From standard monomial theory to semi-toric degenerations via Newton–Okounkov bodies

Abstract

The Hodge algebra structures on the homogeneous coordinate rings of Grassmann varieties provide semi-toric degenerations of these varieties. In this paper we construct these semi-toric degenerations using quasi-valuations and triangulations of Newton-Okounkov bodies.2010 Mathematics Subject Classification. 14M15(primary), and 14M25, 52B20(secondary).

Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties

“…Namely, we construct a lattice isomorphism τ : M → N and show that the Plücker algebra R/I has the structure of an algebra with straightening laws (or Hodge algebra) on N ( [H,DEP2]). We note that these facts were conjectured by Xin Fang and, possibly, other experts in the field (see [FL,Sections 8 and 9] for some related ideas and conjectures). In this respect we also point out that out of several technical combinatorial proofs that could not be avoided, the proof of the second property is the longest and most complicated in this paper.…”

Seshadri stratifications and standard monomial theory

“…In this case the poset A is a distributive lattice, hence for any p ∈ A, the subposet A p is shellable and hence Cohen-Macaulay over any field K ( [4]). Applying Proposition 16.1, we obtain: i) a degeneration of Schubert varieties X(i) ⊆ Gr d K n for i ∈ I d,n into a union of projective spaces using quasi-valuations, recovering the main results in [24]; ii) the projective normality of the Schubert varieties X(i) for i ∈ I d,n in the Plücker embedding; iii) the degree of the embedded Schubert varieties X(i) as the cardinality of C i for i ∈ I d,n ; iv) the Schubert varieties are defined by linear equations in the Grassmann variety; v) the intersection of two Schubert varieties X(i) ∩ X(j) is a reduced union of Schubert varieties. The projective normality of the Schubert varieties in Grassmann varieties are proved by Hochster [27], Laksov [46], and Musili [54] (see also the work of Igusa [29] for the Grassmann varieties themselves).…”

“…Motivated by seeking for an interpretation of the LS-paths in the above setup as vanishing order of functions, the second and the third author in [24] studied the case of Grassmann varieties. Instead of a flag of subvarieties, a web of subvarieties consisting of Schubert varieties is fixed.…”

“…the Stanley-Reisner algebra of the poset arising from the web). It was asked in [24,23] how to generalize this construction to Schubert varieties in a partial flag variety.…”

PBW degenerations, quiver Grassmannians, and toric varieties

“…As an intermediate step one considers the theory of Newton-Okounkov (NO) bodies [105,87]. The connection between the NO bodies and toric degenerations is used in many papers, see, e.g., [4,48,80,86]. The following holds true.…”