Generic pipe dreams and the commuting scheme

Consider the equations XY=YX on a pair of matrices. Do these generate a prime ideal, or, are there secret equations that pairs of commuting matrices satisfy? Mel Hochster asked this in '84 and noone has answered it (past small matrix size). I'll degenerate this scheme into pieces indexed by "generic pipe dreams", thereby giving a formula for its degree as a sum of powers of 2, and use an associated formula to derive both the ordinary and bumpless pipe dream formulae for Schubert polynomials. This work is joint with Paul Zinn-Justin.