Divisor class groups of ladder determinantal varieties (Q2365293)

Let \(X=(X_{ij})\) be an \(r\times s\) matrix of indeterminates over a field \(K\). A ladder \(L\) in \(X\) is a subset of \(X\) which satisfies the following condition: whenever \(X_{ij} \in L\) and \(X_{kl} \in L\) with \(i\leq k\) and \(j\leq l\), then \(X_{il} \in L\) and \(X_{kj} \in L\). Consider the set \(G(X)\) of all \(t_i\times t_i\) minors in the first \(s_i\) columns of \(X\), for \(i=1, \dots, n\), \(1\leq t_1< \cdots <t_n \leq \min (r,s)\), \(1\leq s_1 <\cdots <s_n=s\). Let \(G(L)\) be the subset of \(G(X)\), the elements of which only involve indeterminates of \(L\), and denote by \(I(L)\) the ideal in \(K[L] =K[X_{ij} \mid X_{ij} \in L]\) generated by \(G(L)\). The factor ring \(K[L]/I(L)\) is called a ladder determinantal ring. Ladder determinantal rings are known to be normal Cohen Macaulay domains [\textit{H. Narasimhan}, J. Algebra 102, 162-185 (1986; Zbl 0604.14045); \textit{J. Herzog} and \textit{Ngô Viêt Trung}, Adv. Math. 96, No. 1, 1-37 (1992; Zbl 0778.13022); \textit{J. V. Motwani} and \textit{M. A. Sohoni}, J. Algebra 186, No. 2, 323-337 (1996)]. The authors compute the divisor class group of \(K[L]/I(L)\) in the case in which \(L\) is a one-sided one-corner ladder that is \(L\) has the form \(\{X_{ij} \mid i>u\) or \(j>v\}\) or \(\{X_{ij} \mid i<x\) or \(j<y\}\) for some (``corners'') \((u,v)\) or \((x,y)\), respectively, satisfying \(0\leq u\), \(x\leq r+1\) and \(0\leq v\), \(y\leq s+1\).