In 2018, Hamaker and Reiner generalized the notions of weak order and descent sets from permutations to monotone triangles, which are identified with alternating sign matrices (ASMs). As one associates a matrix Schubert variety to a permutation, one can more generally associate what is called an ASM variety to an ASM. Strong Bruhat order on the lattice of ASMs reflects contaminants among ASM varieties. In this talk, we will review the story for strong Bruhat order and will then describe a comparable relationship between the poset combinatorics and the algebraic geometry determined by ASMs under weak order. We will give applications to the computation of K-polynomials. This is joint work with Laura Escobar and Anna Weigandt.