Thanks. This looks helpful. I don't have time today to think further about it but will do so tomorrow. One quick question: are you overloading $k$ when you say that 'At a point $p\in Y\cap X_k$ the divisor $D$ has equation $z_1\ldots z_k=0$ in local coordinates'?

That the maximal right ring of quotients of $Mat_n(R)$ is $Mat_n(q.f(R))$ (in your notation) follows immediately from Corollary 3.1.6 of McConnell and Robson's book `Noncommutative Noetherian rings': ams.org/bookstore-getitem/item=GSM-30

By the way you should probably see section 1.2.5 of math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf if you haven't already.