Open problems in deformations of Artinian algebras, Hilbert schemes and around (Q6630003)

After an introduction and overview of the area, the author provides a list of 38 open problems related to (deformations of) Artinian algebras and Hilbert schemes. For each problem, the author provides both context and motivation. The problems are broken down into the following ten related areas.\N\begin{itemize}\N\item[1.] Irreducible components of punctual/nested Hilbert schemes and Quot schemes (on smooth varieties)\N\item[2.] Hilbert schemes on singular surfaces\N\item[3.] The locus of very compressed algebras\N\item[4.] Compressed Gorenstein algebras\N\item[5.] Special components of (punctual) Hilbert schemes, , e.g., non-rational, low dimensional, or characteristic specific components\N\item[6.] The tangent spaces to Hilbert schemes\N\item[7.] Punctual Hilbert schemes\N\item[8.] Multigraded Hilbert schemes\N\item[9.] Smoothability of subschemes\N\item[10.] Moduli spaces of bilinear maps\N\end{itemize}\NThe last three topics are also connected to the study of tensors and the complexity of matrix multiplication.\N\NIt is a testament to the high level of interest in these topics and their applications that three of the problems were solved during the preparation and review process of this article.\N\NFor the entire collection see [Zbl 1545.13003].