Determinantal Ideals, Symmetric Determinantal Ideals, and Open Problems

Abstract

In this last chapter, we address for determinantal ideals $I \subset K\left[ {x_0 , \ldots x_n } \right]$, i.e., ideals of codimension c = (p - r + 1)(q - r + 1) generated by the r × r minors of a p × q homogeneous matrix A (see Definition 1.2.3), for symmetric determinantal ideals $I \subset K\left[ {x_0 , \ldots x_n } \right]$, i.e., ideals of codimension $c = \left( {\begin{array}{*{20}c} {m - t + 2} \\ 2 \\ \end{array} } \right)$ generated by the t × t minors of an m × m homogeneous symmetric matrix A (see Definition 1.2.5), and for the three problems considered in the previous chapters for standard determinantal ideals. Namely, we address the following problems:

• (1) CI-liaison class and G-liaison class of determinantal ideals,

• (1′) CI-liaison class and G-liaison class of symmetric determinantal ideals,

• (2) the multiplicity conjecture for determinantal ideals,

• (2′) the multiplicity conjecture for symmetric determinantal ideals,

• (3) unobstructedness and dimension of families of determinantal schemes, and

• (3′) unobstructedness and dimension of families of symmetric determinantal schemes