arXiv subjects
Publications and source records attributed to Zhichen Zhou.
We study graph systems generated by $\Delta$-Y and Y-$\Delta$ moves and prove that, for initial graphs of minimum degree at least $7$, $\Delta$-Y moves alone generate every reachable graph. As a consequence, for $n\ge 8$, the $\Delta$-Y system generated from the complete graph of $n$ vertices is canonically isomorphic as graded directed graphs to the space of partial Steiner systems on $n$ points.
The Brian\c{c}on-Iarrobino conjecture predicts the maximum singularity of the Hilbert scheme of a tetrahedral number of points. As for the maximal singularities of the Hilbert scheme of a non-tetrahedral number of points, the second named author gave some separate conjectural necessary and sufficient conditions. In this paper, we provide a conjectural sufficient condition for the necessary condition, and propose a monotonicity conjecture which predicts that for a fixed colength $l$, the maximal dimension of the tangent space over all the Borel-fixed ideals of colength $l$ is increasing with respect to the smallest pure exponent of the ideal.