Castelnuovo–Mumford regularity and arithmetic Cohen–Macaulayness of complete bipartite subspace arrangements

Abstract

We give the Castelnuovo-Mumford regularity of arrangements of (n − 2)-planes in P n whose incidence graph is a sufficiently large complete bipartite graph, and determine when such arrangements are arithmetically Cohen-Macaulay.

Characteristic-free bases and BGG resolutions of unitary simple modules for quiver Hecke and Cherednik algebras

“…As a consequence, we prove a combinatorial formula for the Betti numbers of the ideal of the m-equals arrangement predicted in [BGS14]. We also calculate the Castelnuovo-Mumford regularity for the coordinate ring of these arrangements, a notoriously difficult problem in general (see [DS02,TT15]).…”

“…We hence provide formulae for the graded Betti numbers and calculate the Castelnuovo-Mumford regularity of these varieties -a notoriously difficult problem in general [DS02,TT15]. Moreover, we also provide formulae for these invariants in the cyclotomic case, where the equalities in the equations defining the above varieties become equalities up to multiplication by an th root of unity.…”

On BGG resolutions of unitary modules for quiver Hecke and Cherednik algebras

“…As a consequence, we prove a combinatorial formula for the Betti numbers of the ideal of the m-equals arrangement predicted in [65]. We also calculate the Castelnuovo-Mumford regularity for the coordinate ring of these arrangements, a notoriously difficult problem in general (see [16,57]).…”

“…We hence provide, see Propositions 8.2 and 8.6, formulae for the graded Betti numbers and calculate the Castelnuovo-Mumford regularity of these varieties -a notoriously difficult problem in general [16,57]. Moreover, we also provide formulae for these invariants in the cyclotomic case, where the equalities in the equations defining the above varieties become equalities up to multiplication by an th root of unity.…”

“…The Castelnuovo-Mumford regularity is a measure of the computational complexity of the module M and it is, in general, incredibly difficult to compute, cf. [16,57].…”