  The geometric naturality of Schubert polynomials and their combinatorial pipe
dream representations was established by Knutson and Miller (2005) via
antidiagonal Gr\"obner degeneration of matrix Schubert varieties. We consider
instead diagonal Gr\"obner degenerations. In this dual setting, Knutson,
Miller, and Yong (2009) obtained alternative combinatorics for the class of
"vexillary'' matrix Schubert varieties. We initiate a study of general diagonal
degenerations, relating them to a neglected formula of Lascoux (2002) in terms
of the $6$-vertex ice model (recently rediscovered by Lam, Lee, and Shimozono
(2018) in the guise of "bumpless pipe dreams'').
