The study of deformations of Artinian algebras is central to the geometry of the Hilbert scheme of points, \(\mathrm{Hilb}^k(\mathbb{C}^n)\). The curvilinear component \(\mathrm{CHilb}_0^k(\mathbb{C}^n)\) plays a distinguished role: it parametrizes subschemes that are locally isomorphic to \(\mathrm{Spec}(k[t]/(t^k))\) inside \(\mathbb{C}^n\).
It is known that every such ideal is the deformation of the curvilinear ideal of the same dimension, but the proof is essentially nonconstructive. Such constructions would be useful for various applications. This problem is to provide an explicit deformation for a specific monomial algebra. The author expects that a solution to this problem would point toward a general strategy for constructing such deformations, which is the true target of this problem.
Warm-up: we ask for an explicit deformation to a simpler algebra, where the solution is known.
Update (2026-03-05): This problem has been removed from the benchmark, as we have determined that a solution does not meet our bar of being a publishable result in its own right. Rather, a solution can be characterized as finding a novel family of worked examples, but it would not be expected to yield a general strategy for producing deformations from arbitrary monomial ideals. We’ve amended our problem sourcing process to catch cases like this in the future. This case came to our attention because, in fact, this problem has been solved! The solution was generated by GPT-5.2 Pro running in a harness developed by David Turturean. Congratulations to David! Stay tuned for a preprint from David and the problem author, Gergely Berczi, describing this nice new family of examples.
Attempts by AI
We have evaluated the following models on this problem. “Warm-up” refers to an easier variant of the problem with a known solution.
AI Prompts
Warm-up
Work over a field $k$ of characteristic $0$. Construct an explicit flat one-parameter deformation from the algebra $k[t]/(t^3)$ to the algebra
\[
A \;=\; k[x,y]/(x,y)^2.
\]
Equivalently, you must give an ideal $I \subset k[x_1,\dots,x_s,\epsilon]$ (for some $s\ge 2$) such that the family
\[
\mathcal{A} \;=\; k[x_1,\dots,x_s,\epsilon]/I
\]
over $k[\epsilon]$ satisfies:
\begin{enumerate}
\item \textbf{Special fiber:} $\mathcal{A}\big|_{\epsilon=0} \cong A$.
In particular, the specialization $I|_{\epsilon=0}$ must cut out an algebra isomorphic to
$k[x,y]/(x,y)^2$ (it is allowed to use more generators $x_1,\dots,x_s$ as long as the fiber is isomorphic to $A$).
\item \textbf{Generic fiber:} for $a \neq 0$, $\mathcal{A}\big|_{\epsilon=a} \cong k[t]/(t^3)$.
\end{enumerate}
\subsection*{Mandatory output format}
Write a Python file using SymPy that defines the polynomial ring variables and the ideal generators.
It should define a function \texttt{solution} that defines SymPy symbols named
\texttt{x1},\dots,\texttt{xs} and \texttt{eps}, where $s\ge 3$. It should return 3 values:
\begin{itemize}
\item A list of generators using the SymPy symbols.
\item A list of all $s$ x-symbols in the same order used in the generators.
\item The symbol \texttt{eps} used in the generators.
\end{itemize}
Other contraints on the solution code:
\begin{itemize}
\item Coefficients must be integers only (no fractions, no floats, no extra parameters besides \texttt{eps}).
\item Use only polynomial operations \texttt{+}, \texttt{-}, \texttt{*}, \texttt{**}; do not use division.
\item Do not include any code at the file level. You may include a "main" block for testing, but it will not be executed by the verifier.
\end{itemize}
An example is given below.
```
import sympy
def solution():
# Example: s = 4. Define x1, x2, x3, x4, and eps (epsilon)
x1, x2, x3, x4, eps = sympy.symbols("x1 x2 x3 x4 eps")
# List of x-symbols to return
x_symbols = [x1, x2, x3, x4]
# List of generators
i_generators = [
x1 * x2,
x1 * x3 - eps * x1**8,
x2 * x3 + eps * x2**8,
x4,
x1**8,
x2**8,
x3**8,
x4**8,
]
return i_generators, x_symbols, eps
``` Full problem
Work over a field $k$ of characteristic $0$. Construct an explicit flat one-parameter deformation from the curvilinear algebra $k[t]/(t^{22})$ to the monomial "spider" algebra
\[
A \;=\; k[x,y,z]/(x^8,\;y^8,\;z^8,\;xy,\;xz,\;yz).
\]
Equivalently, you must give an ideal $I \subset k[x_1,\dots,x_s,\epsilon]$ (for some $s\ge 3$) such that the family
\[
\mathcal{A} \;=\; k[x_1,\dots,x_s,\epsilon]/I
\]
over $k[\epsilon]$ satisfies:
\begin{enumerate}
\item \textbf{Special fiber:} $\mathcal{A}\big|_{\epsilon=0} \cong A$.
In particular, the specialization $I|_{\epsilon=0}$ must cut out an algebra isomorphic to
$k[x,y,z]/(x^8,y^8,z^8,xy,xz,yz)$ (it is allowed to use more generators $x_1,\dots,x_s$ as long as the fiber is isomorphic to $A$).
\item \textbf{Generic fiber:} for $a \neq 0$, $\mathcal{A}\big|_{\epsilon=a} \cong k[t]/(t^{22})$.
\end{enumerate}
\subsection*{Mandatory output format}
Write a Python file using SymPy that defines the polynomial ring variables and the ideal generators.
It should define a function \texttt{solution} that defines SymPy symbols named
\texttt{x1},\dots,\texttt{xs} and \texttt{eps}, where $s\ge 3$. It should return 3 values:
\begin{itemize}
\item A list of generators using the SymPy symbols.
\item A list of all $s$ x-symbols in the same order used in the generators.
\item The symbol \texttt{eps} used in the generators.
\end{itemize}
Other contraints on the solution code:
\begin{itemize}
\item Coefficients must be integers only (no fractions, no floats, no extra parameters besides \texttt{eps}).
\item Use only polynomial operations \texttt{+}, \texttt{-}, \texttt{*}, \texttt{**}; do not use division.
\item Do not include any code at the file level. You may include a "main" block for testing, but it will not be executed by the verifier.
\end{itemize}
An example is given below.
```
import sympy
def solution():
# Example: s = 4. Define x1, x2, x3, x4, and eps (epsilon)
x1, x2, x3, x4, eps = sympy.symbols("x1 x2 x3 x4 eps")
# List of x-symbols to return
x_symbols = [x1, x2, x3, x4]
# List of generators
i_generators = [
x1 * x2,
x1 * x3 - eps * x1**8,
x2 * x3 + eps * x2**8,
x4,
x1**8,
x2**8,
x3**8,
x4**8,
]
return i_generators, x_symbols, eps
``` Mathematician survey
The author assessed the problem as follows.
Number of mathematicians highly familiar with the problem:
a majority of those working on a specialized topic (≈10)
Number of mathematicians who have made a serious attempt to solve the problem:
2–4
Rough guess of how long it would take an expert human to solve the problem:
3–12 months
Notability of a solution:
a solid result in a subfield
A solution would be published:
in a standard specialty journal
Likelihood of a solution generating more interesting math:
fairly likely: the problem is rich enough that most solutions should open new avenues
Probability that the problem is solvable as stated:
80–95%