Toric degenerations of Grassmannians from matching fields

We study the algebraic combinatorics of monomial degenerations of Plucker forms which is governed by matching fields in the sense of Sturmfels and Zelevinsky. We provide a necessary condition for a matching field to yield a SAGBI basis of the Plucker algebra for 3-planes in n-space. When the ideal associated to the matching field is quadratically generated this condition is both necessary and sufficient. Finally, we describe a family of matching fields, called 2-block diagonal, whose ideals are quadratically generated. These matching fields produce a new family of toric degenerations of Gr(3,n).