The one-step Shafarevich gap in embedding dimension five

Abstract

Let $k$ be an algebraically closed field of characteristic zero and let $S=k[x_1,\ldots,x_5]$ with maximal ideal $\mathfrak m=(x_1,\ldots,x_5)$. For a codimension-$r$ subspace $Q\subset S_2$, set $I_Q=(Q)+\mathfrak m^3$. Then $S/I_Q$ has Hilbert function $(1,5,r)$. We prove that the translated one-step locus defined by these ideals is contained in the smoothable component for every $r\in{6,7,\ldots,15}$. We introduce a finite field differential rank certificate proving dominance, for $6\le r\le 14$, of the Erman--Velasco map $\operatorname{GL}*5\times (\mathbb A^5)^r\dashrightarrow \operatorname{Gr}(r,\operatorname{Sym}^2 k^5)$, $(g,a^{(1)},\ldots,a^{(r)})\mapsto g\cdot\langle q(a^{(1)}),\ldots,q(a^{(r)})\rangle$, where $q(a)=\sum*{i=1}^5 a_i y_i^2-\left(\sum_{i=1}^5 a_i y_i\right)^2$. The endpoint $r=15$ is handled separately by a flat degeneration of $21$ general reduced points to the fat point defined by $\mathfrak m^3$. Combined with the known small cases and with the known elementary components for $r=3$ and $r=5$, this gives the complete one-step classification in embedding dimension five: the one-step loci with Hilbert function $(1,5,r)$ are smoothable for all $r\neq 3,5$, and the cases $r=3,5$ are precisely the generically reduced elementary component cases. In this sense the one-step Shafarevich gap in embedding dimension five is completely resolved.