Discriminantal Arrangement, $3\times3$ Minors of Pl\"ucker Matrix and   hypersurfaces in Grassmannian $Gr(3,n)$

S. Sawada; S. Settepanella; S. Yamagata

arXiv:1705.06239·math.AG·February 27, 2018

Discriminantal Arrangement, $3\times3$ Minors of Pl\"ucker Matrix and hypersurfaces in Grassmannian $Gr(3,n)$

S. Sawada, S. Settepanella, S. Yamagata

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TL;DR

This paper explores the relationship between certain degree 2 hypersurfaces in the Grassmannian $Gr(3,n)$ and generic hyperplane arrangements in $\

Contribution

It establishes a connection between discriminantal arrangements with triple intersections and specific hypersurfaces in the Grassmannian $Gr(3,n)$.

Findings

01

Points in these hypersurfaces correspond to hyperplane arrangements with triple intersection multiplicities.

02

The work links algebraic geometry of Grassmannians with combinatorial properties of hyperplane arrangements.

03

Provides a geometric characterization of arrangements via hypersurfaces in Grassmannian.

Abstract

We show that points in specific degree 2 hypersurfaces in the Grassmannian $G r (3, n)$ correspond to generic arrangements of $n$ hyperplanes in $C^{3}$ with associated discriminantal arrangement having intersections of multiplicity three in codimension two.

Figures1

Click any figure to enlarge with its caption.

Equations70

(a_{1}, \dots, a_{k}) \cdot (v_{1}, \dots, v_{k}) = i \sum a_{i} v_{i} .

(a_{1}, \dots, a_{k}) \cdot (v_{1}, \dots, v_{k}) = i \sum a_{i} v_{i} .

α_{L} = i = 1 \sum k + 1 (- 1)^{i} det (α_{s_{1}}, \dots, \overset{α_{s_{i}}}{^}, \dots, α_{s_{k + 1}}) e_{s_{i}},

α_{L} = i = 1 \sum k + 1 (- 1)^{i} det (α_{s_{1}}, \dots, \overset{α_{s_{i}}}{^}, \dots, α_{s_{k + 1}}) e_{s_{i}},

A (A_{\infty}) = (α_{L})_{L \in P_{k + 1} ([n])}

A (A_{\infty}) = (α_{L})_{L \in P_{k + 1} ([n])}

p_{T} (a_{ij}) = J \subset [n] ∣ J ∣ = m \sum det [A_{T, J} (A_{\infty})]^{2}

p_{T} (a_{ij}) = J \subset [n] ∣ J ∣ = m \sum det [A_{T, J} (A_{\infty})]^{2}

γ : G r (k, n) < v_{1}, \dots, v_{k} > \to P (⋀ k C^{n}) \mapsto [v_{1} \land \dots \land v_{k}],

γ : G r (k, n) < v_{1}, \dots, v_{k} > \to P (⋀ k C^{n}) \mapsto [v_{1} \land \dots \land v_{k}],

φ_{x} : C^{n} v \to ⋀ k + 1 C^{n} \mapsto v \land x

φ_{x} : C^{n} v \to ⋀ k + 1 C^{n} \mapsto v \land x

x = I \subseteq [n] ∣ I ∣ = k \sum β_{I} e_{I} = 1 \leq i_{1} < \dots < i_{k} \leq n \sum β_{i_{1} \dots i_{k}} (e_{i_{1}} \land \dots \land e_{i_{k}})

x = I \subseteq [n] ∣ I ∣ = k \sum β_{I} e_{I} = 1 \leq i_{1} < \dots < i_{k} \leq n \sum β_{i_{1} \dots i_{k}} (e_{i_{1}} \land \dots \land e_{i_{k}})

l = 0 \sum k (- 1)^{l} β_{i_{1} \dots i_{k - 1} j_{l}} β_{j_{0} \dots \hat{j_{l}} \dots j_{k}} = 0

l = 0 \sum k (- 1)^{l} β_{i_{1} \dots i_{k - 1} j_{l}} β_{j_{0} \dots \hat{j_{l}} \dots j_{k}} = 0

A (A_{\infty}) = M_{x} .

A (A_{\infty}) = M_{x} .

A_{T} (A_{\infty}) = α_{L_{1}} α_{L_{2}} α_{L_{3}} = - β_{234} - β_{256} 0 β_{134} β_{156} 0 - β_{124} 0 - β_{456} β_{123} 0 β_{356} 0 - β_{126} - β_{346} 0 β_{125} β_{345}, β_{ij k} = det a_{i 1} a_{i 2} a_{i 3} a_{j 1} a_{j 2} a_{j 3} a_{k 1} a_{k 2} a_{k 3}

A_{T} (A_{\infty}) = α_{L_{1}} α_{L_{2}} α_{L_{3}} = - β_{234} - β_{256} 0 β_{134} β_{156} 0 - β_{124} 0 - β_{456} β_{123} 0 β_{356} 0 - β_{126} - β_{346} 0 β_{125} β_{345}, β_{ij k} = det a_{i 1} a_{i 2} a_{i 3} a_{j 1} a_{j 2} a_{j 3} a_{k 1} a_{k 2} a_{k 3}

(I) ⎩ ⎨ ⎧ - β_{456} (β_{134} β_{256} - β_{234} β_{156}) = 0 β_{356} (β_{134} β_{256} - β_{234} β_{156}) = 0 - β_{346} (β_{134} β_{256} - β_{234} β_{156}) = 0 β_{345} (β_{134} β_{256} - β_{234} β_{156}) = 0 - β_{256} (β_{124} β_{356} - β_{123} β_{456}) = 0 β_{156} (β_{124} β_{356} - β_{123} β_{456}) = 0 - β_{126} (β_{124} β_{356} - β_{123} β_{456}) = 0 β_{125} (β_{124} β_{356} - β_{123} β_{456}) = 0 - β_{234} (β_{125} β_{346} - β_{126} β_{345}) = 0 β_{134} (β_{125} β_{346} - β_{126} β_{345}) = 0 - β_{124} (β_{125} β_{346} - β_{126} β_{345}) = 0 β_{123} (β_{125} β_{346} - β_{126} β_{345}) = 0 \mbox an d (I I) ⎩ ⎨ ⎧ β_{234} β_{126} β_{456} + β_{124} β_{256} β_{346} = 0 - (β_{234} β_{125} β_{456} + β_{124} β_{256} β_{345}) = 0 - (β_{234} β_{126} β_{356} + β_{123} β_{256} β_{346}) = 0 β_{234} β_{125} β_{356} + β_{123} β_{256} β_{345} = 0 - (β_{134} β_{126} β_{456} + β_{124} β_{156} β_{346}) = 0 β_{134} β_{125} β_{456} + β_{124} β_{156} β_{345} = 0 β_{134} β_{126} β_{356} + β_{123} β_{156} β_{346} = 0 - (β_{134} β_{125} β_{356} + β_{123} β_{156} β_{345}) = 0

(I) ⎩ ⎨ ⎧ - β_{456} (β_{134} β_{256} - β_{234} β_{156}) = 0 β_{356} (β_{134} β_{256} - β_{234} β_{156}) = 0 - β_{346} (β_{134} β_{256} - β_{234} β_{156}) = 0 β_{345} (β_{134} β_{256} - β_{234} β_{156}) = 0 - β_{256} (β_{124} β_{356} - β_{123} β_{456}) = 0 β_{156} (β_{124} β_{356} - β_{123} β_{456}) = 0 - β_{126} (β_{124} β_{356} - β_{123} β_{456}) = 0 β_{125} (β_{124} β_{356} - β_{123} β_{456}) = 0 - β_{234} (β_{125} β_{346} - β_{126} β_{345}) = 0 β_{134} (β_{125} β_{346} - β_{126} β_{345}) = 0 - β_{124} (β_{125} β_{346} - β_{126} β_{345}) = 0 β_{123} (β_{125} β_{346} - β_{126} β_{345}) = 0 \mbox an d (I I) ⎩ ⎨ ⎧ β_{234} β_{126} β_{456} + β_{124} β_{256} β_{346} = 0 - (β_{234} β_{125} β_{456} + β_{124} β_{256} β_{345}) = 0 - (β_{234} β_{126} β_{356} + β_{123} β_{256} β_{346}) = 0 β_{234} β_{125} β_{356} + β_{123} β_{256} β_{345} = 0 - (β_{134} β_{126} β_{456} + β_{124} β_{156} β_{346}) = 0 β_{134} β_{125} β_{456} + β_{124} β_{156} β_{345} = 0 β_{134} β_{126} β_{356} + β_{123} β_{156} β_{346} = 0 - (β_{134} β_{125} β_{356} + β_{123} β_{156} β_{345}) = 0

p_{T} (a_{ij}) = [det (α_{i} \times α_{i + 1})]^{2} = {(a_{12} a_{23} - a_{13} a_{22}) Δ_{11} + (a_{11} a_{23} - a_{13} a_{21}) Δ_{12} + (a_{11} a_{22} - a_{12} a_{2}) Δ_{13}}^{2},

p_{T} (a_{ij}) = [det (α_{i} \times α_{i + 1})]^{2} = {(a_{12} a_{23} - a_{13} a_{22}) Δ_{11} + (a_{11} a_{23} - a_{13} a_{21}) Δ_{12} + (a_{11} a_{22} - a_{12} a_{2}) Δ_{13}}^{2},

U_{i, j}^{⊥} = {v \in R^{k} ∣ v \cdot α_{τ} = 0, τ \in L_{i} \cap L_{j}},

U_{i, j}^{⊥} = {v \in R^{k} ∣ v \cdot α_{τ} = 0, τ \in L_{i} \cap L_{j}},

W_{T} = {R^{k} {v \in R^{k} ∣ v \cdot α_{τ} = 0, τ \in T} (T = \emptyset) (T \neq = \emptyset) .

W_{T} = {R^{k} {v \in R^{k} ∣ v \cdot α_{τ} = 0, τ \in T} (T = \emptyset) (T \neq = \emptyset) .

A_{W_{T}} = {H \cap τ \in T ⋂ H_{τ} ∣ H \in A \ {H_{i}}_{i \in T}}

A_{W_{T}} = {H \cap τ \in T ⋂ H_{τ} ∣ H \in A \ {H_{i}}_{i \in T}}

A = a_{11} ⋮ a_{61} a_{12} ⋮ a_{62} a_{13} ⋮ a_{63}

A = a_{11} ⋮ a_{61} a_{12} ⋮ a_{62} a_{13} ⋮ a_{63}

φ_{x} : C^{6} v \to ⋀ 4 C^{6} \mapsto v \land x,

φ_{x} : C^{6} v \to ⋀ 4 C^{6} \mapsto v \land x,

β_{i_{1} i_{2} k_{0}} β_{k_{1} k_{2} k_{3}} - β_{i_{1} i_{2} k_{1}} β_{k_{0} k_{2} k_{3}} + β_{i_{1} i_{2} k_{2}} β_{k_{0} k_{1} k_{3}} - β_{i_{1} i_{2} k_{3}} β_{k_{0} k_{1} k_{2}} = 0.

β_{i_{1} i_{2} k_{0}} β_{k_{1} k_{2} k_{3}} - β_{i_{1} i_{2} k_{1}} β_{k_{0} k_{2} k_{3}} + β_{i_{1} i_{2} k_{2}} β_{k_{0} k_{1} k_{3}} - β_{i_{1} i_{2} k_{3}} β_{k_{0} k_{1} k_{2}} = 0.

β_{124} β_{356} - β_{123} β_{456} + β_{125} β_{436} - β_{126} β_{435} = 0,

β_{124} β_{356} - β_{123} β_{456} + β_{125} β_{436} - β_{126} β_{435} = 0,

P l_{T} (D_{L_{4}}) = (α_{L_{i}})_{1 \leq i \leq 4} .

P l_{T} (D_{L_{4}}) = (α_{L_{i}})_{1 \leq i \leq 4} .

β_{246} \cdot (e) - β_{245} \cdot (d) = 0 .

β_{246} \cdot (e) - β_{245} \cdot (d) = 0 .

β_{256} β_{234} \cdot (c) - β_{246} \cdot (g) + β_{245} \cdot (f) = 0 .

β_{256} β_{234} \cdot (c) - β_{246} \cdot (g) + β_{245} \cdot (f) = 0 .

P l_{T} (D_{{2, 3, 5, 6}}) = - β_{234} - β_{256} 00 β_{134} β_{156} 0 - β_{356} - β_{124} 0 - β_{456} β_{256} β_{123} 0 β_{356} 0 0 - β_{126} - β_{346} - β_{236} 0 β_{125} β_{345} β_{235}

P l_{T} (D_{{2, 3, 5, 6}}) = - β_{234} - β_{256} 00 β_{134} β_{156} 0 - β_{356} - β_{124} 0 - β_{456} β_{256} β_{123} 0 β_{356} 0 0 - β_{126} - β_{346} - β_{236} 0 β_{125} β_{345} β_{235}

P l_{T} (D_{{1, 4, 5, 6}}) = - β_{234} - β_{256} 0 - β_{456} β_{134} β_{156} 00 - β_{124} 0 - β_{456} 0 β_{123} 0 β_{356} β_{156} 0 - β_{126} - β_{346} - β_{146} 0 β_{125} β_{345} β_{145}

P l_{T} (D_{{1, 4, 5, 6}}) = - β_{234} - β_{256} 0 - β_{456} β_{134} β_{156} 00 - β_{124} 0 - β_{456} 0 β_{123} 0 β_{356} β_{156} 0 - β_{126} - β_{346} - β_{146} 0 β_{125} β_{345} β_{145}

P l_{T} (D_{{1, 3, 5, 6}}) = - β_{234} - β_{256} 0 - β_{356} β_{134} β_{156} 00 - β_{124} 0 - β_{456} β_{156} β_{123} 0 β_{356} 0 0 - β_{126} - β_{346} - β_{136} 0 β_{125} β_{345} β_{135}

P l_{T} (D_{{1, 3, 5, 6}}) = - β_{234} - β_{256} 0 - β_{356} β_{134} β_{156} 00 - β_{124} 0 - β_{456} β_{156} β_{123} 0 β_{356} 0 0 - β_{126} - β_{346} - β_{136} 0 β_{125} β_{345} β_{135}

(a) = 0 : β_{134} β_{256} - β_{234} β_{156} = 0.

(a) = 0 : β_{134} β_{256} - β_{234} β_{156} = 0.

β_{L \ {s_{i}}} M_{i} \pm β_{L \ {s_{j}}} M_{j} = 0

β_{L \ {s_{i}}} M_{i} \pm β_{L \ {s_{j}}} M_{j} = 0

M_{i} = \pm M_{j} \cdot \frac{β _{L \ {s_{j}}}}{β _{L \ {s_{i}}}}

M_{i} = \pm M_{j} \cdot \frac{β _{L \ {s_{j}}}}{β _{L \ {s_{i}}}}

σ . T = {{i_{1}, i_{2}, i_{3}, i_{4}}, {i_{1}, i_{2}, i_{5}, i_{6}}, {i_{3}, i_{4}, i_{5}, i_{6}}}

σ . T = {{i_{1}, i_{2}, i_{3}, i_{4}}, {i_{1}, i_{2}, i_{5}, i_{6}}, {i_{3}, i_{4}, i_{5}, i_{6}}}

β_{i_{1} i_{3} i_{4}} β_{i_{2} i_{5} i_{6}} - β_{i_{2} i_{3} i_{4}} β_{i_{1} i_{5} i_{6}} = 0 .

β_{i_{1} i_{3} i_{4}} β_{i_{2} i_{5} i_{6}} - β_{i_{2} i_{3} i_{4}} β_{i_{1} i_{5} i_{6}} = 0 .

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Full text

Discriminantal Arrangement, $3\times 3$ Minors of Plücker Matrix and hypersurfaces in Grassmannian $Gr(3,n)$ .

S. Sawada

,

S. Settepanella

and

S. Yamagata

Department of Mathematics, Hokkaido University, Japan.

[email protected]

Abstract.

We show that points in specific degree 2 hypersurfaces in the Grassmannian $Gr(3,n)$ correspond to generic arrangements of $n$ hyperplanes in $\mathbb{C}^{3}$ with associated discriminantal arrangement having intersections of multiplicity three in codimension two.

Key words and phrases:

discriminantal arrangements, Intersection lattice, Plücker relations, Grassmannian

1991 Mathematics Subject Classification:

52C35 05B35 14M15

The second named author was supported by JSPS Kakenhi Grant Number 26610001.

1. Introduction

In 1989 Manin and Schechtman (cf.[10]) considered a family of arrangements of hyperplanes generalizing classical braid arrangements which they called the discriminantal arrangements (cf. [10] p.209). Such an arrangement ${\mathcal{B}}(n,k),n,k\in{\bf N}$ for $k\geq 2$ depends on a choice $H^{0}_{1},...,H^{0}_{n}$ of collection of hyperplanes in general position in ${\mathbb{C}}^{k}$ . It consists of parallel translates of $H_{1}^{t_{1}},...,H_{n}^{t_{n}},(t_{1},...,t_{n})\in{\mathbb{C}}^{n}$ which fail to form a generic arrangement in ${\mathbb{C}}^{k}$ . ${\mathcal{B}}(n,k)$ can be viewed as a generalization of the pure braid group arrangement (cf. [12]) with which ${\mathcal{B}}(n,1)$ coincides. These arrangements have several beautiful relations with diverse problems including combinatorics (cf. [10], [1], [3] and also [4], which is an earlier appearance of discriminantal arrangmements), Zamolodchikov equation with its relation to higher category theory (cf. Kapranov-Voevodsky [7]), and vanishing of cohomology of bundles on toric varieties (cf. [13]). Paper [10] concerns with arrangements ${\mathcal{B}}(n,k)$ which combinatorics is constant on a Zariski open set $\mathcal{Z}$ in the space of generic arrangements $H^{0}_{i},i=1,...,n$ but does not describe the set $\mathcal{Z}$ explicitly. In 1994 (see [5]) Falk showed that, contrary to what was frequently stated (see for instance [11], sect. 8, [12] or [8]), the combinatorial type of ${\mathcal{B}}(n,k)$ depends on the arrangement ${\mathcal{A}}$ of hyperplanes $H^{0}_{i},i=1,...,n$ by providing an example of ${\mathcal{A}}$ for which the corresponding discriminantal arrangement has combinatorial type distinct from the one which occurs when ${\mathcal{A}}$ varies within the Zariski open set $\mathcal{Z}$ . In 1997 Bayer and Brandt ( cf. [3] ) called the arrangements ${\mathcal{A}}$ in $\mathcal{Z}$ very generic and conjectured the full description of intersection lattice of ${\mathcal{B}}(n,k)$ if ${\mathcal{A}}\in\mathcal{Z}$ . In 1999 Athanasiadis proved their conjecture (cf. [1]). In particular, for the case of arrangement ${\mathcal{A}}$ in $\mathbb{R}^{k}$ , endowed with standard metric, he introduced a degree $m$ polynomial $p_{\mathbb{T}}(a_{ij})$ ( section 1 in [1] and subsection 2.3 in this paper ) in the indeterminates $(a_{ij})$ where $\alpha_{i}=(a_{ij})$ is the normal vector to hyperplane $H_{i}^{0}$ , $i\in L_{h}\in\mathbb{T}$ , $L_{h}$ is a subset of cardinality $k+1$ of $\{1,\ldots,n\}$ and $\mathbb{T}$ is a set of cardinality $m$ . Since null space of this polynomial corresponds to intersection of hyperplanes in ${\mathcal{B}}(n,k)$ , he provided, in case of very generic arrangements, a full description of sets $\mathbb{T}$ such that $p_{\mathbb{T}}(a_{ij})=0$ (cf. Theorem 3.2 in [1]). In particular all codimension $2$ intersections of hyperplanes in ${\mathcal{B}}(n,k)$ have multiplicity $2$ or $k+2$ if ${\mathcal{A}}$ is very generic.

More recently, in 2016 ( cf. [9]), Libgober and second author gave a sufficient geometric condition for an arrangement ${\mathcal{A}}$ not to be very generic. In particular they gave a necessary and sufficient condition for multiplicity $3$ codimension $2$ intersections of hyperplanes in ${\mathcal{B}}(n,k)$ to appear ( Theorem 3.8 [9] and Theorem 2.2 in this paper).

The purpose of this short note is double. From one side to rewrite the result obtained in [9] in terms of the polynomial $p_{\mathbb{T}}(a_{ij})$ introduced by Athanasiadis and prove that, in case of non very generic arrangements, if $\mathbb{T}$ is a set of cardinality $3$ such that $p_{\mathbb{T}}(a_{ij})=0$ , then polynomial $p_{\mathbb{T}}(a_{ij})$ has a simpler polynomial expression $\widetilde{p}_{\mathbb{T}}(a_{ij})$ .

On the other side to show, by mean of a more algebraic point of view, that non very generic arrangements ${\mathcal{A}}$ of cardinality $n$ in ${\mathbb{C}}^{3}$ are points in a well defined degree $2$ hypersurface in the projective Grassmannian $Gr(3,n)$ . Indeed the space of generic arrangements of $n$ lines in $({\mathbb{P}}^{2})^{n}$ is Zariski open set $U$ in the space of all arrangements of $n$ lines in $({\mathbb{P}}^{2})^{n}$ . On the other hand in $Gr(3,n)$ there is open set $U^{\prime}$ consisting of $3$ -spaces intersecting each coordinate hyperplane transversally (i.e. having dimension of intersection equal $2$ ). One has also one set $\tilde{U}$ in $Hom({\mathbb{C}}^{3},{\mathbb{C}}^{n})$ consisting of embeddings with image transversal to coordinate hyperplanes and $\tilde{U}/GL(3)=U^{\prime}$ and $\tilde{U}/({\mathbb{C}}^{*})^{n}=U$ . Hence generic arrangements can be regarded as points in $Gr(3,n)$ .

The content of paper is the following.

In section 2 we recall definition of discriminantal arrangement from [10], basic results in [9], definition of $p_{\mathbb{T}}(a_{ij})$ in [1] and basic notions on Grassmannian ( cf. [6] ). In section 3 we give a full description of main example ${\mathcal{B}}(6,3)$ of $6$ hyperplanes in ${\mathbb{R}}^{3}$ . Section 4 contains the result stating equivalence of polynomial $p_{\mathbb{T}}(a_{ij})$ with its reduced form $\widetilde{p}_{\mathbb{T}}(a_{ij})$ (cf. Theorem 4.4). The last section contain the last result of this paper (cf. Theorem 5.4) describing a family of hypersurfaces in projective Grassmannian $Gr(3,n)$ in terms of non very generic arrangements ${\mathcal{A}}$ in ${\mathbb{C}}^{3}$ . Notice that in Sections 3 and 4 ${\mathcal{A}}$ is an arrangement in $\mathbb{R}^{k}$ while in Sections 5 ${\mathcal{A}}$ is an arrangement in $\mathbb{C}^{k}$ .

Finally, authors want to thank A. Libgober and an anonymous referee for very useful comments.

2. Preliminaries

2.1. Discriminantal arrangement

Let $H^{0}_{i},i=1,...,n$ be a generic arrangement in ${\mathbb{C}}^{k},k<n$ i.e. a collection of hyperplanes such that ${\rm dim}\bigcap_{i\in K,\mid K\mid=k}H_{i}^{0}=0$ . Space of parallel translates $\SS(H_{1}^{0},...,H_{n}^{0})$ (or simply $\SS$ when dependence on $H_{i}^{0}$ is clear or not essential) is the space of $n$ -tuples $H_{1},...,H_{n}$ such that either $H_{i}\cap H_{i}^{0}=\emptyset$ or $H_{i}=H_{i}^{0}$ for any $i=1,...,n$ . One can identify $\SS$ with $n$ -dimensional affine space ${\mathbb{C}}^{n}$ in such a way that $(H^{0}_{1},...,H^{0}_{n})$ corresponds to the origin. In particular, an ordering of hyperplanes in ${\mathcal{A}}$ determines the coordinate system in $\SS$ (see [9]).

We will use the compactification of ${\mathbb{C}}^{k}$ viewing it as ${\mathbb{P}}^{k}\setminus H_{\infty}$ endowed with collection of hyperplanes $\bar{H}^{0}_{i}$ which are projective closures of affine hyperplanes $H^{0}_{i}$ . Condition of genericity is equivalent to $\bigcup_{i}H^{0}_{i}$ being a normal crossing divisor in ${\mathbb{P}}^{k}$ .

For a generic arrangement ${\mathcal{A}}$ in ${\mathbb{C}}^{k}$ formed by hyperplanes $H_{i},i=1,...,n$ the trace at infinity (denoted by ${\mathcal{A}}_{\infty}$ ) is the arrangement formed by hyperplanes $H_{\infty,i}=\bar{H}^{0}_{i}\cap H_{\infty}$ .

The trace $\mathcal{A}_{\infty}$ of an arrangement $\mathcal{A}$ determines the space of parallel translates $\mathbb{S}$ (as a subspace in the space of $n$ -tuples of hyperplanes in $\mathbb{P}^{k}$ ). For a $t$ -tuple $H_{i_{1}},\dots,H_{i_{t}}$ ( $t\geq 1$ ) of hyperplanes in $\mathcal{A}$ , recall that the arrangement which is obtained by intersections of hyperplanes $H\in\mathcal{A},H\neq H_{i_{s}}$ , $s=1,\dots,t$ with $H_{i_{1}}\cap\dots\cap H_{i_{t}}$ , is called the restriction of $\mathcal{A}$ to $H_{i_{1}}\cap\dots\cap H_{i_{t}}$ .

For a generic arrangement $\mathcal{A}_{\infty}$ , consider the closed subset of $\mathbb{S}$ formed by those collections which fail to form a generic arrangement. This subset is a union of hyperplanes with each hyperplane $D_{L}$ corresponding to a subset $L=\{i_{1},\dots,i_{k+1}\}\subset[n]:=\{1,\dots,n\}$ and consisting of $n$ -tuples of translates of hyperplanes $H_{1}^{0},\dots,H_{n}^{0}$ in which translates of $H_{i_{1}}^{0},\dots,H_{i_{k+1}}^{0}$ fail to form a generic arrangement. The arrangement $\mathcal{B}(n,k,\mathcal{A}_{\infty})$ of hyperplanes $D_{L}$ is called $the$ $discriminantal$ $arrangement$ and has been introduced by Mannin and Schectman (see [10]). Notice that since combinatorics of discriminantal arrangement depends on the arrangement $\mathcal{A}_{\infty}$ rather than ${\mathcal{A}}$ , we denote it by $\mathcal{B}(n,k,\mathcal{A}_{\infty})$ following notation in [9].

2.2. Good 3s-partitions

Given $s\geq 2$ and $n\geq 3s$ , consider the set $\mathbb{T}=\{L_{1},L_{2},L_{3}\}$ , with $L_{i}$ subsets of $[n]$ such that $|L_{i}|=2s$ , $|L_{i}\cap L_{j}|=s$ ( $i\neq j$ ), $L_{1}\cap L_{2}\cap L_{3}=\emptyset$ (in particular $|\bigcup L_{i}|=3s$ ) with a choice $L_{1}=\{i_{1},\dots,i_{2s}\},L_{2}=\{i_{s+1},\dots,i_{3s}\},L_{3}=\{i_{1},\dots,i_{s},i_{2s+1},\dots,i_{3s}\}$ . We call the set $\mathbb{T}=\{L_{1},L_{2},L_{3}\}$ a good $3$ s-partition.

Given a generic arrangement $\mathcal{A}$ in $\mathbb{C}^{k}$ , subsets $L_{i}$ define hyperplanes $D_{L_{i}}$ in the discriminantal arrangement $\mathcal{B}(n,k,\mathcal{A}_{\infty})$ . In the rest of the paper we will always use $D_{L}$ to denote hyperplanes in discriminantal arrangement. With above notations the following lemma holds.

Lemma 2.1.

(Lemma 3.1 [9]) Let $s\geq$ 2, $n=3s$ , $k=2s-1$ and $\mathcal{A}$ be a generic arrangement of $n$ hyperplanes in $\mathbb{C}^{k}$ . Given a good $3s$ -partition $\mathbb{T}=\{L_{1},L_{2},L_{3}\}$ of [ $n$ ] = [ $3s$ ], consider the triple of codimension $s$ subspaces $H_{\infty,i,j}=\bigcap_{t\in L_{i}\cap L_{j}}H_{\infty,t}$ of the hyperplane at infinity $H_{\infty}$ . Then $H_{\infty,i,j}$ span a proper subspace in $H_{\infty}$ if and only if the codimension of $D_{L_{1}}\cap D_{L_{2}}\cap D_{L_{3}}$ is 2.

In [9] authors define a notion of dependency for a generic arrangement $\mathcal{A}_{\infty}=\{W_{\infty,1},\dots,W_{\infty,3s}\}$ in $\mathbb{P}^{2s-2},s\geq 2$ based on Lemma 2.1 as follows. If there exists a partition $I_{1},I_{2}$ and $I_{3}$ of [ $3s$ ] such that $P_{i}=\bigcap_{t\in I_{i}}W_{\infty,t}$ span a proper subspace in $\mathbb{P}^{2s-2}$ , then $\mathcal{A}_{\infty}$ is called dependent. Remark that if $\{L_{1},L_{2},L_{3}\}$ is a good $3s$ -partition and we set $I_{1}=L_{1}\cap L_{2}$ , $I_{2}=L_{1}\cap L_{3}$ , $I_{3}=L_{2}\cap L_{3}$ then the the assumption of Lemma 2.1 is that the trace at infinity $\mathcal{A}_{\infty}$ of ${\mathcal{A}}$ is dependent and the following theorem holds .

Theorem 2.2.

(Theorem 3.8 [9]) Let $\mathcal{A}$ be a generic arrangement of $n$ hyperplanes in $\mathbb{C}^{k}$ and $\mathcal{A}_{\infty}$ the trace at infinity of $\mathcal{A}$ .

1. The arrangement $\mathcal{B}(n,k,\mathcal{A}_{\infty})$ has $\binom{n}{k+2}$ codimension 2 strata of multiplicity $k+2$ .

2. There is one to one correspondence between

(a) restriction arrangements of $\mathcal{A}_{\infty}$ which are dependent, and

(b) triples of hyperplanes in $\mathcal{B}(n,k,\mathcal{A}_{\infty})$ for which the codimension of their intersection is equal to 2.

3. There are no codimension 2 strata having multiplicity 4 unless $k=3$ . All codimension 2 strata of $\mathcal{B}(n,k,\mathcal{A}_{\infty})$ not mentioned in part 1, have multiplicity either 2 or 3.

4. Combinatorial type of $\mathcal{B}(n,2,\mathcal{A}_{\infty})$ is independent of $\mathcal{A}$ .

2.3. Matrices $A(\mathcal{A}_{\infty})$ and $A_{\mathbb{T}}(\mathcal{A}_{\infty})$

Let $\alpha_{i}=(a_{i1},\dots,a_{ik}$ ) be the normal vectors of hyperplanes $H^{0}_{i}$ , $1\leq i\leq n$ , in the generic arrangement $\mathcal{A}$ in $\mathbb{C}^{k}$ . Normal here is intended with respect to the usual dot product

[TABLE]

Then the normal vectors to hyperplanes $D_{L}$ , $L=\{s_{1}<\dots<s_{k+1}\}\subset$ [ $n$ ] in $\mathbb{S}\simeq\mathbb{C}^{n}$ are nonzero vectors of the form

[TABLE]

where $\{e_{j}\}_{1\leq j\leq n}$ is the standard basis of $\mathbb{C}^{n}$ (cf. [1]).

Let $\mathcal{P}_{k+1}([n])=\{L\subset[n]\mid|L|=k+1\}$ be the set of cardinality $k+1$ subsets of $[n]$ , we denote by

[TABLE]

the matrix having in each row the entries of vectors $\alpha_{L}$ normal to hyperplanes $D_{L}$ and by $A_{\mathbb{T}}(\mathcal{A}_{\infty})$ the submatrix of $A(\mathcal{A}_{\infty})$ with rows $\alpha_{L}$ , $L\in\mathbb{T}$ , $\mathbb{T}\subset\mathcal{P}_{k+1}([n])$ of cardinality $m$ .

2.4. Polynomial $p_{\mathbb{T}}(a_{ij})$

Construction in Subsection 2.3 naturally holds also in real case, i.e. ${\mathcal{A}}$ arrangement in ${\mathbb{R}}^{k}$ . In this case Athanasiadis (see [1] ) defined the polynomial

[TABLE]

in the variable $a_{ij}$ given by the sum of the squares of determinants of the $m\times m$ submatrices $A_{\mathbb{T},J}$ of $A_{\mathbb{T}}(\mathcal{A}_{\infty})$ obtained considering columns $j\in J$ . Notice that if ${\mathcal{A}}$ is a generic arrangement in ${\mathbb{R}}^{k}$ , if $\mathbb{T}=\{L_{1},L_{2},L_{3}\}$ is a good $3s$ -partition then condition in Lemma 2.1 is equivalent to $p_{\mathbb{T}}(a_{ij})=0$ .

2.5. Grassmannian $Gr(k,n)$

Let $Gr(k,n)$ be the Grassmannian of $k$ -dimensional subspaces of ${\mathbb{C}}^{n}$ and

[TABLE]

the Plücker embedding. Then $[x]\in\mathbb{P}(\bigwedge^{k}{\mathbb{C}}^{n})$ is in $\gamma(Gr(k,n))$ if and only if the map

[TABLE]

has kernel of dimension $k$ , i.e. ker $\varphi_{x}=<v_{1},\dots,v_{k}>$ . If $e_{1},\dots,e_{n}$ is a basis of ${\mathbb{C}}^{n}$ then $e_{I}=e_{i_{1}}~{}\wedge~{}\dots~{}\wedge~{}e_{i_{k}}$ , $I=\{i_{1},\dots,i_{k}\}\subset[n],i_{1}<\dots<i_{k}$ , is a basis for $\bigwedge^{k}{\mathbb{C}}^{n}$ and $x\in\bigwedge^{k}{\mathbb{C}}^{n}$ can be written uniquely as

[TABLE]

where homogeneous coordinates $\beta_{I}$ are the Plücker coordinates on $\mathbb{P}(\bigwedge^{k}{\mathbb{C}}^{n})=\mathbb{P}^{\binom{n}{k}-1}$ associated to the ordered basis $e_{1},\dots,e_{n}$ of ${\mathbb{C}}^{n}$ . With this choice of basis for ${\mathbb{C}}^{n}$ the matrix $M_{x}=(b_{ij})$ associated to $\varphi_{x}$ is the $\binom{n}{k+1}\times n$ matrix with rows indexed by ordered subsets $I\subseteq[n],|I|=k$ , and entries $b_{ij}=(-1)^{i}\beta_{I\cup\{j\}\setminus\{i\}}$ if $i\in I$ , $b_{ij}=0$ otherwise. Plücker relations, i.e conditions for dim(ker $\varphi_{x}$ ) = $k$ , are vanishing conditions of all $(k+1)\times(k+1)$ minors of $M_{x}$ . It is well known (see for instance [6]) that Plücker relations are degree 2 relations and they can also be written as

[TABLE]

for any $2k$ -tuple $(i_{1},\dots,i_{k-1},j_{0},\dots,j_{k})$ .

Remark 2.3.

Notice that vectors $\alpha_{L}$ in equation (1) normal to hyperplanes $D_{L}$ correspond to rows $I=L$ in the Plücker matrix $M_{x}$ , that is

[TABLE]

For this reason in the rest of the paper we will call $A(\mathcal{A}_{\infty})$ Plücker coordinate matrix. Notice that, in particular, $\det(\alpha_{s_{1}},\dots,\hat{\alpha_{s_{i}}},\dots,\alpha_{s_{k+1}})$ is the Plücker coordinate $\beta_{I},I=\{s_{1},s_{2},\dots,s_{k+1}\}\backslash\{s_{i}\}$ .

In the following section we give an example to illustrate the general Theorem in section 4. This example appears also in [5], [9] and, in the context of oriented matroids, in [2].

3. Example $\mathcal{B}$ (6, 3, $\mathcal{A}_{\infty}$ ) in real case

Consider $\mathcal{A}=\{H_{1}^{0},H_{2}^{0},\dots,H_{6}^{0}\}$ be a generic arrangement of hyperplanes in $\mathbb{R}^{3}$ with normal vectors $\alpha_{i}=(a_{i1},a_{i2},a_{i3})$ , $1\leq i\leq 6$ and $H_{i}^{t_{i}}$ be hyperplane obtained by translating $H_{i}^{0}$ along direction $\alpha_{i}$ , i.e., $H_{i}^{t_{i}}=H_{i}^{0}+t_{i}\alpha_{i}$ , $t_{i}\in\mathbb{R}$ . Let $\mathbb{T}=\{L_{1},L_{2},L_{3}\}$ be the good $6$ -partition with $L_{1}=\{1,2,3,4\},L_{2}=\{1,2,5,6\}$ and $L_{3}=\{3,4,5,6\}$ , then

[TABLE]

is a submatrix of the Plücker coordinate matrix $A(\mathcal{A}_{\infty})$ .

Let $\alpha_{i}\times\alpha_{j}$ be the cross product of $\alpha_{i},\alpha_{j}$ corresponding to the direction orthogonal to both $\alpha_{i}$ and $\alpha_{j}$ and denote by $(\alpha_{i}\times\alpha_{i+1})$ the matrix $\begin{pmatrix}\alpha_{1}\times\alpha_{2}\\ \alpha_{3}\times\alpha_{4}\\ \alpha_{5}\times\alpha_{6}\end{pmatrix}$ . Then $\alpha_{i}\times\alpha_{j}$ is the direction of the line $H_{i}\cap H_{j}$ , since $\alpha_{i}$ and $\alpha_{j}$ are, respectively, directions orthogonal to $H_{i}$ and $H_{j}$ and ${\rm rank}A_{\mathbb{T}}(\mathcal{A}_{\infty})=2$ if and only ${\rm rank}(\alpha_{i}\times\alpha_{i+1})=2$ . Indeed ${\rm rank}(A_{\mathbb{T}}(\mathcal{A}_{\infty}))=2$ is equivalent to codim ( $D_{L_{1}}\cap D_{L_{2}}\cap D_{L_{3}}$ ) = 2, hence by Lemma 2.1, the points ${\displaystyle\bigcap_{i\in L_{1}\cap L_{2}}}\bar{H}_{i}^{t_{i}}\cap H_{\infty}=\bar{H}_{3}^{t_{3}}\cap\bar{H}_{4}^{t_{4}}\cap H_{\infty},{\displaystyle\bigcap_{i\in L_{1}\cap L_{3}}}\bar{H}_{i}^{t_{i}}\cap H_{\infty}=\bar{H}_{1}^{t_{1}}\cap\bar{H}_{2}^{t_{2}}\cap H_{\infty}$ , and ${\displaystyle\bigcap_{i\in L_{2}\cap L_{3}}}\bar{H}_{i}^{t_{i}}\cap H_{\infty}=\bar{H}_{5}^{t_{5}}\cap\bar{H}_{6}^{t_{6}}\cap H_{\infty}$ are collinear, that is directions of $H_{i}^{t_{i}}\cap H_{i+1}^{t_{i+1}}$ are dependent and hence ${\rm rank}(\alpha_{i}\times\alpha_{i+1})=2$ (see Fig. 1 ).

Rank of $A_{\mathbb{T}}(\mathcal{A}_{\infty})$ is equal to 2 if and only if $\beta_{ijk}$ are solutions of the system:

[TABLE]

and polynomial $p_{\mathbb{T}}(a_{ij})$ is

$p_{\mathbb{T}}(a_{ij})={\displaystyle\sum_{\begin{subarray}{c}J\subset[6]\\ |J|=3\end{subarray}}}\det(A_{\mathbb{T},J})^{2}=$ $(\beta_{134}\beta_{256}-\beta_{234}\beta_{156})^{2}({\displaystyle\sum_{\begin{subarray}{c}I_{1}\subset\{3,4,5,6\}\\ |I_{1}|=3\end{subarray}}}\beta_{I_{1}}^{2})$ + $(\beta_{124}\beta_{356}-\beta_{123}\beta_{456})^{2}({\displaystyle\sum_{\begin{subarray}{c}I_{2}\subset\{1,2,5,6\}\\ |I_{2}|=3\end{subarray}}}\beta_{I_{2}}^{2})$ + $(\beta_{125}\beta_{346}-\beta_{126}\beta_{345})^{2}({\displaystyle\sum_{\begin{subarray}{c}I_{3}\subset\{1,2,3,4\}\\ |I_{3}|=3\end{subarray}}}\beta_{I_{3}}^{2})$ + ${\displaystyle\sum_{\begin{subarray}{c}i=5,6\\ j=3,4\end{subarray}}}(\beta_{234}\beta_{12i}\beta_{j56}+\beta_{12j}\beta_{256}\beta_{34i})^{2}$ + ${\displaystyle\sum_{\begin{subarray}{c}i=5,6\\ j=3,4\end{subarray}}}(\beta_{134}\beta_{12i}\beta_{j56}+\beta_{12j}\beta_{156}\beta_{34i})^{2}$ .

On the other hand the condition ${\rm rank}(\alpha_{i}\times\alpha_{i+1})=2$ is simply $\det(\alpha_{i}\times\alpha_{i+1})=0$ and if we define

[TABLE]

$\Delta_{1l}$ cofactors of $(\alpha_{i}\times\alpha_{i+1})$ , then $p_{\mathbb{T}}(a_{ij})=0$ if and only if $\widetilde{p}_{\mathbb{T}}(a_{ij})=0$ . That is polynomial $\widetilde{p}_{\mathbb{T}}(a_{ij})$ is a polynomial of, in general, lower degree than $p_{\mathbb{T}}(a_{ij})$ with the same set of zeros.

4. Polynomial $\widetilde{p}_{\mathbb{T}}(a_{ij})$ in $\mathcal{B}(n,k,\mathcal{A}_{\infty})$ in real case

4.1. Case $\mathcal{B}$ ( $n$ , 3, $\mathcal{A}_{\infty}$ )

It is straightforward to generalize the example in section 3 to the case of $n$ hyperplanes in $\mathbb{R}^{3}$ . Denote by $(\alpha_{i_{j}}\times\alpha_{i_{j+1}})$ the matrix $\begin{pmatrix}\alpha_{i_{1}}\times\alpha_{i_{2}}\\ \alpha_{i_{3}}\times\alpha_{i_{4}}\\ \alpha_{i_{5}}\times\alpha_{i_{6}}\end{pmatrix}$ , the following Theorem holds.

Theorem 4.1.

Let $\mathcal{A}$ be a generic arrangement of $n$ hyperplanes in $\mathbb{R}^{3}$ with normal vectors $\alpha_{i}=(a_{i1},a_{i2},a_{i3})$ . Let $\mathbb{T}=\{L_{1},L_{2},L_{3}\}$ be a good 6-partition with a choice $L_{1}=\{i_{1},i_{2},i_{3},i_{4}\},L_{2}=\{i_{3},i_{4},i_{5},i_{6}\}$ and $L_{3}=\{i_{1},i_{2},i_{5},i_{6}\}$ and $A_{\mathbb{T}}(\mathcal{A_{\infty}})$ be the matrix with rows $\alpha_{L_{1}}$ , $\alpha_{L_{2}}$ , $\alpha_{L_{3}}$ . Then the followings are equivalent:

(1)

${\rm rank}A_{\mathbb{T}}(\mathcal{A}_{\infty})=2$ ; 2. (2)

$p_{\mathbb{T}}(a_{ij})=0$ ; 3. (3)

${\rm rank}(\alpha_{i_{j}}\times\alpha_{i_{j+1}})=2$ ; 4. (4)

$\widetilde{p}_{\mathbb{T}}(a_{ij})=[\det(\alpha_{i_{j}}\times\alpha_{i_{j+1}})]^{2}=0$ .

Proof.

The equivalences (1) $\Leftrightarrow$ (2) and (3) $\Leftrightarrow$ (4) are obvious by definitions of $p_{\mathbb{T}}(a_{ij})$ and $\widetilde{p}_{\mathbb{T}}(a_{ij})$ . The proof that (1) $\Leftrightarrow$ (3) can be obtained by remarks in Section3 relabeling indices $1,\dots,6$ with $i_{1},\dots,i_{6}$ . ∎

Remark 4.2.

Notice that since $\widetilde{p}_{\mathbb{T}}(a_{ij})=[\det(\alpha_{i_{j}}\times\alpha_{i_{j+1}})]^{2}$ , then $\widetilde{p}_{\mathbb{T}}(a_{ij})=0$ if and only if $\det(\alpha_{i_{j}}\times\alpha_{i_{j+1}})=0$ and equivalence of conditions $(1),(3)$ and $(4)$ in Theorem 4.1 holds also for generic arrangements in ${\mathbb{C}}^{3}$ .

4.2. Generalization to $\mathcal{B}(n,k,\mathcal{A}_{\infty})$

Let $\mathcal{A}=\{H_{1},\dots,H_{n}\}$ be a generic arrangement of hyperplanes in $\mathbb{R}^{k}$ and $\mathbb{T}=\{L_{1},L_{2},L_{3}\}$ be a good $3s$ -partition of indices in [ $n$ ]. If $\alpha_{\tau}$ are normal vectors to $H_{\tau}\in\mathcal{A}$ , $\tau=1,\dots,n$ , $T=\{j_{1},\cdots,j_{t}\}$ a subset of [ $n$ ] which has empty intersection with $L_{1}\cup L_{2}\cup L_{3}$ , define vector spaces

[TABLE]

where $v\cdot\alpha_{\tau}$ is the scalar product of $v$ and $\alpha_{\tau}$ , and

[TABLE]

Then $W_{T}$ is the vector space associated to ${\displaystyle\bigcap_{\tau\in T}}H_{\tau}$ and $U_{i,j}^{\perp}\cap W_{T}=\{v\in\mathbb{R}^{k}\ |\ v\cdot\alpha_{\tau}=0,\tau\in(L_{i}\cap L_{j})\cup T\}$ is a vector space of dimension $k-(s+t)$ , where $s$ and $t$ are, respectively, cardinalities of $L_{i}\cap L_{j}$ and $T$ . With above notations, define the polynomial

$\widetilde{p}_{\mathbb{T},T}(a_{ij})={\displaystyle\sum_{\begin{subarray}{c}U\end{subarray}\in\mathbb{U}_{\mathbb{T},T}}}[\det U]^{2}$ ,

where $\mathbb{U}_{\mathbb{T},T}$ is the set of all $k\times k$ submatrices of the $3(k-s-t)\times k$ matrix having as rows the vectors spanning $U_{i,j}^{\perp}\cap W_{T}$ .

If $k=2s-1$ and $n=3s$ , $s\geq 2$ , we have $T=\emptyset$ and hence $U_{i,j}^{\perp}\cap W_{T}=U_{i,j}^{\perp}$ is a space of dimension ${\rm dim}U_{i,j}^{\perp}=s-1$ . $\mathbb{U}_{\mathbb{T},\emptyset}$ is the set of all $(2s-1)\times(2s-1)$ submatrices of the $3(s-1)\times(2s-1)$ matrix having as rows the vectors spanning $U_{i,j}^{\perp}$ and the following lemma equivalent to Lemma 2.1 holds.

Lemma 4.3.

Let $s\geq 2$ , $n=3s$ , $k=2s-1$ , i.e. $T=\emptyset$ , and ${\mathcal{A}}$ be a generic arrangement of $n$ hyperplanes in ${\mathbb{R}}^{k}$ . Given a good $3s$ -partition $\mathbb{T}=\{L_{1},L_{2},L_{3}\}$ of $[3s]=[n]$ , $U_{i,j}^{\perp}$ span a proper subspace of $\mathbb{R}^{k}$ if and only if the rank of $A_{\mathbb{T}}(\mathcal{A}_{\infty})$ is 2. That is $\widetilde{p}_{\mathbb{T},\emptyset}(a_{ij})=0$ if and only if $p_{\mathbb{T}}(a_{ij})=0$ .

Proof.

Since $\mathbb{T}$ is a good $3s$ -partition and $A_{\mathbb{T}}(\mathcal{A}_{\infty})=(\alpha_{L})_{L\in\mathbb{T}}$ is a $3\times n$ matrix, the rank of the matrix $A_{\mathbb{T}}(\mathcal{A}_{\infty})$ is equal to 2 if and only if $\alpha_{L},L\in\mathbb{T}$ , are linearly dependent that is the intersection $D_{L_{1}}\cap D_{L_{2}}\cap D_{L_{3}}$ of hyperplanes in $\mathcal{B}(n,k,\mathcal{A}_{\infty})$ is a space of codimension 2. Then by Lemma 2.1 this corresponds to $H_{\infty,i,j}=\bigcap_{\tau\in L_{i}\cap L_{j}}\bar{H}_{\tau}\cap H_{\infty}$ $\subset H_{\infty}$ span a proper subspace in $H_{\infty}$ . Let $V_{\tau}$ be the associated vector spaces to hyperplanes $H_{\tau}$ , hence $V_{i,j}=\bigcap_{\tau\in L_{i}\cap L_{j}}V_{\tau}$ are the associated vector spaces to $H_{i,j}=\bigcap_{\tau\in L_{i}\cap L_{j}}H_{\tau}$ and $V_{i,j}=U_{i,j}^{\perp}$ since $v\in V_{i,j}$ if and only if $v\cdot\alpha_{\tau}=0$ for any $\tau\in L_{i}\cap L_{j}$ . It follows that $H_{\infty,i,j}$ span a proper subspace of $H_{\infty}$ if and only if $U_{i,j}^{\perp}$ span a proper subspace of $\mathbb{R}^{k}$ . That is $\det U=0$ for any $U\in\mathbb{U}_{\mathbb{T},\emptyset}$ or, equivalently, $\widetilde{p}_{\mathbb{T},\emptyset}(a_{ij})=0$ . ∎

Notice that if $s$ = 2, i.e. case $\mathcal{B}$ (6, 3, $\mathcal{A}_{\infty}$ ), $\widetilde{p}_{\mathbb{T},\emptyset}(a_{ij})$ coincides with $\widetilde{p}_{\mathbb{T}}(a_{ij})$ defined in Section 3. In this case 1-dimensional subspaces $U_{1,2}^{\perp}$ , $U_{1,3}^{\perp}$ and $U_{2,3}^{\perp}$ are spanned, respectively, by $\alpha_{1}\times\alpha_{2},\alpha_{3}\times\alpha_{4}$ and $\alpha_{5}\times\alpha_{6}$ , that is they are the lines drown in Figure 1.

Analogously to [9] we call a generic arrangement $\mathcal{A}$ = $\{W_{1},\cdots,W_{3s}\}$ in $\mathbb{R}^{2s-1}$ , $s\geq 2$ , dependent if there exists a good 3 $s$ -partition such that $U_{i,j}^{\perp}$ span a proper subspace of $\mathbb{R}^{2s-1}$ . With this notation, by Lemma 2.1 and Theorem 2.2, the following theorem holds.

Theorem 4.4.

Let ${\mathcal{A}}$ be a generic arrangement of $n$ hyperplanes in $\mathbb{R}^{k}$ , $\mathbb{T}$ a good $3s$ -partition, $3s\leq n$ , and $T=[n]\setminus\cup_{L\in\mathbb{T}}L$ . If $W_{T}$ is the vector space defined in equation (10), then rank of $A_{\mathbb{T}}(\mathcal{A_{\infty}})$ is equal to $2$ if and only if the restriction arrangement

[TABLE]

is dependent. With this choice of $T$ and $\mathbb{T}$ we get that $p_{\mathbb{T}}(a_{ij})=0$ if and only if $\widetilde{p}_{\mathbb{T},T}(a_{ij})=0.$

Remark 4.5.

*For a fixed good $3s$ -partition $\mathbb{T}$ , equation $p_{\mathbb{T}}(a_{ij})=0$ corresponds to $n\choose 3s$ $3s\choose s$ nonlinear relations on Plücker coordinates $\beta_{I}$ , $(2s-1)\times(2s-1)$ minors of the matrix $A=(a_{ij})$ .

On the other hand $\widetilde{p}_{\mathbb{T},T}(a_{ij})=0$ is equivalent to vanishing of $(2s-1)\times(2s-1)$ minors of the matrix with rows given by solutions of system $A_{I}\cdot x=0$ , $A_{I}=(a_{ij})_{i\in I}$ , i.e. $n\choose 3s$ $3s-3\choose 2s-1$ equations on $a_{ij}$ . That is $\widetilde{p}_{\mathbb{T},T}(a_{ij})=0$ is reduced form of $p_{\mathbb{T}}(a_{ij})=0$ .*

5. Hypersurfaces in complex Grassmannian $Gr(3,n)$

Let now ${\mathcal{A}}$ be a generic arrangement of $6$ hyperplanes in $\mathbb{C}^{3}$ (i.e. the example in Section 3 in $\mathbb{C}^{3}$ instead of $\mathbb{R}^{3}$ ) and

[TABLE]

be the matrix having in each row normal vectors $\alpha_{i}$ to hyperplanes $H_{i}^{0}\in{\mathcal{A}}$ . Since ${\mathcal{A}}$ is generic, columns of $A$ are independent vectors in ${\mathbb{C}}^{6}$ and they span a subspace of dimension $3$ in $\mathbb{C}^{6}$ , i.e. an element in the Grassmannian $Gr(3,6)$ . The non null $3\times 3$ minors of $A$ are Plücker coordinates $\beta_{ijk}$ and the matrix $A(\mathcal{A}_{\infty})$ is the matrix of the map

[TABLE]

where $x=\sum_{1\leq i<j<k\leq n}\beta_{ijk}(e_{i}\wedge e_{j}\wedge e_{k})$ . If ${\mathcal{A}}_{\infty}$ is dependent then $\beta_{ijk}$ have to satisfy both, classical Plücker relations and relations in equation (8) (notice that since relations in equation (8) come directly from condition ${\rm rank}A_{\mathbb{T}}(\mathcal{A}_{\infty})=2$ , we get exactly same relations in real and complex case) . The latter can be simplified as:

$(I):\,\begin{cases}(a):\beta_{134}\beta_{256}-\beta_{234}\beta_{156}=0\\ (b):\beta_{124}\beta_{356}-\beta_{123}\beta_{456}=0\\ (c):\beta_{125}\beta_{346}-\beta_{126}\beta_{345}=0\end{cases}$ and $(II):\,\begin{cases}(d):\beta_{234}\beta_{126}\beta_{456}+\beta_{124}\beta_{256}\beta_{346}=0\\ (e):\beta_{234}\beta_{125}\beta_{456}+\beta_{124}\beta_{256}\beta_{345}=0\\ (f):\beta_{234}\beta_{126}\beta_{356}+\beta_{123}\beta_{256}\beta_{346}=0\\ (g):\beta_{234}\beta_{125}\beta_{356}+\beta_{123}\beta_{256}\beta_{345}=0\\ (h):\beta_{134}\beta_{126}\beta_{456}+\beta_{124}\beta_{156}\beta_{346}=0\\ (i):\beta_{134}\beta_{125}\beta_{456}+\beta_{124}\beta_{156}\beta_{345}=0\\ (j):\beta_{134}\beta_{126}\beta_{356}+\beta_{123}\beta_{156}\beta_{346}=0\\ (k):\beta_{134}\beta_{125}\beta_{356}+\beta_{123}\beta_{156}\beta_{345}=0\quad.\end{cases}$

Where equation $(I)(a)$ is obtained dividing the first four equations in system $(I)$ in (8) respectively by $-\beta_{456},\beta_{356},-\beta_{346},\beta_{345}\neq 0$ and, similarly, equations $(I)(b)$ and $(c)$ are obtained dividing, respectively, equations from $5$ to $8$ and equations from $9$ to $12$ in system $(I)$ in (8) by, respectively, $-\beta_{256},\beta_{156},-\beta_{126},\beta_{125}\neq 0$ and $-\beta_{234},\beta_{134},-\beta_{124},\beta_{123}\neq 0$ . While equations in $(II)$ (8) are left unchanged except for a change of sign. Remark that this is only possible since ${\mathcal{A}}$ is a generic arrangement which implies that all $\beta_{ijk}\neq 0$ and hence we can divide equations in (8) $(I)$ opportunely by them. In the following we refer to equations in $(I)$ and $(II)$ by using corresponding letters, for example $(a)$ will refers to equation $\beta_{134}\beta_{256}-\beta_{234}\beta_{156}$ . Plücker relations in equation (7) for $k=3$ becomes:

[TABLE]

Fixing $i_{1}=1,i_{2}=2,k_{0}=4,k_{1}=3,k_{2}=5,k_{3}=6$ , we obtain

[TABLE]

that is $(b)=(c)$ , and fixing $i_{1}=5,i_{2}=6,k_{0}=2,k_{1}=1,k_{2}=3,k_{3}=4$ we get $(a)=(b)$ . That is relations in (I) are equivalent.

Next we focus on type (II) relations and vanishing of all $4\times 4$ minors of Plücker matrix . Fixed a good 6-partition $\mathbb{T}=\{L_{1},L_{2},L_{3}\}$ , for any subset $L_{4}\subset[6]$ of cardinality $4$ such that $L_{4}\notin\mathbb{T}$ , define the submatrix

[TABLE]

of $A(\mathcal{A}_{\infty})$ . The matrix $Pl_{\mathbb{T}}(D_{L_{4}})$ is obtained adding one row to the matrix $A_{\mathbb{T}}(\mathcal{A}_{\infty})$ . Hence since relations in equation (8) correspond to vanishing of $3\times 3$ minors of $A_{\mathbb{T}}(\mathcal{A}_{\infty})$ , $\mathbb{T}=\{\{1,2,3,4\},\{1,2,5,6\},\{3,4,5,6\}\}$ , then zero of $4\times 4$ minors of matrix $Pl_{\mathbb{T}}(D_{L_{4}})$ for same fixed $\mathbb{T}$ naturally give rise to relations among relations in (8). For example $(d)=0$ and $(e)=0$ correspond to vanishing of minors obtained considering, respectively, 1st, 3rd and 5th columns and 1st, 3rd and 6th columns of $A_{\mathbb{T}}(\mathcal{A}_{\infty})$ . Adding to $A_{\mathbb{T}}(\mathcal{A}_{\infty})$ the normal vector to the hyperplane $D_{\{2,4,5,6\}}$ as 4th row we get

$Pl_{\mathbb{T}}(D_{\{2,4,5,6\}})=$ $\begin{pmatrix}-\beta_{234}&\beta_{134}&-\beta_{124}&\beta_{123}&0&0\\ -\beta_{256}&\beta_{156}&0&0&-\beta_{126}&\beta_{125}\\ 0&0&-\beta_{456}&\beta_{356}&-\beta_{346}&\beta_{345}\\ 0&-\beta_{456}&0&\beta_{256}&-\beta_{246}&\beta_{245}\end{pmatrix}$

and calculating the determinant of submatrix obtained by 1st, 3rd, 5th and 6th columns we get the relation among (e) and (d) :

[TABLE]

Anagously vanishing of minor obtained by 1st, 4th, 5th and 6th columns gives:

[TABLE]

Applying similar considerations to opportunely chosen $L_{4}\notin\mathbb{T}$ we get the following additional syzygies.

Vanishing of minors obtained considering 1st, 4th, 5th and 6th columns and 1st, 3rd, 5th and 6th columns of

[TABLE]

correspond, respectively, to relations $\beta_{236}\cdot(g)-\beta_{235}\cdot(f)=0$ and $\beta_{256}\beta_{234}\cdot(c)+\beta_{236}\cdot(e)-\beta_{235}\cdot(d)=0$ . Those relations, jointly with the one in equations (13) and (14), state dependency of $(d),(e),(f)$ and $(g)$ from $(c)$ which, in turn, is equivalent to $(a)$ , i.e. they are all zero if and only if $(a)$ is zero.

By vanishing of minors given by 2nd, 3rd, 5th and 6th columns and 2nd, 4th, 5th and 6th columns of submatrix

[TABLE]

we get, respectively, $\beta_{146}\cdot(i)-\beta_{145}\cdot(h)=0$ and $\beta_{156}\beta_{134}\cdot(c)-\beta_{146}\cdot(k)+\beta_{145}\cdot(j)=0$ .

Finally vanishing of minors given by 2nd, 4th, 5th and 6th columns and 2nd, 3rd, 5th and 6th columns of

[TABLE]

give relations $\beta_{136}\cdot(k)-\beta_{135}\cdot(j)=0$ and $-\beta_{156}\beta_{134}\cdot(c)-\beta_{136}\cdot(i)+\beta_{135}\cdot(h)=0$ .

That is relations in equation (8) are all equivalent and we are left with only one independent relation

[TABLE]

This degree 2 homogeneous polynomial defines a degree 2 hypersurface on the projective variety $Gr(3,6)$ .

The above computations are a direct consequence of the following more general Lemma.

Lemma 5.1.

Let $A({\mathcal{A}}_{\infty})$ be the Plücker matrix associated to a generic arrangement ${\mathcal{A}}$ of $n$ hyperplanes in ${\mathbb{C}}^{3}$ and ${\mathbb{T}}$ a good 6-partition of indices $i_{1},\ldots,i_{6}\in[n]$ . If entries $\beta_{I}$ of the matrix $A({\mathcal{A}}_{\infty})$ satisfy Plücker relations, then ${\rm rank}A_{\mathbb{T}}({\mathcal{A}}_{\infty})=2$ if and only if one of its 3 $\times$ 3 minor vanishes.

Proof.

$\Rightarrow$ ) Since ${\rm rank}A_{\mathbb{T}}({\mathcal{A}}_{\infty})=2$ if and only if all 3 $\times$ 3 minors of $A_{\mathbb{T}}({\mathcal{A}}_{\infty})$ vanish, it is obvious.

$\Leftarrow$ ) Entries $\beta_{I}$ of $A({\mathcal{A}}_{\infty})$ satisfy Plücker relations if and only if any $4\times 4$ minor in $A({\mathcal{A}}_{\infty})$ vanishes. For any 4 columns $s_{1}<s_{2}<s_{3}<s_{4}\in\{i_{1},\ldots,i_{6}\}$ of matrix $A({\mathcal{A}}_{\infty})$ let $M_{i}$ and $M_{j}$ be the two $3\times 3$ minors in $A_{\mathbb{T}}({\mathcal{A}}_{\infty})$ obtained considering, respectively, columns $\{s_{1},s_{2},s_{3},s_{4}\}\backslash\{s_{i}\}$ and $\{s_{1},s_{2},s_{3},s_{4}\}\backslash\{s_{j}\}$ . If we add to submatrix $A_{\mathbb{T}}({\mathcal{A}}_{\infty})$ the row of $A({\mathcal{A}}_{\infty})$ corresponding to vector $\alpha_{L}$ , $L=\{s_{i},s_{j},s_{5},s_{6}\}$ , with $\{s_{5},s_{6}\}=\{i_{1},\ldots,i_{6}\}\backslash\{s_{1},s_{2},s_{3},s_{4}\}$ , then the $4\times 4$ minor of the matrix $\begin{pmatrix}A_{\mathbb{T}}({\mathcal{A}}_{\infty})\\ \alpha_{L}\end{pmatrix}$ obtained considering columns $\{s_{1},s_{2},s_{3},s_{4}\}$ vanishes, that is

[TABLE]

where $\beta_{L\backslash\{s_{t}\}}$ is the entry of the row $\alpha_{L}$ in the column $s_{t},t=i,j$ . Dividing by $\beta_{L\backslash\{s_{i}\}}\neq 0$ (entries of $A({\mathcal{A}}_{\infty})$ are all not zero by ${\mathcal{A}}$ generic) we get

[TABLE]

that is $M_{i}=0$ if and only if $M_{j}=0$ . Applying the above considerations to any subset $\{s_{1}<s_{2}<s_{3}<s_{4}\}\subset\{i_{1},\ldots,i_{6}\}$ and transitivity of equality, we get that if a 3 $\times$ 3 minor of $A({\mathcal{A}}_{\infty})$ vanishes then all minors vanish. ∎

Remark 5.2.

*Recall that if ${\mathcal{A}}$ is an arrangement of $n$ hyperplanes in ${\mathbb{C}}^{3}$ then the matrix $A(\mathcal{A}_{\infty})$ is an ${n\choose 4}\times n$ matrix such that for any $L=\{s_{1}<s_{2}<s_{3}<s_{4}\}$ , entries $(x_{1},\ldots,x_{n})$ of row vector $\alpha_{L}$ are all zeros except $x_{i_{j}}=(-1)^{j}\beta_{I_{j}},I_{j}=L\setminus\{s_{j}\},j=1,\ldots,4$ . Hence for any fixed $6$ indices $s_{1}<\ldots<s_{6}\in[n]$ we get a ${6\choose 4}\times 6$ submatrix of $A(\mathcal{A}_{\infty})$ obtained considering all rows $\alpha_{L}$ , $L\subset\{s_{1},\ldots,s_{6}\},\mid L\mid=4$ and columns $\{s_{1},\ldots,s_{6}\}$ ( all columns $j\notin\{s_{1},\ldots,s_{6}\}$ of the matrix $(\alpha_{L})_{L\subset\{s_{1},\ldots,s_{6}\},\mid L\mid=4}$ are zero). It follows that the general case of $n$ hyperplanes in ${\mathbb{C}}^{3}$ essentially reduce to the case $n=6$ .

On the other hand it is an easy remark that, if $s_{1}<\ldots<s_{6}\in[n]$ are $6$ fixed indices and $\mathbb{T}=\{\{s_{1},s_{2},s_{3},s_{4}\}$ , $\{s_{1},s_{2},s_{5},s_{6}\},\{s_{3},s_{4},s_{5},s_{6}\}\}$ ( analogous of good $6$ -partition $\{\{1,2,3,4\}$ , $\{1,2,5,6\},\{3,4,5,6\}\}$ of indices $\{1,\ldots 6\}$ ), then any other good $6$ -partition on indices $\{s_{1},\ldots,s_{6}\}$ is of the form*

[TABLE]

where $i_{j}=\sigma(s_{j})$ , $\sigma\in\mathbf{S_{6}}$ , $\mathbf{S_{6}}$ being the group of all permutations of indices $\{s_{1},\ldots,s_{6}\}$ . Notice that in general $i_{j}$ are not ordered and we can have $i_{j}>i_{j+1}$ .

The following Lemma holds.

Lemma 5.3.

Let ${\mathcal{A}}$ be an arrangement of $n$ hyperplanes in ${\mathbb{C}}^{3}$ and $\sigma.\mathbb{T}=\{\{i_{1},i_{2},i_{3},i_{4}\}$ , $\{i_{1},i_{2},i_{5},i_{6}\},\{i_{3},i_{4},i_{5},i_{6}\}\}$ , a good $6$ -partition of indices $s_{1}<\ldots<s_{6}\in[n]$ such that ${\rm rank}A_{\sigma.\mathbb{T}}(\mathcal{A}_{\infty})=2$ then ${\mathcal{A}}$ is a point in the hypersurface

[TABLE]

Proof.

Let $\sigma.\mathbb{T}=\{L_{1}^{\prime}=\{i_{1},i_{2},i_{3},i_{4}\},L_{2}^{\prime}=\{i_{1},i_{2},i_{5},i_{6}\},L_{3}^{\prime}=\{i_{3},i_{4},i_{5},i_{6}\}\}$ be a good $6$ -partition of indices $s_{1}<\ldots<s_{6}\in[n]$ and denote by $(L_{1}^{\prime})=(i_{1},i_{2},i_{3},i_{4})$ , $(L_{2}^{\prime})=(i_{1},i_{2},i_{5},i_{6})$ and $(L_{3}^{\prime})=(i_{3},i_{4},i_{5},i_{6})$ the ordered $4$ -uples of indices. Then there exist unique permutations $\tau_{i}$ , $i=1,2,3$ of indices $s_{1}<\ldots<s_{6}$ such that $\tau_{i}$ fixes indices outside $L_{i}^{\prime}$ and, if $L_{i}^{\prime}=\{s_{j_{1}}<s_{j_{2}}<s_{j_{3}}<s_{j_{4}}\}$ , then $(L_{i}^{\prime})=\tau_{i}.L_{i}^{\prime}=(\tau_{i}(s_{j_{1}}),\tau_{i}(s_{j_{2}}),\tau_{i}(s_{j_{3}}),\tau_{i}(s_{j_{4}}))$ , $i=1,2,3$ . By determinant rule on permutations of columns we have that

[TABLE]

Hence, if we define the matrix $\sigma.A_{\mathbb{T}}$ as the matrix having in its rows respectively the coefficients of the three vectors

[TABLE]

with respect to the ordered basis $\{e_{i_{1}},\ldots,e_{i_{6}}\}$ , then $i$ -th row of $\sigma.A_{\mathbb{T}}$ is obtained from $i$ -th row of $A_{\sigma.\mathbb{T}}(\mathcal{A}_{\infty})$ by permutation $\sigma$ of columns and multiplication by ${\rm sign}(\tau_{i})$ (notice that $\sigma_{\mid_{L_{i}^{\prime}}}=\tau_{i}$ ). That is ${\rm rank}\sigma.A_{\mathbb{T}}={\rm rank}A_{\sigma.\mathbb{T}}(\mathcal{A}_{\infty})$ and, more in details, the $3\times 3$ minor given by columns $\{i,j,k\}$ in $A_{\sigma.\mathbb{T}}(\mathcal{A}_{\infty})$ vanishes if and only if the $3\times 3$ minor of columns $\{\sigma(i),\sigma(j),\sigma(k)\}$ in $\sigma.A_{\mathbb{T}}$ vanishes. Hence, by Lemma 5.1 ${\rm rank}A_{\sigma.\mathbb{T}}(\mathcal{A}_{\infty})={\rm rank}\sigma.A_{\mathbb{T}}=2$ if and only if one minor vanishes. In particular the first three columns $\{i_{1},i_{2},i_{3}\}$ in $\sigma.A_{\mathbb{T}}$ are of the form

[TABLE]

from which we get that the $3\times 3$ minor corresponding to them vanishes if and only if

[TABLE]

( recall that all entries $\beta_{I}$ in the matrix $A(\mathcal{A}_{\infty})$ verify $\beta_{I}\neq 0$ ). ∎

By Remark 5.2 and Lemma 5.3, the following main Theorem follows.

Theorem 5.4.

The set of generic arrangements ${\mathcal{A}}$ of $n$ hyperplanes in ${\mathbb{C}}^{3}$ that contains a dependent sub-arrangement is the set of points in an hypersurface in Grassmannian $Gr(3,n)$ such that each component is intersection of Grassmannian with a quadric.

Bibliography13

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] C. A. Athanasiadis. The Largest Intersection Lattice of a Discriminantal Arrangement, Beiträge Algebra Geom., 40 (1999), no. 2, 283–289.
2[2] A. Bachemand and W. Kern, Adjoints of oriented matroids, Combinatorica 6 (1986) 299–308.
3[3] M. Bayer and K.Brandt, Discriminantal arrangements, fiber polytopes and formality, J. Algebraic Combin. 6 (1997), 229–246.
4[4] H.Crapo, Concurrence geometries , Adv. in Math., 54 (1984), no. 3, 278–301.
5[5] M. Falk, A note on discriminantal arrangements, Proc. Amer. Math. Soc., 122 (1994), no.4, 1221-1227
6[6] Joe Harris, Algebraic Geometry: A First Course, Springer-Verlag
7[7] M. Kapranov, V.Voevodsky, Braided monoidal 2-categories and Manin-Schechtman higher braid groups, Journal of Pure and Applied Algebra, 92 (1994), no. 3, 241–267.
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Discriminantal Arrangement, 3×33\times 33×3 Minors of Plücker Matrix and hypersurfaces in Grassmannian Gr(3,n)Gr(3,n)Gr(3,n).

Abstract.

Key words and phrases:

1991 Mathematics Subject Classification:

1. Introduction

2. Preliminaries

2.1. Discriminantal arrangement

2.2. Good 3s-partitions

Lemma 2.1**.**

Theorem 2.2**.**

2.3. Matrices A(A∞)A(\mathcal{A}_{\infty})A(A∞​) and AT(A∞)A_{\mathbb{T}}(\mathcal{A}_{\infty})AT​(A∞​)

2.4. Polynomial pT(aij)p_{\mathbb{T}}(a_{ij})pT​(aij​)

2.5. Grassmannian Gr(k,n)Gr(k,n)Gr(k,n)

Remark 2.3**.**

3. Example B\mathcal{B}B(6, 3, A∞\mathcal{A}_{\infty}A∞​) in real case

4. Polynomial p~T(aij)\widetilde{p}_{\mathbb{T}}(a_{ij})p​T​(aij​) in B(n,k,A∞)\mathcal{B}(n,k,\mathcal{A}_{\infty})B(n,k,A∞​) in real case

4.1. Case B\mathcal{B}B(nnn, 3, A∞\mathcal{A}_{\infty}A∞​)

Theorem 4.1**.**

Proof.

Remark 4.2**.**

4.2. Generalization to B(n,k,A∞)\mathcal{B}(n,k,\mathcal{A}_{\infty})B(n,k,A∞​)

Lemma 4.3**.**

Proof.

Theorem 4.4**.**

Remark 4.5**.**

5. Hypersurfaces in complex Grassmannian Gr(3,n)Gr(3,n)Gr(3,n)

Lemma 5.1**.**

Proof.

Remark 5.2**.**

Lemma 5.3**.**

Proof.

Theorem 5.4**.**

Discriminantal Arrangement, $3\times 3$ Minors of Plücker Matrix and hypersurfaces in Grassmannian $Gr(3,n)$ .

Lemma 2.1.

Theorem 2.2.

2.3. Matrices $A(\mathcal{A}_{\infty})$ and $A_{\mathbb{T}}(\mathcal{A}_{\infty})$

2.4. Polynomial $p_{\mathbb{T}}(a_{ij})$

2.5. Grassmannian $Gr(k,n)$

Remark 2.3.

3. Example $\mathcal{B}$ (6, 3, $\mathcal{A}_{\infty}$ ) in real case

4. Polynomial $\widetilde{p}_{\mathbb{T}}(a_{ij})$ in $\mathcal{B}(n,k,\mathcal{A}_{\infty})$ in real case

4.1. Case $\mathcal{B}$ ( $n$ , 3, $\mathcal{A}_{\infty}$ )

Theorem 4.1.

Remark 4.2.

4.2. Generalization to $\mathcal{B}(n,k,\mathcal{A}_{\infty})$

Lemma 4.3.

Theorem 4.4.

Remark 4.5.

5. Hypersurfaces in complex Grassmannian $Gr(3,n)$

Lemma 5.1.

Remark 5.2.

Lemma 5.3.

Theorem 5.4.