorname{in}_w(J_c)=(y,x^3). \end{array} \] Their common wall is $w_y=2w_x$. The special local algebra has embedding dimension two, whereas for $c\neq0$ it is $k[x]/(x^3)$, of embedding dimension one; the endpoint algebras are not isomorphic. Thus even the number of maximal restricted cones changes within this single irreducible family, in every characteristic. The raw restricted fan is not constant on a Hilbert-scheme component.

The aid separately reports generic constancy: for an irreducible finite-type family, the fan geometry together with its maximal monomial-initial-ideal labels is constant on a dense open subset, after fixing the relevant weight and coordinate conventions. This is a generic statement, not constancy at every specialization. It then isolates the surviving component question: after choosing both a general component point and general coordinates, does the resulting double-generic fan distinguish components? The summary supplies neither a full definition of the allowed coordinate action and fan equivalence nor an injectivity theorem. Consequently this generic formulation is not marked refuted or solved merely because the raw invariant jumps.

Short English statement

A flat family of three points already has a Gröbner fan that changes from one maximal positive-weight cone to two. This defeats pointwise constancy inside a component. The supplied generic-constancy result leaves a different question: whether suitably double-generic fans identify components.

Sources

• Local UnsolvedMath registry, permanent ID $20000816$, source code catalogue entry, supplied complete source snapshot and research summary.
• American Institute of Mathematics, 2010 workshop Components of Hilbert Schemes, Problem 15, https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf; canonical local record aim-arithmetic-geometry-notes.json, notes[61]. The broken OCR spelling is read as Gröbner; no PDF was retrieved.
• The original literature field is empty. No full authoritative bibliographic citation for the counterexample, localization/generic-constancy assertion or double-generic formulation is supplied.
• Research attempt 1: counterexample; problem refuted at the recorded run, registry status solved and classification SOLVED-IN-LITERATURE; supplied rigor label "proved", novelty confidence "low". These labels concern the raw reading. The machine-generated aid is not peer reviewed and requires independent verification before citation.