A Representation Theorem for Generic Line Arrangements with Global Cyclicity in the Plane
OON LINE ARRANGEMENTS OVER FIELDS WITH − ad STRUCTURE
C.P. ANIL KUMAR
Abstract.
In this article we explore local and global gonality principle present in the linearrangements of the plane. We prove two main results, Theorems [1.1,1.2]. Firstly we as-sociate invariants such as permutation cycles and local cycles at infinity with 2 − standardconsecutive structures (refer to Definitions [5.9, 5.11, 5.26]) to a line arrangement (referto Definition 2.3) which has global cyclicity (refer to Definition 5.7) over fields with 1 − ad structure (refer to Definition 2.1) to describe the gonality structures (refer to Definition 3.1)in Theorem 1.1 when there exists a local permutation chart where the intersections pointscorresponding to simple transpositions satisfy One Sided Property 1.1. We construct agraph of isomorphism of classes of line arrangements over fields with 1 − ad structure usingthe associated invariants and Elementary Collineation Transformations (ECT) in Theo-rem 1.2 and in Note 5.32. Secondly here we prove another main theorem of the article, therepresentation Theorem 1.2, where we represent each isomorphism class with lines havinga given set of distinct slopes. We also prove two isomorphism Theorems [5.35,5.36] forthe line arrangement collineation maps using quadrilateral substructures (refer to Defini-tion 5.33) based on the theme of central points of triples of intersection points. At the endof the article we ask some open questions on line-folds (refer to Definition 6.1). Introduction
The line arrangements (refer to Definition 2.3) has been studied by various authors likeA. Dimca [1], S.Papadima, A.D.R Choudary, A.Suciu, P. Orlik, H. Terao [2] in variouscontexts over fields Q , R , C and finite fields F q , q a prime power. This field has applicationsin fields like Combinatorics, Braids and Confiugurations Spaces, Computer Science andPhysics. Here we study the features of a finite set of linear inequalities in two variables overfields with 1 − ad structure (refer to Definition 2.1) exploring the local and global gonalityprinciple with a purpose to associate invariants which describe the gonality structures (referto Definition 3.1) in a line arrangement (refer to Definition 2.3).1.1. Main Results.
We prove two main results Theorem 1.1 and Theorem 1.2 in thisarticle. We state the theorems here but refer the reader to the required definitions in thearticle.Exploring the local and global gonality principle the first main Theorem 1.1 gives a com-binatorial description of the polygonal regions formed in a line arrangement by associatingcombinatorial invariants which respect geometrical notions. The following Theorem 1.1 isproved after the statement of Definition 5.26. Now we state the theorem.
Theorem 1.1 (Criterion for Existence of a Local k − Gonality and Sufficiency) . Let F be a field with − ad structure (refer to Definition 2.1). Let L F n = { L , L , . . . , L n } be a line arrangement (refer to Definition 2.3) in the plane F . • (Existence: Local − standard consecutive structure) Suppose L i −→ L i −→ L i −→ . . . −→ L i k − −→ L i k −→ L i Mathematics Subject Classification.
Primary: 51A20,51H10,51G05,52C30,14P10,12J15 Secondary:14P25.
Key words and phrases.
Ordered Fields, Linear Inequalities in Two Variables, Line Arrangments,Collineation Isomorphism. a r X i v : . [ m a t h . C O ] J a n C.P. ANIL KUMAR is a k − gonality (refer to Definition 3.1) in the arrangement then there exists a localpermutation coordinate chart such that the local cycle at infinity (refer to Defini-tions [ 5.9, 5.26]) obtained by deleting the subscripts in { , , . . . , n }\{ i , i , . . . , i k } lies in T k (refer to Definition 5.20). • (Sufficiency: One Sided Property) If there exists a local coordinate chart such thatthe intersection points corresponding to simple transpositions lie on only one halfside for each of the lines L i j : j = 1 , , . . . , k then the lines L i −→ L i −→ L i −→ . . . −→ L i k − −→ L i k −→ L i form a k − gonality and if the local cycle chart respects the slope property (refer toDefinition 5.25) (we also say that the chart preserves the orientation here) then this k − gonality is given in this anti-clockwise cyclic manner. The second main Theorem 1.2 is regarding a representation of a line arrangement isomor-phically, (refer to Definition 4.4), by some set of lines forming a line arrangment with agiven set of distinct slopes, of same cardinality, which is useful to pick an element in thesame isomorphism class by fixing a finite set of slopes. The representation theorem is provedafter the proof of Lemma 5.31. Now we state the theorem.
Theorem 1.2 (Representation Theorem) . Let F be a field which has the − ad structure. Let { m , m , . . . , m n } ⊂ F ∪ {∞} be a set of n distinct slopes. In any isomorphism class (refer to Definition 4.4) of linearrangement L F n there exists a set of lines which represents exactly this slope set. Structure of the paper.
Here we mention about the structure of this paper bymentioning about the results proved in various sections.In Section 2 we define the features of a line arrangement over a field with 1 − ad structureand prove Lemma 2.7 which counts the number of sets of strict inequalities which havesolutions over the same field arising out of all possibilities in a Venn-Diagram.In Section 3 we define the gonality structures (refer to Definition 3.1) and two possible orders(refer to Definition 3.2) the slope order and the order of intersection points on a line inducedby other lines. The slope order is used for defining 2 − standard consecutive structures whichrespect slope property(refer to Definitions 5.11,5.25) and the order of intersection points isused in proving isomorphism Theorem 4.6 for line arrangements.In Section 4 we define line arrangement collineation isomorphism (refer to Definition 4.2)and isomorphism between two line arrangements (refer to Definition 4.4) to prove the iso-morphism Theorem 4.6 using the order of intersection points on a line with the remaininglines.Section 5 is the important section of this article. In this section we define the importantstructures associated to a line arrangement and then describe the gonality structures bydescribing the combinatorial structure of the cycle at infinity when a certain type of gonalityexists and also give a sufficient criterion as to when a certain subset of lines form a gonality.In this section we prove our first main Theorem 1.1, the local analogue after proving theglobal version in Theorems 5.14, 5.17, 5.19. We also describe in Theorems 5.22, 5.24 thepossible gonalities present when there is global cyclicity (refer to Definition 5.7) and alsoa give a count of them. This is done by identifying certain topological, geometric andcombinatorial features of line arrangement when there is global cyclicity.In the same section we describe the graph of isomorphism classes in Theorems 5.30, 1.2and in Note 5.32. Later in this section we describe quadrilateral structures as unique nook N LINE ARRANGEMENTS OVER FIELDS WITH 1 − ad STRUCTURE 3 point structures and prove that any bijection of two line arrangements which is identityon subscripts where the lines are indexed with increasing angles or slopes in the order0 −→ ∞ −→ −∞ −→
Line Arrangements over Fields with − ad Structure
We begin the section with the theme of Venn diagrams on how lines divide the plane intovarious regions over a field with 1 − ad structure.2.1. Theme of Venn Diagrams, Regions and − ad Structure on the Field.
Thetheme of Venn diagrams is a very well known combinatorial way of partitioning an union of n − sets and its complement into 2 n sets. We do have a partition of the plane associated to n − lines in a plane. We define below in the next few sections more precisely this partitionby seeking solutions to a set of n − inequalities.2.1.1. 1 − ad Structure on the Field and − ad Structured Subsets of F and thePlane F . First we define a 1 − ad structure on a field F . Definition 2.1.
Let ( F , ≤ ) be a totally ordered field. We say F has a − ad structure if inaddition the total order satisfies the following properties. • If x, y, z ∈ F then x ≤ y ⇒ x + z ≤ y + z . • If x, y ∈ F then x ≥ , y ≥ ⇒ xy ≥ . We say a set S ⊂ F is a 1 − ad structured subset if there exists an element a ∈ S such that x ∈ S ⇐⇒ x ≥ a or x ∈ S ⇐⇒ x ≤ a . We say that in both cases S has a 1 − ad structureat the point a with respect to the set S . Let F + = { x | x ≥ } , F − = { x | x ≤ } .We say a set in S ⊂ F has a 1 − ad structure of dimension one if there exists points v, w ∈ F such that S = { v + tw | t ≥ , t ∈ F } . We say a set S ⊂ F is a set with1 − ad structure of dimension 2 if there exists an affine functional f : F −→ F such that S = { ( x, y ) ∈ F | f ( x, y ) ∈ F + } . We say that the set S has 1 − ad structure at each pointof ( x, y ) ∈ S such that f ( x, y ) = 0. Also see the below note. Note 2.2. • In linear programming problems, sets with − ad structures naturallyarise. Also the definition of the region R given below happens to be the intersectionof inverse images of − ad structured subsets of the field F with − ad structure atthe origin. • In the case of manifolds (Euclidean Spaces) over reals, the − ad structured subsetsof dimension n are the half spaces of dimension n . • “ ad ” stands for adjacency. − ad means there is only one side. For example locallythe point a ∈ R in the interval [ a, ∞ ] ⊂ R has only one side. • It is possible to develop a notion (not in this article) called ( p − − ad structure fora finite field F p with p points. Note here we say the origin ∈ F p as p − adjacentsides. Each side of contains just one other point. C.P. ANIL KUMAR
Definition of Region Using the − ad Structure.
We start this section with adefinition.
Definition 2.3 (Lines in Generic Position in the Plane F or Line Arrangement) . Let F be a field with − ad structure. We say a finite set L F n = { L , L , . . . , L n } of lines in F are in generic position or forms a line arrangement if the following two conditions hold.(1) No two lines are parallel.(2) No three lines are concurrent.In this case we say that L F n is a line arrangement. We denote the line arrangement by L n if the field F = R . The most general theme is that of Venn Diagrams. A Venn Diagram on n − sets gives riseto 2 n − disjoint sets. In the case of a line arrangement L F n each line L : ax + by = c, a, b, c ∈ F gives rise to two “regions” given by R = { ( x, y ) | ax + by ≤ c } , R = { ( x, y ) | ax + by ≥ c } both of which just include the line L in common. Definition 2.4 (Definition of a Region (a Polygonal Region)) . Let L F n be a line arrangement. Suppose an equation for L i is given by a i x + b i y = c i with a i , b i , c i ∈ F . A region R is defined to be a set of solutions to these inequalities { ( x, y ) ∈ F | a i x + b i y ≤ , ≥ c i } A region R has non-empty interior if there exists ( x, y ) ∈ R which satisfies strict inequalities. Definition 2.5 (Definition of Bounded/Unbounded Regions) . We say the region R is unbounded if there exists v, w ∈ R such that either { v + t ( w − v ) | t ≥ } ⊂ R or { v + t ( w − v ) | t ≤ } ⊂ R . Otherwise we say the region R is bounded. Note 2.6.
Out of the n − possibilities (exponential in n ) of sets of inequalities, in the caseof line arrangements we will see in the next lemma the sets that have solutions in x, y over the field F are a sparse sub-collection of possibilities of inequalities having polynomial ( O ( n )) − cardinality. We mention the following two lemmas [2.7,2.9] without proof as they are straight forward.
Lemma 2.7 (Count of Number of Regions) . Let F be a field with − ad structure. Let L F n be a line arrangement. Then we have n unbounded regions, (cid:0) n − (cid:1) bounded regions and (cid:0) n +12 (cid:1) + 1 total number of regions. Inparticular a set of lines generated by n − points in the plane has at most n − n + 3 n − n + 88 regions. Figure 1 represents intersection regions at an intersection point over a field which has 1 − ad structure. Definition 2.8 (Definition of Crossing Number) . Let F be a field with − ad structure. Let L F n be a line arrangement. Let R , R be twodifferent polygonal regions formed by the line arrangement. Let L be a new line which isgenerically placed meeting the regions R and R . Then we define the crossing number C ( R , R ) between the regions R and R as the number of intersection vertices of the line N LINE ARRANGEMENTS OVER FIELDS WITH 1 − ad STRUCTURE 5
Figure 1.
The Regions at an Intersection L between an interior point of R and an interior point of R with the original given set oflines. This is well defined because of the following lemma. Lemma 2.9 (Crossing Number between Two Regions) . Let F be a field with − ad structure. Let L F n be a line arrangement. Let R , R be twodifferent polygonal regions. Then the crossing number C ( R , R ) is well defined and it isindependent of the choice of the generic line used for the definition. Definition of Gonality Structures and Orders
In this section we introduce gonality structures and orders. First we begin with a definition.
Definition 3.1 (Definition of a k − Gonality) . Let n ≥ , k ≥ , k ≤ n be positive integers.Let L F n = { L , L , . . . , L n } be a line arrangement. We say a subset { L i , L i , . . . , L i k } forma bounded k − gonality if there exists a bounded region R and end vertices v i j , w i j ∈ L i j ⊂ F , ≤ i ≤ n such that • L i ∩ R (cid:54) = ∅ ⇒ i = i j for some ≤ j ≤ k . • L i j ∩ R = [ v i j , w i j ] = { (1 − t ) v i j + tw i j | ≤ t ≤ } . • The vertices v i j , w i j are consecutive i.e. there is no intersection vertex in the linesegment ( v i j , w i j ) .We say the k − gonality is unbounded if there exists an unbounded region R which satisfiesthe same conditions except for only two of the lines L i j , L i j , ≤ j (cid:54) = j ≤ k there isexactly one intersection vertex in each of the sets L i j ∩ R, L i j ∩ R as the end vertex whichneed not be common. Now below we define two orders the slope order on all the lines and the intersection orderwith respect to a line on the remaining lines.
Definition 3.2 (Two Definitions of Orders on the Lines: Slope Order, Order Induced byLines) . Let F be a field with − ad structure. Assigning a linear coordinate system to a lineas { v + tw | t ∈ F } we obtain an order of the intersection points and hence on the set of other lines. We obtaintwo possibilities which are mutually inverses of each other. The slope order is defined as C.P. ANIL KUMAR usual using the total order on F and using positive elements and negative elements in F corresponding to positive and negative slopes. Also the vertical line corresponds to infiniteslope lies in between positive slopes and negative slopes as first positives and then negativesin accordance with angles. +0 −→ ∞ −→ −∞ −→ − . Isomorphism Theorem of Line Arrangements
In this section we state and prove an isomorphism theorem for line arrangements. We beginwith definitions.
Definition 4.1.
Let F be a field. Let P , P be two sets of points in the plane. We say amap φ : P −→ P is a collineation if for any three points P , P , P which are collinear thepoints φ ( P ) , φ ( P ) , φ ( P ) are collinear. If in addition the map φ is a bijection then we say φ is a collineation isomorphism. Definition 4.2 (Line Arrangement Collineation Isomorphism) . Let F be a field. Let ( L F n ) , ( L F n ) be two line arrangements. For k = 1 , let P k = { L i ∩ L j | L i , L j ∈ ( L F n ) k , ≤ i < j ≤ n } . We say T : ( L F n ) −→ ( L F n ) is a line arrangement collineation isomorphism if T : P −→ P is bijective and if P i , P j , P k are collinear by some line L t ∈ ( L F n ) then T ( P i ) , T ( P j ) , T ( P k ) is collinear by some line M s ∈ ( L F n ) with ≤ s, t ≤ n . Example 4.3.
A permutation of lines gives rise to a line arrangement collineation au-tomorphism of a line arrangement. For n = 3 any collineation automorphism is a linearrangement collineation isomorphism as this holds true vacuously. Definition 4.4.
Let F be a field equipped with − ad structure. Let ( L F n ) , ( L F n ) be twoline arrangements. We say that they are isomorphic if there exists a piece-wise linearautomorphism of F which takes one line arrangement to another. Note 4.5.
A piece-wise linear automorphism which takes one line arrangement ( L F n ) toanother ( L F n ) by taking intersection points to intersection points and lines to lines is notonly a line arrangement collineation isomorphism but also takes regions which are k − gons tocorresponding regions which are k − gons for k = 3 , , . . . , n . It also preserves the adjacencyof the intersection vertices. Now we state the isomorphism theorem below and mention its proof.
Theorem 4.6 (Line Arrangement Isomorphism Theorem) . Let F be a field equipped with − ad structure. Let ( L F n ) , ( L F n ) be two line arrangements.Let φ : ( L F n ) −→ ( L F n ) be a line arrangement collineation isomorphism such that φ preserves the order of intersec-tion points on each line. Then there exists a piece-wise linear automorphism of the planewhich takes one line arrangement to the other.Proof. Let φ be a line arrangement collineation isomorphism preserving the orders of inter-section vertices on each line. Then the following holds.(1) (Bijection on the set of Lines): The line arrangement collineation isomorphisminduces a bijection on the lines.(2) (Preserves Adjacency): If there are k − intermediate intersection vertices betweentwo intersection vertices on a line then there are also k − intermediate intersectionvertices between the image of the two intersection vertices on the image line. N LINE ARRANGEMENTS OVER FIELDS WITH 1 − ad STRUCTURE 7 (3) (Preservation of Central point): This can be derived from the previous step. If thereare three points P , P , P on a line L such that P is in between P and P then φ ( P ) is in between φ ( P ) and φ ( P ) on φ ( L ).(4) (Preservation of Sidedness): We have only two sides of any line in the plane. Supposewe have a partition of intersection points P = P L (cid:97) P Side L (cid:97) P Side L and P = P φ ( L ) (cid:97) P Side φ ( L ) (cid:97) P Side φ ( L ) where L , L represent either sides of the line L and φ ( L ) , φ ( L ) represent eithersides of the line φ ( L ) then we have • φ ( P L ) = P φ ( L ) i.e. the points on the line L goes to points on the line φ ( L ). • Either φ ( P Side L ) = P Side φ ( L ) and φ ( P Side L ) = P Side φ ( L ) . • Or the other way φ ( P Side L ) = P Side φ ( L ) and φ ( P Side L ) = P Side φ ( L ) .Here P Side L , P Side L are the discrete half sides of the discrete line P L . Similarlyfor the line P φ ( L ) = φ ( P L ).(5) (Preservation of the Regions): Regions are mapped to regions as the regions areformed by intersection points of lines as finite intersections of discrete half sides ofthe lines. • If a half side of one line intersects with a half side of another line then thecorresponding intersection of the half sides in the image is also non-empty. • More importantly for a bounded region the number of half sides (without repe-tition) whose intersection is considered is the same as the number of intersectionvertices which forms the region which is the intersection set provided we in-clude in the intersection the adjacent intersection vertices of the lines whosehalf sides are considered. • For an unbounded region the number of half sides (without repetition) whoseintersection is considered is one more than the number of intersection verticeswhich forms the region which is the intersection set provided we include in theintersection the adjacent intersection vertices of the lines whose half sides areconsidered. • The map φ preserves all these properties of the regions.Finally since the regions are convex and these regions occupy as a jigsaw puzzle for theplane, we have, after further similar triangulation, by convex extension, there is a piece-wise linear bijection of the two dimensional plane F to F . This proves the theorem thatthe two line arrangements are isomorphic. We note however that this piece-wise linearbijection need not be an orientation preserving map of the plane F . (cid:4) On some Structures in a Line Arrangement
We begin with a definition of a simplicial map/path, adjacent simplicial map/path, jordancurve in this section.
Definition 5.1 (Simplicial Map/Path, Adjacent Simplicial Map/Path) . Let ( F , ≤ ) denote a field equipped with a − ad structure. Let L F n = { L , L , . . . , L n } be a line arrangement. Let P = { L i ∩ L j | ≤ i < j ≤ n } . Let X = L ∪ L ∪ . . . ∪ L n .For x, y ∈ F let [ x, y ] = { z ∈ F | x ≤ z ≤ y } . We say a map σ : [0 , −→ F is simplicialif σ ( t ) = (1 − t ) σ (0) + tσ (1) . We say a map σ : [0 , −→ F is simplicial if there exists C.P. ANIL KUMAR v = σ (0) , w = σ (1) ∈ F such that σ ( t ) = (1 − t ) v + tw . We say two vertices v, w ∈ P areadjacent if the following holds. • There exists a subscript ≤ i ≤ n such that v, w ∈ L i . • The simplicial map σ : [0 , −→ X given by σ ( t ) = (1 − t ) v + tw has the propertythat σ ([0 , ∩ P = { v, w } . We say σ is an adjacent simplicial map if σ (0) , σ (1) ∈ P and are adjacent. Definition 5.2 (Polygonal Jordan Curve) . Let F be a field with − ad structure. Let [0 ,
1] = { x ∈ F | ≤ x ≤ } . We say acurve σ : [0 , −→ F is piece-wise simplicial if there exists a finite sequence of vertices v , v , v , . . . , v n ∈ F and field elements t < t < t < . . . < t n = 1 such that σ | [ ti,ti +1] ( t ) = (1 − t ) v i + tv i +1 , ≤ i ≤ n − . We say the piece-wise simplicial curve is closed if v n = v . We say the curve is jordan if itis piece-wise simplicial, closed and injective on [0 , . We introduce the next two definitions regarding any polygon made up of lines. The useful-ness of the first definition lies in identifying convex polygons in a line arrangement.
Definition 5.3 (2-Standard Consecutive Structure) . Let F be a field with − ad structure.We say a certain set of slopes ( m , m , . . . , m n ) ∈ ( F ∪ {∞} ) n = ( PF F ) n has a − standard consecutive structure if the following occurs. • ≤ m < m < . . . < m i ≤ ∞• m i +1 < m i +2 < . . . < m j ≤ • < m j +1 < m j +2 < . . . < m k ≤ ∞• m k +1 < m k +2 < . . . < m n < ≤ i < j < k ≤ n . The last sequence of slopes may be empty. i.e. k = n . If theslopes m i : 1 ≤ i ≤ n arise from a line arrangement then the slopes are distinct as any twodistinct lines meet. The structure is considered as 2 − standard by referring to usual anglesinstead of slopes. Theorem 5.4.
Let F be a field with − ad structure. Let L F n be a line arrangement. Let T bea closed traversal of distinct vertices except the first and last one by adjacent simplicial pathssuch that no two consecutive paths cyclically considered lie on the same line and there existsa consecutive set upto cyclic permutation of slopes which satisfy a − standard consecutivestructure. Then the interior domain of the jordan curve T is a region.Proof. Using the 2 − standard consecutive structure and the fact that no two consecutivepaths lie on the same line we have that the interior domain of the jordan curve satisfiesthat it lies on only one side of each line and hence it is a region. This proves the theorem.This 2 − standard consecutive structure is needed for existence of solutions to a set of linearinequalities in a line arrangement by identifying the regions uniquely with respect to theset of inequalities. (cid:4) In Figure 2 we depict a domain whose boundary simplicial lines of the colored region satisfyall properties of Theorem 5.4 except the 2 − standard consecutive structure for slopes. Wenote that the movement at an intersection point is from one simplicial line to anothersimplicial line and the domain in the figure is not a region. The cyan colored triangularregion is enclosed on all the three sides by green, yellow and orange regions. N LINE ARRANGEMENTS OVER FIELDS WITH 1 − ad STRUCTURE 9
Figure 2.
Slopes do not have a 2 − Standard Consecutive StructureNow we introduce the second definition. This second definition is useful in classifying thecycles at infinity of a line arrangement L F n = { L , L , . . . , L n } which has an n − gonality in this anti-clockwise cyclic order L −→ L −→ . . . −→ L n −→ L Definition 5.5 (Opposite Vertex of a Side in a Convex Polygon) . Let L F n = { L , L , . . . , L n } be a line arrangement. Let { L i −→ L i −→ . . . −→ L i r −→ L i } be a convex polygon with r − sides in this anticlockwise cyclic order. We assume that by rota-tion (such rotations exist) that L i has the least non-negative slope. Suppose the − standardconsecutive structure for this polygon is given by • ≤ m < m < . . . < m i ≤ ∞• m i +1 < m i +2 < . . . < m j < • < m j +1 < m j +2 < . . . < m k ≤ ∞• m k +1 < m k +2 < . . . < m r < for some ≤ i < j < k ≤ n where m j is the slope of the line L i j . Then the opposite vertexof L i is defined to be L i j ∩ L i j +1 . We just note that m j < ≤ m < m j +1 . The definition of the opposite vertex for the remaining sides of the polygon is similar usingrotations and making the given side as having least non-negative slope with X − axis. Note 5.6.
Let F be a field with − ad structure. Hence F is of characteristic zero with Q ⊂ F . Let T AN = { m ∈ F + | m = (cid:3) } . Given a slope m > there exists an m ∈ T AN such that (cid:54) = m < m . On Global Cyclicity.
We introduce this section with the main definition.
Definition 5.7 (Existence of Global Cyclicity) . Let L F n = { L , L , . . . , L n } be a line arrangement. We say that there exists global cyclicityif all the lines give rise to an n − gonality (refer to Definition 3.1). Now we give a criterion as to when a global cyclicity exists in a line arrangement. Beforewe state the theorem we need another definition. In general for an arbitrary field F with1 − ad structure the equation x = a > Definition 5.8 (Orientation of the plane) . We further introduce the notion of quadrants and hence the standard forms of lines. • First Quadrant: { ( x, y ) ∈ F | x > , y > } . • Second Quadrant: { ( x, y ) ∈ F | x < , y > } . • Third Quadrant: { ( x, y ) ∈ F | x < , y < } . • Fourth Quadrant: { ( x, y ) ∈ F | x > , y < } .An orientation of the plane (more precisely an orientation at the origin) with respect to thequadrants is given by I −→ II −→ III −→ IV −→ I We make an important observation that any line not parallel to any one of the axes meetsonly three quadrants and misses exactly one of them. Their equations are given as followsin standard forms with orientations.
IV, I, II, : xa + yb = 1 , a > , b > II, III, IV : xa + yb = 1 , a < , b < I, II, III : xa + yb = 1 , a < , b > III, IV, I : xa + yb = 1 , a > , b < ax + by = c > , abc (cid:54) = 0is oriented such that the origin is on the left side of the line. If ab = 0 then the orientationcan be coherently induced as well on these lines with this definition provided c (cid:54) = 0 givenas I, II or II, III or III, IV or IV, I . Definition 5.9 (Cycle (Local Cycle) at infinity: An Element of the Symmetric Group) . Let L F n be a line arrangement. Let L be any line with slope different from that of the linesof the arrangement and not passing through origin. We assume that the line L is genericto obtain the line arrangement L F n ∪ { L } and right side (non origin side) of the discrete half sides of L is empty. Then the cycle atinfinity is defined as the sequence of subscripts of the lines, a permutation ( i i . . . i n ) ∈ S n corresponding to the intersections of the lines in L F n with L in the direction of the orientationof L .A local cycle with respect to a subset A ⊂ L F n is the cycle at infinity obtained by droppingsubscripts of the lines not in the set A . Also refer to Definition 5.26 Now we introduce a structure on a permutation as follows.
N LINE ARRANGEMENTS OVER FIELDS WITH 1 − ad STRUCTURE 11
Definition 5.10.
We say an n − cycle ( a = 1 , a , . . . , a n ) is an i − standard cycle if thereexists a way to write the integers a i : i = 1 , . . . , n as i sequences of inequalities as follows: a < a < . . . < a j a < a < . . . < a j a < a < . . . < a j ... a i < a i < . . . < a ij i where { a st | ≤ s ≤ i, ≤ t ≤ j s } = { a , a , . . . , a n } = { , , . . . , n } , j + j + . . . + j i = n and i is minimal .i.e. there exists no smaller integer with such property and a s ( t +1) occursto the right of a st for every ≤ s ≤ i and ≤ t ≤ j s − in this cycle arrangement ( a = 1 , a , . . . , a n ) . Definition 5.11.
We say an n − cycle ( a = 1 , a , . . . , a n ) is a consecutive i − standardcycle or a i − standard consecutive cycle if we have a s < a s < . . . < a sj s and in addition a st = a s + ( t − , ≤ t ≤ j s , ≤ s ≤ i where { a st | ≤ s ≤ i, ≤ t ≤ j s } = { a , a , . . . , a n } = { , , . . . , n } , j + . . . + j s = n and a s ( t +1) occurs to the right of a st for every s = 1 , . . . , i and ≤ t ≤ j s − in this cycle arrangement ( a = 1 , a , . . . , a n ) and i is minimal .i.e. there exists no smaller integer with such property. Example 5.12.
For example if we consider the − cycle (1 , , , , it is a − standardconsecutive cycle. However it has the following two − standard structures. • < < , < (not consecutive). • < < , < (consecutive). Now we prove a lemma on the existence and uniqueness of the i − standard consecutivestructure on an n − cycle. Lemma 5.13 (Existence and Uniqueness of the Consecutive i − Standard Structure on an n − cycle) . The consecutive i − standard structure exists on an n − cycle and is unique.Proof. We prove this by induction on i, n as follows. If i = n = 1 then there is nothing toprove. The position of the element n is uniquely determined as it should appear in one ofthem at the end and ( n −
1) appears before n if ( n −
1) appears before n in the n − cycleand appears as a single element of standardness if ( n −
1) appears after n . Now we remove n from the cycle. The remaining cycle is either i − standard on ( n − − elements or ( i − n − − elements. This proves the lemma.We can actually build this structure in an unique way for the given n − cycle as follows. Write1 first. Then write 1 < (cid:4) Now we state the theorem.
Theorem 5.14.
Let L F n = { L , L , . . . , L n } be a line arrangement which gives rise to an n − gonality in this anticlockwise manner L −→ L −→ . . . −→ L n −→ L then the cycle at infinity has a − standard consecutive structure. Proof.
Assume by rotation that L has the least non-negative slope. Suppose L i ∩ L i +1 is the opposite vertex for L . Then on the n − cycle at infinity we have the following unique2 − standard consecutive structure. • < < . . . < i • i + 1 < i + 2 < . . . < n This proves the theorem. (cid:4)
We mention a note below.
Note 5.15.
The converse need not be true. Consider a line arrangement L F n with n = 5 where there exist a − standard consecutive structure on the cycle at infinity however thearrangement does not have a pentagonality. After a parallel translation of the lines we havea pentagonality structure. However we have the following theorem for the converse. We need a definition.
Definition 5.16 (Simple Transposition) . Let S n be a symmetric group on n − letters { , , . . . , n } . We say a transposition is simpleif it is of the form ( i ( i + 1)) where ≤ i ≤ n − or the transposition is ( n which is alsodenoted by ( n ( n + 1) ≡ n ) . Now we state the theorem about existence of global cyclicity.
Theorem 5.17 (Existence of Global Cyclicity) . Let L F n be a line arrangement. For any of the lines L i : i = 1 , , . . . , n the intersectionvertices corresponding to simple transpositions apart from (( i − i ) , ( i ( i + 1)) lie on only one discrete half side of L i . Then there exists an n − gonality giving rise to globalcyclicity.Proof. We need to prove that the vertices (( i − i ) , ( i ( i + 1)) on the line L i are adjacent.Suppose we have L i ∩ L j as an intermediate vertex then the vertex ( i ( i + 1)) and the vertex(( i − i ) are on either side of the line L j which is a contradiction. Now the proof is immediateas the vertices { L i ∩ L i +1 | ≤ i ≤ n } form a region. However the existence of 2 − standardconsecutive structure does not guarantee the clockwise or anticlockwise cyclicity of the lines L i : i = 1 , , . . . , n . (cid:4) Example 5.18.
For n = 4 , if the cycle at infinity is (1324) then we have both possibilitiesof − gonalities of lines L −→ L −→ L −→ L −→ L , L −→ L −→ L −→ L −→ L . Now we prove a theorem regarding the global cyclicity which says that the opposite verticesof the line segments of the global gonality and the cycle at infinity which respects the slopeproperty (refer to Definition 5.25) can be obtained from one other and each one of themdetermines the line arrangement up to isomorphism.
Theorem 5.19 (Cycle at Infinity and the Opposite Vertices of sides of the Global Gonal-ity) . Let L F n = { L −→ L −→ . . . −→ L n −→ L } be a line arrangement in a plane giving rise to an n − gonality in this anticlockwise cyclicorder. Then the cycle at infinity having the − standard structure which respects the slopeproperty determines uniquely the opposite vertex for any side in the n − gon. Conversely if we N LINE ARRANGEMENTS OVER FIELDS WITH 1 − ad STRUCTURE 13 know the opposite vertex for any side in this n − gon then the cycle at infinity is determineduniquely and its − standard structure respects the slope property (refer to Definition 5.25).Proof. Respecting the slope property is a given, once the cycle at infinity is determined.The cycle at infinity determines uniquely the opposite vertex for any side in the n − gonas we can use the 2 − standard structures which respects the slope property on each of thecyclically permuted conjugate cycles of the cycle at infinity.Conversely if we know the opposite vertex for any side we need to determine the slopeordering of the lines L , L , . . . , L n . First we can assume without loss of generality that L has the least non-negative slope with respect to X − axis. This slope order gets determinedbecause of the following reason. If we know the opposite vertex for a side on a line L of the arrangement then the sequence of intersections gets determined including the end-points using the opposite vertex on the line L . Hence the complete gonality structure getsdetermined using isomorphism Theorem 4.6. Hence this determines the cycle at infinity aswell. (cid:4) Definition 5.20 (2-standard consecutive n − cycles) . Let T n ⊂ S n be the set of − standard consecutive n − cycles in S n . Lemma 5.21. • We have T n ) = 2 n − − n. • The number of non-isomorphic global gonality structures arising from a line ar-rangement L F n is given by Z /n Z (cid:8) T n ) . Proof.
This follows by counting the cardinality of T n and the isomorphism class does notchange under cyclic renumbering of the subscripts of the lines. (cid:4) On the Gonality Function when there is Global Cyclicity.
Now we prove be-low a theorems which give the gonality functions of the bounded regions and the unboundedregions when there is global cyclicity.
Theorem 5.22 (A Theorem on Bounded Gonalities) . Let n ≥ be a positive integer. Let L F n be a line arrangement which give rise to a global n − gonality say in the anticlockwise order L −→ L −→ . . . −→ L n −→ L . Let k denote the number of lines L i : i = 1 , , . . . , n such that the following occurs. For eachline L i the vertices corresponding to simple transpositions lie on one discrete half side of L i and the vertex L i − ∩ L i +1 lies on the other side of L i . Then we have • k − triangular regions. • − bounded n − gonality. • (cid:0) n − (cid:1) − k − quadrilaterals.Proof. This theorem follows because of the observation that the triangles if they exist areadjacent to the global gonality and the rest of the bounded regions are all quadrilateralsapart from the n − gonality. (cid:4) Definition 5.23 (The inner coordinates of a point on a line) . Let L F n be a line arrangement. Let P = L i ∩ L j . Then the point P acquires a pair of innercoordinates of the form ( ± a mod n, ± b mod n ) depending on the position of the pointon the lines L i , L j . These are called inner coordinates of the point. It is well defined upto asign modulo n and if the lines are oriented then it is a well defined pair of integers in the set { , , . . . , n − } × { , , . . . , n − } . We say a point P is an outer point if one of the inner coordinates of the pair is ± n . Otherwise it is called a non-outer point. We say apoint P is an extreme point if the inner coordinates is given by ( ± n, ± n ) . Theorem 5.24 (A Theorem on Unbounded Gonalities) . Let n ≥ be a positive integer. Let L F n be a line arrangement which give rise to a global n − gonality. Let R i : 1 ≤ i ≤ (cid:18) n − (cid:19) be the set of bounded regions. Since the plane is coherently orientable at every point let T denote a cyclic traversal of the boundary line segments of the domain ( n − ) (cid:91) i =1 R i where all the regions are coherently oriented. Let r denote the number of extreme points.Let k denote the non-outer points of cyclic traversal T of the boundary line segments. Thenwe have • r − unbounded − gonalities (One extreme point). • k − unbounded − gonalities (One non-outer point in between two outer points on thetraversal T ). • (2 n − r − k ) − unbounded − gonalities (Two adjacent outer points on the traversal T ). • There are no unbounded gonalities higher than unbounded − gonalities.Proof. This theorem follows from the observation that unbounded 2 − gonalities are formedby extreme points. The unbounded 4 − gonalities are formed due to non-outer points onthe cyclic traversal T . If L a i ∩ L b i and L a i +1 ∩ L b i +1 are two consecutive extreme points inthe clock-wise order then a non-outer point if it exists i.e. a i (cid:54) = b i +1 in between them isgiven by L a i ∩ L b i +1 . Remaining points will be outer points on these two lines L a i , L b i +1 .This observation also proves that the remaining unbounded gonalities are all unbounded3 − gonalities and there are no higher > (cid:4) On Local Gonality Structures and Local Gonality Cycles at Infinity.
Asa continuation from the previous Example 5.18 we prove a theorem of existence of localgonality structures with respect to the 2 − standard structures on local cycles at infinity ina local permutation chart in this section. First we need a definition. Definition 5.25 (Slope Property) . We say that the − standard consecutive structure on a permutation n − cycle associated toa line arrangement L F n respects the slope property if the following occurs. If the -standardconsecutive structure is given by • < < < . . . < j . • j + 1 < j + 2 < . . . < n .then we have modulo a rotation of the plane F • ≤ m < m < . . . < m i ≤ ∞• m i +1 < m i +2 < . . . < m j ≤ • < m j +1 < m j +2 < . . . < m k ≤ ∞• m k +1 < m k +2 < . . . < m n < with m j < ≤ m < m j +1 . N LINE ARRANGEMENTS OVER FIELDS WITH 1 − ad STRUCTURE 15
In Example 5.18 if in addition the 2 − standard consecutive structure respects the slopeproperty then the clockwise 4 − gonality can be eliminated.In the definition below we introduce the concept of a local cycle at infinity and local per-mutation (coordinate) charts for local cycles at infinity. Definition 5.26 (Definition of a Local Cycle, Permutation (Coordinate) Charts for theLocal Cycles at Infinity) . Let σ ∈ S n be an n − cycle. Let A c ⊂ { , , . . . , n } be a subset of cardinality n − k . Bydropping the letters in the subset A c in the cycle σ we obtain a cycle σ k an element inthe symmetric group on k − letters. This is called a local cycle at infinity on k − letters. If A = { i , i , . . . , i k } in some order. Let τ = (cid:18) i i . . . i k . . . k (cid:19) . Then a cycle τ σ k τ − is the local cycle of σ k at infinity in a local permutation (coordinate) chart. Usually oneconsiders the local cycles where the order of the numbers i , i , . . . , i k to construct the cycle τ is such that there exists a local k − gonality L i −→ L i −→ . . . −→ L i k −→ L i in this anticlockwise order among just these k − lines. Now we prove the first main Theorem 1.1, a more general theorem of local gonality struc-tures.
Proof.
The proof is short after having proved Theorems 5.14, 5.17. The first main theoremfollows from Theorems 5.14, 5.17 by applying locally. (cid:4)
Note 5.27 (Global and Local Gonality Principle) . Once global cyclicity structure is known then in a general situation these “global structures”are present as “local structures” among the lines of their respective structures and theselocal structures embed in a certain way in the general situation.
Finite Graph of Isomorphism Classes.
We prove in this section a representationTheorem 1.2 for the isomorphism classes. First we observe that for any given two line ar-rangements a bijection of the lines give rise to a line arrangement collineation isomorphism.So these types of collineation maps are not useful to distinguish the isomorphism classes ofline arrangements.We begin with the definition of an Elementary Collineation Transformation (ECT) of twoline arrangments.
Definition 5.28 (Definition of ECT) . Let ( L F n ) , ( L F n ) be two line arrangements in the plane such that ( L F n ) ∩ ( L F n ) = { L , L , . . . , L k − , L k +1 , . . . , L n } . In addition the k th − line is given by ( L k ) : ax + by = c ∈ ( L F n ) , ( L k ) : ax + by = c ∈ ( L F n ) with c < c . Consider the three lines L i , L j , ( L k ) ∈ ( L F n ) with intersection vertices L j ∩ L i , ( L k ) ∩ L j , L i ∩ ( L k ) in the anticlockwise order and the three lines L i , L j , ( L k ) ∈ ( L F n ) with intersection vertices L i ∩ L j , ( L k ) ∩ L i , L j ∩ ( L k ) in the clockwise orderWe assume that in the space F the parallel strip { ( x, y ) | c < ax + by < c } contains only one intersection point L i ∩ L j of both the line arrangements. Then the ECT E ijk : ( L F n ) −→ ( L F n ) is defined to be the transformation induced by mapping the lines L i −→ L i for ≤ i ≤ n, i (cid:54) = k and by mapping the line ( L k ) to ( L k ) on the intersection points. Lemma 5.29 (Swap of the Inner Coordinates, Changes in Gonalities) . With the notations as in Definition 5.28 if the points and inner coordinates in the anticlock-wise order are given by L j ∩ L i = ( B j = Q j − , A i = X i + 1) , ( L k ) ∩ L j = (( P k ) = ( Y k ) − , Q j = B j + 1) ,L i ∩ ( L k ) = ( X i = A i − , ( Y k ) = ( P k ) + 1) then after applying E ijk we obtain the following points with inner coordinates in the clockwiseorder (so that the orientation of the lines are unchanged) as L i ∩ L j = ( X i = A i − , Q j = B j + 1) , ( L k ) ∩ L i = (( P k ) = ( Y k ) − , A i = X i + 1) ,L j ∩ ( L k ) = ( B j = Q j − , ( Y k ) = ( P k ) + 1) The gonalities change in this manner from ( n , n , n , n , n , n ) −→ ( n ± , n ∓ , n ± , n ∓ , n ± , n ∓ with bounded gonalities remain bounded and unbounded gonalities remain unbounded. Thetriangle L i L j ( L k ) becomes ( L k ) L j L i .Proof. The proof is immediate. (cid:4)
Now we prove the following theorem on line arrangements with global cyclicity which differby parallel translations.
Theorem 5.30 (Transitivity on the n − cycles which have 2 − standard consecutive struc-tures by parallel translations) . Let L F n be a line arrangement which gives rise to an n − gonality with a cycle τ at infinity hav-ing a − standard consecutive structure which respects slope property with the anticlockwise n − gonality L −→ L −→ . . . −→ L n −→ L . Let σ be another n − cycle having a − standard consecutive structure. Assume that L hasthe least non-negative slope. Then we can move the lines L , L , . . . , L n by parallel translations into another line arrangement which also gives rise to an n − gonalitywhich after a permutation of subscripts , , . . . , n N LINE ARRANGEMENTS OVER FIELDS WITH 1 − ad STRUCTURE 17 has the the cycle at infinity σ having a − standard consecutive structure which respectsslope property with the anticlockwise n − gonality L −→ L −→ . . . −→ L n −→ L . Proof.
Consider the n − cycles τ and σ which have the 2 − standard consecutive structures.We claim that there exists a parallel translation of any set of lines which give rise to an n − gonality which gives any n − cycle with a 2 − standard consecutive structure where the n − gon is given by L −→ L −→ . . . −→ L n −→ L in this anticlockwise order. To observe this fact first we consider over the field of reals inwhich we consider arbitrary n − distinct angles in [0 , π ) in the increasing order correspondingto n − lines in the real plane. 0 = θ < θ < . . . < θ n < π It does not matter what the exact angles are, however what matters is the order of theangles with respect to subscripts. Now the possibilities of the n − gons are precisely all thepossibilities which satisfy the following. • α < α < . . . < α i < π • < α i +1 < α i +2 < . . . < α n < π .where { α = 0 , α , . . . , α n } = { θ = 0 , θ , . . . , θ n } . The lines with slopes α i : i = 1 , , . . . , n gives an anticlockwise n − gon L −→ L −→ . . . −→ L n −→ L where L i makes an angle α i with respect to X − axis. The permutation λ of the subscriptscorresponding to θ i = α λ ( i ) are precisely those λ which have the 2 − standard consecutivestructures. This observation can be extended to any field F which has a 1 − ad structure.This proves the theorem. (cid:4) Lemma 5.31 (ECT Application Lemma) . Let L F n be a line arrangement. Let E ijk be an elementary collineation transformation (ECT).Suppose L i L j L k is a triangular region of the line arrangement with vertices L i ∩ L j , L k ∩ L i , L j ∩ L k oriented anticlockwise. Then by moving all the other lines L t for t (cid:54) = i, j, k parallely away from the fixed triangle L i L j L k we can make the ECT E ijk applicable withthe required condition on the parallel strip that arises from L k .Proof. This is straight forward. (cid:4)
Now we prove the Representation Theorem 1.2.
Proof.
First we observe that if we have two line arrangements with global cyclicity and hasthe same cycle at infinity then they are isomorphic. Now using the previous Theorem 5.30we can move from one cycle at infinity to another using parallel translations and also byapplying elementary collineation transformations and their inverses using Lemma 5.31. Sofrom an arbitrary line arrangement which may not have a global gonality we apply ECT’sto obtain an arrangement which has a global gonality. This proves the theorem. (cid:4)
Note 5.32.
Now we have a finite graph with possibly multiple edges and loop edges onisomorphism classes of line arrangements L F n . The vertices of this graph are the isomorphismclasses and the edges are denoted by elementary collineation transformations which can bemade applicable for an isomorphism class to go to another isomorphism class. On Quadrilateral Structures as Unique Nook Point Structures, Automor-phism group of the Quadrilateral Structure.
We begin this section with definition ofthe quadrilateral structure and the nook point.
Definition 5.33 (Quadrilateral Structure, Nook Point, Extreme Nook Point, End Points,Central Pair) . A quadrilateral structure Q is defined to be any four line arrangement L F n in the plane. Thenook point is defined to be the unique point none of whose inner coordinates is ± .The opposite vertex of the nook point is defined to be the extreme nook point. The remainingtwo extreme points are called end points. The remaining two points are defined as centralpair of points. Note 5.34.
Let A col ( Q ) denote the line arrangement collineation automorphism group of Q and let A Nook ( Q ) denote the line arrangement collineation group of automorphisms whichpreserve the nook points. Then we have A Nook ( Q ) ∼ = Z × Z . We prove the following isomorphism theorem which preserve the pair of central points andthe nook point and hence the extreme nook point. There is exactly a unique non-trivial suchautomorphism for a quadrilateral structure. So the group here is Z and hence non-trivial.This non-triviality gives rise to the following two non-trivial theorems 5.35 5.36. Theorem 5.35.
Let ( L F n ) , ( L F n ) be two line arrangements. Let φ : ( L F n ) −→ ( L F n ) be a line arrangement collineation isomorphism. Suppose for every quadrilateral substruc-ture of the line arrangement the map φ preserves nook points and the pair of central pointswith respect to the image quadrilateral substructure. Then φ is an isomorphism of line ar-rangements. Also conversely any isomorphism of the line arrangement preserves the nookpoints and the central points of any quadrilateral substructure and its image substructure.Proof. If the nook point and central points are preserved for every quadrilateral substructurethen the isomorphism has the property that for any three intersection points P , P , P ona line of the line arrangement if P is in between P , P then φ ( P ) is in between φ ( P )and φ ( P ). So it preserves the order of intersection points on each line. This proves thetheorem. (cid:4) Theorem 5.36 (Preservation of Nook Points Under Isomorphisms which preserve SlopeOrder) . Let F be a field with − ad structure. Let ( L n ) F = { L , L , . . . , L n } , ( L n ) F = { L , L , . . . , L n } are two line arrangements in the plane where the lines are indexed in the order of increasingslopes −→ ∞ −→ −∞ −→ . Then a bijection φ : ( L n ) F −→ ( L n ) F which is identityon the subscripts is an isomorphism of line arrangements if and only if for any pair offour subsets { L ti , L tj , L tk , L tl } , t = 1 , the map φ preserves the nook points. Moreover in thiscase there is a cyclic renumbering of any one of the arrangements such that the nook pointintersection of any pair of corresponding four subsets is identical.Proof. The proof is similar to the previous theorem based on the theme of central point ofthree points on any line. (cid:4)
Invariants of Line Arrangements.
The following is a list of invariants of any typefor line arrangements. • The planarity crossing numbers. • The slopes of the equations of the lines.
N LINE ARRANGEMENTS OVER FIELDS WITH 1 − ad STRUCTURE 19 • The cycle at infinity, local cycles at infinity with 2 − standard consecutive structureswhich respects the slope property. • The nook points of quadrilateral substructures. • The gonality structures and the opposite vertices of the sides of any gonality. • The modified simplicial homology groups.6.
Line-Folds
In this section we define line-folds and later ask some open questions.
Definition 6.1 (Line-Folds) . Let F be a field with − ad structure. We say a set of lines LF F n = { L , L , . . . , L n } in the plane F is called a line-fold. We say two line-folds ( LF F n ) , ( LF F n ) are isomorphic if there is a piecewise linear bijectionof the plane which map one line-fold to another line-fold. We define the following for aline-fold. Definition 6.2.
Let LF F n be a line-fold. We say a point is k − fold concurrency point if itis the intersection concurrency of k − lines. The intersection points of the line-fold is calledthe zero skeleton of the line-fold. The cardinality of the line-fold LF F n is defined to be n .A gonality of a line-fold is defined to be a convex region given by a set of n − inequalitiessimilarly as before. It is bounded if there does not exist a subset which has a − ad structureof dimension one. It is unbounded otherwise. On the Complement of Zero Sets of Certain Polynomials.
Here we prove atheorem on the exact number of regions present in the complement of the zero set of apolynomial which corresponds to a line-fold. The author Prof. Milnor J. has obtainedbounds for the betti numbers associated to the complement of the zero set of polynomialsin the article [3]. Now we state theorem which has its generalizations to higher dimensionsas follows.
Theorem 6.3.
Let F be a field with − ad structure. Let f ( x, y ) ∈ F [ x, y ] be a polynomialwhich factorizes completely into linear factors over the field F corresponding to lines inthe plane F . Let l , l , . . . , l r be the number of lines in each equivalence class under theparallel equivalence relation. Let p = ( x , y ) ∈ F and let M p = ( x − x , y − y ) denoteits corresponding maximal ideal. Let k p denote the order of vanishing of f red at p i.e. f red ∈ M k p \M k p +1 where f red is the reduced polynomial associated to f . Let d be thedegree of the reduced polynomial f red . Then we have • The total number of regions is given by d + (cid:18) d (cid:19) − (cid:88) p ∈ F (cid:18) k p − (cid:19) − r (cid:88) i =1 (cid:18) l i (cid:19) . • If l i = 1 , ≤ i ≤ r then the total number of bounded regions is given by (cid:18) d − (cid:19) − (cid:88) p ∈ F (cid:18) k p − (cid:19) . • Also in this case the total number of unbounded regions is given by d .Proof. This follows by subtracting the number of bounded local gonalities formed by non-degenerate perturbation of the concurencies of order k p when k p >
2. This proves thetheorem. (cid:4)
Open Questions on Line-Folds.
We have the following combinatorial, topological,geometric and number theoretic open questions.
Question 6.4. (1) Classify two line-folds upto isomorphism by associating invariants?(2) How many isomorphism classes of line-folds whose cardinality is n are there?(3) How many isomorphism classes of line arrangements whose cardinality is n arethere?(4) What are the possible cardinalities of the zero skeleton of a line-fold whose cardinalityis n ?(5) What are the possible cardinalities of the zero skeleton of a line-fold whose cardinalityis n which has no parallel lines?(6) How many non-isomorphic line-folds are there passing through a given set of n − generic points? Acknowledgements
The author is supported by a research grant and facilities provided by Center for study ofScience, Technology and Policy (CSTEP), Bengaluru, INDIA for this research work.
References [1] A. Dimca,
Hyperplane Arrangements: An Introduction, Universitext, ISBN: 978-3-319-56220-9, doi:10.1007/978-3-319-56221-6[2] P. Orlik, H. Terao, Arrangements of Hyperplanes, Grundlehren der mathematischen Wissenschaften ,Springer-Verlag ISBN 978-3-662-02772-1[3] J. Milnor, On the betti numbers of real varieties, Proc. Amer. Math. Soc. (1964), no. 2, 275-280
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