Unobstructed Stanley-Reisner degenerations for dual quotient bundles on \(G(2,n)\) (Q308149)

Let \(G(2,n)\) be the Grassmannian parametrizing 2-dimensional linear subspaces of an \(n\)-dimensional vector space. We denote by \(Q\) the tautological quotient bundle on \(G(2,n)\) and let \(Q^*\) be its dual. The bundle \(Q^*\) comes with an embedding in \(\mathbb{P}^{\binom{n}{2}-1}\times \mathbb{A}^n\). Let \(J_n\) be the ideal of \(Q^*\) in this embedding.NEWLINENEWLINEThe authors in their first main result, construct a term order such that under this construction, the initial ideal of \(J_n\) is the Stanley-Reisner ideal associated to the join of a \((2n-4)\)- dimensional simplex with a simplicial complex \(K_n\). As immediate corollary the authors conclude that the coordinate ring \(S_n/J_n\) of \(Q^*\) is Cohen-Macaulay. They use this knowledge to study the syzygies of the ideal \(J_n\).NEWLINENEWLINEIn their second main result, the authors are proving that the ring \(S_n/J_n\) is rigid for all \(n\geq 6\). The authors apply their result to several geometric questions, for example, from the last result, the authors are able to show that both the projectivization and the projective closure of \(Q^*\) are also rigid.