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Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

Consider the affine space consisting of pairs of matrices (A,B) of fixed size, and its closed subvariety given by the rank conditions rank A ≤ a, rank B ≤ b, and rank (A ⋅ B) ≤ c, for three non-negative integers a,b,c. These varieties are precisely the orbit closures of representations for the equioriented A3 quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of two Schur functors of tautological bundles on a 2-step flag variety. We provide several techniques for the determination of these groups, which is of independent interest. [ABSTRACT FROM AUTHOR]

Let R denote a 6-dimensional subspace of the ring M4(k) of 4 × 4 matrices over an algebraically closed field k. Fix a vector space isomorphism M4(k) ≅ k4 ⊗ k4. We associate to R a closed subscheme XR of the Grassmannian of 2-dimensional subspaces of k4, where the reduced subscheme of XR is the set of 2-dimensional subspaces Q ⊆ k4 such that (Q ⊗ k4) ∩ R ≠ 0. Our main result is that if XR has minimal dimension (namely, one), then its degree is 20 when it is viewed as a subscheme of P5 via the Plücker embedding. We present several examples of XR that illustrate the wide range of possibilities for it; there are reduced and non-reduced examples. Two examples involve elliptic curves: in one case, XR is a P1-bundle over an elliptic curve the second symmetric power of the curve; in the other, it is a curve having seven irreducible components, three of which are quartic elliptic space curves, and four of which are smooth plane conics. These two examples arise naturally from a problem having its roots in quantum statistical mechanics. The scheme XR appears in non-commutative algebraic geometry: under appropriate hypotheses, it is isomorphic to the line scheme L of a certain graded algebra determined by R. In that context, it has been an open question for several years to describe such L of minimal dimension, i.e., those L of dimension one. Our main result implies that if (L) = 1, then, as a subscheme of P5 under the Plücker embedding, deg (L) = 20. [ABSTRACT FROM AUTHOR]