
TL;DR
This paper investigates the algebraic properties of ideals generated by principal minors of generic matrices, revealing structural isomorphisms and component decompositions, especially focusing on cases with fixed rank and specific matrix sizes.
Contribution
It introduces new isomorphisms between principal minor ideals in localized rings and analyzes the geometric structure of their algebraic sets, including component decomposition and linkage for specific cases.
Findings
For rank n, ideals of principal minors are isomorphic after localization.
The algebraic set defined by principal minors has a codimension n component.
In the case n=4, the components are linked, with implications for algebraic geometry.
Abstract
A minor is principal means it is defined by the same row and column indices. Let be a square generic matrix, the polynomial ring in entries of , over an algebraically closed field, . For fixed , let denote the ideal generated by the size principal minors of . When the resulting quotient ring is a normal complete intersection domain. When we break the problem into cases depending on a fixed rank, , of . We show when for any , the respective images of and in the localized polynomial ring, where we invert , are isomorphic. From that we show the algebraic set given by has a codimension component, plus a codimension 4 component defined by the determinantal ideal (which is given by all the submaximal minors of ). When the…
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