The expected codimension of a matroid variety

Abstract

Matroid varieties are the closures in the Grassmannian of sets of points defined by specifying which Plücker coordinates vanish and which do not. In general these varieties are very ill-behaved, but in many cases one can estimate their codimension by keeping careful track of the conditions imposed by the vanishing of each Plücker coordinates on the columns of the matrix representing a point of the Grassmannian. This paper presents a way to make this procedure precise, producing a number for each matroid variety called its expected codimension that can be computed combinatorially solely from the list of Plücker coordinates that are prescribed to vanish. We prove that for a special, well-studied class of matroid varieties called positroid varieties, the expected codimension coincides with the actual codimension.

The foundations of $(2n,k)$-manifolds

“…Following the approach described above (see the details in [6]), the embeddings I σ : F σ → F can be described using the notion of coordinates in a chart. We do it in the chart M 12 1), I σ14 (F σ14 ) = I σ23 (F σ23 ) = (0 : 1), I σ13 (F σ13 ) = I σ24 (F σ24) = (1 : 0) and I σ14,24 (F σ14,23 ) = (0 : 1), I σ13,24 (F σ13,24 ) = (1 : 0). There are also the strata W σ that do not belong to the chart M 12 but for which…”

“…Following [6], let (c i,12 : c ′ i,12 ), 1 ≤ i ≤ 3 be such coordinates in (CP 1 ) 3 that the main stratum in the local coordinates of the chart M 12 is given by the system of equations c ′ 1,12 z 12 11 z 12 22 = c 1,12 z 12 21 z 12 12 , c ′ 2,12 z 12 11 z 12 32 = c 2,12 z 12 31 z 12 12 , c ′ 3,12 z 12 21 z 12 32 = c 3,12 z 12 31 z 12 22 . The condition that z 12 11 , z 12 31 → 0 implies that inF 12 we have that (c 1,12 : c ′ 1,12 ) = (0 : 1), (c 3.12 : c ′ 3,12 ) = (1 : 0), while the limit of the points , (c 2,12 : c ′ 2,12 ) inF 12 is not defined. HereF 12 is the closure of the space of parameters F 12 of the main stratum in CP 1 × CP 1 × CP 1 considered in the chart M 12 .…”

Positroids and non-crossing partitions

“…, S t ) of [n], and then putting the structure of a connected positroid on each block S i . The first statement was also discovered in [OPS], where it is stated without proof, and in [For13]. We also give an alternative description of this non-crossing partition in terms of Kreweras complementation.…”

Geometric Equations for Matroid Varieties

“…We give a second proof that F is in I x using the Grassmann-Cayley algebra in the next section. Ford [3] claims that I x = N x + F though this is difficult to check without exhaustively computing the closure of Γ x .…”

“…We also claim that the three cubics are independent. Let If g 7 were a C[Gr(3, 9)]-combination of g 8 and g 9 , then we would have polynomials ϕ, ψ ∈ C[Gr (3,9)] such that g 7 = ϕg 8 + ψg 9 at all points in Gr(3, 9). Since points 8 and 9 are zero, g 8 (z) = g 9 (z) = 0 and (ϕg 8 +ψg 9 )(z) = 0.…”