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 KeywordsExact SequenceComplete IntersectionHomogeneous PolynomialPure ResolutionMaximal MinorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.