Desingularization of generic symmetric and generic skew-symmetric determinantal singularities (Q6564499)

Resolution of singularities of generic skew-symmetric and generic symmetric determinantal singularities is discussed. More precisely, let \(R_0\) be a commutative regular ring, let \N\[\NR:=R_0[x_{i,j}: 1\leq i, j\leq m]\N\]\Nbe the polynomial ring with \(m^2\) independent variables and coefficients in \(R_0\), and set \N\[\N{\mathcal J}_{m,r}:=\langle f_{I,J}: I,J\subseteq \{1,\ldots, m\}: \# I=\# J=r\rangle\subset R,\N\]\Nwhere \(f_{I,J}:=\text{det}(M_{I,J})\), and \(M_{I,J}:=(x_{i,j})_{i\in I, j\in J}\). If \(\text{char}(R_0)\neq 2\), define: \N\[\NI_{\text{skew}}:=\langle x_{i,j}+x_{j,i}, x_{k,k}: 1\leq i<j\leq m, 1\leq k\leq m\rangle,\N\]\Nand \N\[\NS_{\text{skew}}:=R/I_{\text{skew}}, \ \ \ Z_{\text{skew}}:=\text{Spec}(S_{\text{skew}}), \ \ {\mathcal I}_{m,r}^{\text{skew}}:={\mathcal J}_{m,r}+ I_{\text{skew}}, \ \ \ Y_{m,r}^{\text{skew}}: =\text{Spec}(R/{\mathcal I}_{m,r}^{\text{skew}})\subseteq Z_{\text{skew}}. \N\]\NIn addition, define: \N\[\NI_{\text{sym}}:=\langle x_{i,j}-x_{j,i}: 1\leq i<j\leq m\rangle,\N\]\N\[\NS_{\text{sym}}:=R/I_{\text{sym}}, \ \ \ Z_{\text{sym}}:=\text{Spec}(S_{\text{sym}}), \ \ \ {\mathcal I}_{m,r}^{\text{sym}}:={\mathcal J}_{m,r}+I_{\text{sym}},\N\]\N\[\NY_{m,r}^{\text{sym}}:=\text{Spec}(R/{\mathcal I}_{m,r}^{\text{sym}})\subset Z_{\text{sym}}.\N\]\NThe following theorems are proven:\N\N{Theorem A.} Let \(m,\ell\in {\mathbb Z}_{+}\) be positive integers with \(2\ell\leq m\), let \(R_0\) be a commutative regular ring with \(\text{char}(R_0)\neq 2\). The following sequence of blowing ups is an embedded resolution of singularities for the reduction of the generic skew-symmetric determinantal singularity \(Y^{\text{skew}}_{m,2\ell}\subseteq Z_{\text{skew}}\): \N\[\N\begin{array}{ccccccc} Z_{\text{skew}}=:Z_0 & \stackrel{{\pi_1}}{\longleftarrow} & Z_1 & \stackrel{{\pi_2}}{\longleftarrow} & \ldots & \stackrel{{\pi_{\ell-1}}}{\longleftarrow} & Z_{\ell-1}, \end{array} \N\] \Nwhere \(\pi_{\alpha}: Z_{\alpha}\to Z_{\alpha-1}\) is the blowing up with center the strict transform of \((Y_{m,2\alpha}^{\text{skew}})_{\text{red}}\simeq (Y_{m,2\alpha-1}^{\text{skew}})_{\text{red}}\) in \(Z_{\alpha-1}\), for \(\alpha\in \{1,\ldots, \ell-1\}\). \N\N\N{Theorem B.} Let \(m,r\in {\mathbb Z}_{+}\) be positive integers with \(r\leq m\), let \(R_0\) be a commutative regular ring. The following sequence of blowing ups is an embedded resolution of singularities for the generic symmetric determinantal singularity \(Y^{\text{sym}}_{m,r}\subseteq Z_{\text{sym}}\): \N\[\N\begin{array}{ccccccc} Z_{\text{sym}}=:Z_0 & \stackrel{{\pi_1}}{\longleftarrow} & Z_1 & \stackrel{{\pi_2}}{\longleftarrow} & \ldots & \stackrel{{\pi_{\ell-1}}}{\longleftarrow} & Z_{r-1}, \end{array}\N\]\Nwhere \(\pi_{\alpha}: Z_{\alpha}\to Z_{\alpha-1}\) is the blowing up with center the strict transform of \(Y_{m,\alpha}^{\text{sym}}\) in \(Z_{\alpha-1}\), for \(\alpha\in \{1,\ldots, r-1\}\).