  We relate a classic algebro-geometric degeneration technique, dating at least
to [Hodge 1941], to the notion of vertex decompositions of simplicial
complexes. The good case is when the degeneration is reduced, and we call this
a "geometric vertex decomposition".
  Our main example in this paper is the family of vexillary matrix Schubert
varieties, whose ideals are also known as (one-sided) ladder determinantal
ideals. Using a diagonal term order to specify the (Gr\"obner) degeneration, we
show that these have geometric vertex decompositions into simpler varieties of
the same type. From this, together with the combinatorics of the pipe dreams of
[Fomin--Kirillov 1996], we derive a new formula for the numerators of their
multigraded Hilbert series, the double Grothendieck polynomials, in terms of
"flagged set-valued tableaux". This unifies work of [Wachs 1985] on flagged
tableaux, and [Buch 2002] on set-valued tableaux, giving geometric meaning to
both.
  This work focuses on diagonal term orders, giving results complementary to
those of [Knutson--Miller 2004], where it was shown that the generating minors
form a Gr\"obner basis for any antidiagonal term order and any matrix Schubert
variety. We show here that under a diagonal term order, the only matrix
Schubert varieties for which these minors form Gr\"obner bases are the
vexillary ones, reaching an end toward which the ladder determinantal
literature had been building.
