Gröbner Degenerations of Determinantal Ideals with an Application to Toric Degenerations of Grassmannians

The concept of the Gröbner fan for a polynomial ideal, introduced by Mora and Robbiano in 1988, provides a robust polyhedral framework where maximal cones correspond to the reduced Gröbner bases of the ideal. Within this geometric structure resides the tropical variety, a subcomplex of the Gröbner fan, utilized across various mathematical domains. Despite its significance, the computational complexity associated with tropical varieties often limits practical computations to smaller instances. In this note, we revisit a family of monomial ideals, called matching field ideals, from the context of Gröbner degenerations. We show that they can be obtained as weighted initial ideals of determinantal ideals. We explore the algebraic properties of these ideals, with a particular emphasis on minimal free resolutions and Betti numbers.