Blow-up of Straightening-Closed Ideals in Ordinal Hodge Algebras

Abstract

We study a class of ideals $I$ in graded ordinal Hodge algebras $A$. These ideals are distinguished by the fact that their powers have a canonical standard basis. This leads to Hodge algebra structures on the Rees ring and the associated graded ring. Furthermore, from a natural standard filtration one obtains a depth bound for $A/{I^n}$ which, under certain conditions, is sharp for $n$ large. Frequently one observes that ${I^n}= {I^{(n)}}$. Under suitable hypotheses it is possible to calculate the divisor class group of the Rees algebra. Our main examples are ideals of "virtual" maximal minors and ideals of maximal minors "fixing a submatrix".