The Linear Span of Four Points in the Pl\"ucker's Quadric in ${\mathbb P}^5$

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 7, No. 4, 2014, 472-485

ISSN 1307-5543 – www.ejpam.com

The Linear Span of Four Points in the Plücker’s Quadric in

P

Jacqueline Rojas

_{, Ramón Mendoza}

1_{CCEN - Departamento de Matemática - UFPB Cidade Universitária, 58051-900,}

João Pessoa - PB - Brasil

2_{CCEN - Departamento de Matemática - UFPE Cidade Universitária, 50740-540, Recife - PE - Brasil}

Abstract. Given four (distinct) lines ℓ₁,ℓ₂,ℓ₃,ℓ₄ inP3_{. Let P}

i (i = 1, . . . , 4) be the image ofℓi in the Plücker’s quadricQ ⊂P5_{under the Plücker embedding} P _{(in (1)). SetΛ =P}

1, . . . ,P4

be the linear span of those four points inP5_{. The purpose of this article is to write specifically what kind of} quadricΛ∩ Qcan be, taking under considerations all possible configurations of these four lines inP3_. In particular, having in mind the classical problem in Schubert Calculus:How many lines in 3-space meet four given lines in general position? whose answer is 2 (see p. 272 in[3]or p. 746 in[4]). We verified that four lines inP3_{are in general position if and only ifΛis a 3-plane andΛ}∩ Q_{is an irreducible} quadric surface. In fact, we prove that there are exactly two solutions if and only ifΛis a 3-plane and

Λ∩ Qis a nonsingular quadric.

2010 Mathematics Subject Classifications: AMS 14N05, 14N20

Key Words and Phrases: Plücker’s quadric, linear span, 4-line problem.

1. Introduction

Plücker’s coordinates were introduced by the German geometer Julius Plücker (1801-1868) in the 19th century, as a way to assign six homogenous coordinates to each line in the complex projective 3-spaceP3_{. Since they satisfy a homogeneous quadratic equation, it} follows an embedding of the 4-dimensional space of lines inP3_{(denoted by}G₁₍P3_{)and called}

grassmannian of lines inP3_{) onto a nonsingular quadric hypersurface}Q _in P5 _(see Proposi-tion 3). Thus, reminding the Schubert’s classical enumerative problem: How many lines in 3-space meet four given lines in general position? Whose answer can be found in many texts and is given by: “there are two lines in P3 _{which meet 4 given lines in general position” (see} Example 14.7.2 at p. 272 in [3]). In fact, if you want to know an algorithm to determine explicit solutions, then see[7]. On the other hand, any beginner in the art of solving enumer-ative problems will ask: what does general positionmeans? In Algebraic Geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the

∗_{Corresponding author.}

Email addresses:[email protected](J. Rojas),[email protected](R. Mendoza),

general case situation, as opposite to some more special or coincident cases that are possible. Its precise meaning differs in different settings. For example, in[8]the authors imposed the conditionℓi∩ℓj=;(1≤i< j≤4) to the four given linesℓ1,ℓ2,ℓ3,ℓ4inP3and, even under this assumption they found (in one case) infinitely many solutions for the 4-lines problem (cf. 3.1in the last subsection). So, this condition is not enough for the four given lines to be in general position.

In this article, we use the identification between lines in P3 _{and points in the quadric} hypersurfaceQ to explain what is the precise meaning of general position for that problem. Nevertheless, the emphasis in our work lies on to take under considerations all possible config-urations of these four lines inP3_{and write specifically what kind of quadricΛ}∩Q_{can be. One} key ingredient in this work lies on the well known description of all linear subspaces contained in a quadric hypersurface inP3_andP5 _{(see subsection 2.1 and 3.1).}

Finally, we note that Plücker was a geometer that firmly believed in the importance of the applications of Mathematics to the physical sciences. So, in 1847 he turned to Physics, accepting the chair of Physics at Bonn and working on magnetism, electronics and atomic physics. He anticipated Gustav Kirchhoff and Robert Wilhelm Bunsen in announcing that the lines of the spectrum were characteristic of the chemical substance which emitted them, and in indicating the value of this discovery in chemical analysis. According to Johann Hittorf he was the first who saw the three lines of the hydrogen spectrum, which a few months after his death were recognized in the spectrum of the solar protuberances.

2. Notations and Preliminary Results

We denote by C _{the field of complex numbers. Let V be an n-dimensional vector space} overC_{. Denote by[v1, . . . , vk]the subspace ofV generated by the vectors v1, . . . , vk}∈_V.

Thek-grassmannian associated to the vector spaceV. For each integerk,

0≤k≤n=dimV, we denote byGk(V)the set of allk-dimensional linear subspaces ofV and

call it the k-grassmannianassociated to V. In the particular casek =1, the 1-grassmannian associated toV it is also calledprojective space associated to V and it is denoted byP_(V)(i.e. P_{(V) := G1(V)). We use the notation} Pn _{instead of} P₍Cn+1_{) and p = [a0 : . . . : an] for}

p= [(a₀, . . . ,a_n)]∈Pn_{, just for the sake of simplicity.}

IfW ∈G_k+1(V)thenP_(W)⊆P_{(V)will be calledk-linear subspaceof}P_{(V). The set of allk} -linear subspaces ofP_{(V)will be denoted by}G_k(P_{(V)), thegrassmannian of k-linear subspaces} ofP_{(V). Moreover, we shall call}G₁₍P_(V)), G₂₍P_{(V)) and}G_n−1(P_{(V))the grassmannian of} lines, planes and hyperplanes inP_{(V), respectively. So, since a lineℓin}P3 _{is equal to}P_(W) for someW ∈G2(C4), we have the correspondence

G₂(C4₎ −→ G₁₍P3₎

W 7−→ P(W),

Next we introduce the notion of algebraic projective set inPn_{. We will see in Proposition 1} thatk-linear subspaces ofPn _{are examples of algebraic sets.}

Algebraic projective sets inPn_{. Let}C_{[X] =}C_{[X0, . . . ,Xn]be the polynomial ring over}C_in the variables X₀, . . . ,X_n. Now, for each integer d ≥ 0 consider the vector subspace C_[X]d, generated by all monomials in X0, . . . ,Xn of degreed. Each element in C[X]d will be called

an homogeneous polynomial of degreed. IfF ∈C_{[X]d, then we define,}Z_{(F),the zero setof}

F inPn_by

Z(F) =¦[v]∈Pn|_{F(v) =0}©_.

For example, if L∈C_{[X]1, then}Z_{(L) =}P_{(W)withW =}{v∈Cn+1|_{L(v) =}0}. Therefore, Z(L)is a hyperplane ofPn_{, if L}6_{=0 else}Z_{(L) =}Pn_.

Ifd≥1, then an element[F]in the projectivization ofC_{[X]d will be calledhypersurface of}

degree d inPn_{. From here on, when we say: LetX} ⊂Pn _{be the reduced hypersurface defined} byF ∈C_{[X]d, that means thatX=}Z_(F)⊂Pn _{andF is square-free.}

A subset X ofPn _{will be calledalgebraic projective set, if there exist homogeneous} polyno-mialsF₁, . . . ,F_k inC_{[X]such thatX =}Z_(F1)∩ · · · ∩ Z_(Fk).

Next we show that any r-linear subspace inPn _{is the intersection of exactly n}−_r hyper-planes.

Proposition 1. LetΛ be an r-linear subspace inPn _{with n> r. Then, there are exactly n}−_r

linearly independent linear forms L₁,. . . ,L_n−rinC_{[X]such that}

Λ =Z(L₁)∩ · · · ∩ Z(L_n−r).

Proof. Assume thatΛ =P_(W)withW ∈_Gr+1(Cn+1_{). Letα=}{_{e1, . . . ,en+1}}_{be the} canon-ical base ofCn+1 _{and letβ =}{_e∗

1, . . . ,e

∗

n+1}be the associated dual base of(C

n+1₎∗_{. Next we}

consider the linear isomorphism

ϕ: (Cn+1₎∗ → C_[X]1

e∗_i 7→ Xi−1.

Now, letW0be the annihilator ofW, thenϕ(W0)is an(n−r)-dimensional linear subspace of C_{[X]1. Finally, an easy verification show that any base}{_{L1, . . . ,Ln−r}}_ofϕ(W0_{)verified that}

Λ =Z(L₁)∩ · · · ∩ Z(L_n−r).

Incidence ofr-linear subspaces with reduced hypersurfaces inPn_{. The following} proposi-tion will play an important role in our investigaproposi-tions.

Proposition 2. Let Z⊂Pn _{be the reduced hypersurface defined by the homogeneous polynomial}

F of degree d andΛbe an r-linear subspace ofPn_{with r}≥_{1. Then the following conditions are}

(1) Z∩Λ6=;;

(2) Z∩Λconsists of infinitely many points, ifΛ⊂ Z or r≥2, else Z∩Λconsists of at most d points.

Proof. To arrive at statements (1)and first part of(2) have in mind that dimZ = n−1, dimΛ = r and apply Theorem 7.2 at p. 48 in[5]. Already, the last part of statement (2)

follows easily from the fundamental theorem of algebra.

Projective Tangent Space and Nonsigular Reduced hypersurface. LetZ⊂Pn_{be the reduced} hypersurface defined byF ∈C_{[X]d andp= [v]}∈_{Z. LetF}′

v:C

n+1_{−→Cbe the differential of}

F at v. We define theprojective tangent space,T_{pZ, to the hypersurface Zat pby}

T_pZ=P_(ker(F′

v)) =

¦

[u0: . . . :un]∈Pn| n

X

i=0 ∂F

∂X_i(v)·ui=0

©

.

Now, note that:

• follows from the Euler relationPn_i=0∂∂_XF

i ·Xi=d F that p∈

T_pZ.

• T_{pZis a hyperplane in}Pn_{if and only if p}6∈ ∩n

i=0Z(

∂F

∂Xi).

A pointp∈Zsatisfying the last condition above will be called anonsingular pointofZ, elsep

will be called asingular pointofZ. If all the points inZare nonsingular, thenZwill be called anonsingularhypersurface inPn_.

For example, after a linear change of coordinates (see Theorem 4 at p. 411 in [2]) we concluded thatZ(X₀2+X₁2+X₂2+X₃2)is the unique nonsingular quadric surface inP3_{, whereas} the singular reduced quadric surfaces inP3_{correspond either to the union of two planes or a} quadric cone. Moreover, it is straightforward to show that the vertex of a quadric cone is its unique singular point. In the case of the union of two (distinct) planes, it is verified that the points in the line where the two planes meet are their singular points.

2.1. Lines on Reduced Quadrics Surfaces in

P

Next, we present some observations about lines contained on reduced quadrics surfaces in P3_.

• Union of two planes. Of course any line in this surface will be contained in one of the planes. Thus this surface could have at most two disjoint lines.

• Quadric cone. Any line in this surface passes through its vertex. Thus this surface does not contain disjoint lines.

Lemma 1. Let Q be a nonsingular quadric surface inP3_{. Then there exist two families of lines} L ={Lp}p∈P1 andM={M_p}_p∈P1in Q such that

(1) Lp∩Lq=;and Mp∩Mq=;for all Lp,Lq∈ L, Mp,Mq∈ M and p6=q∈P1. (2) L_p∩M_q6=;for all L_p∈ L, M_q∈ M and p,q∈P1_.

(3) Ifℓis a line contained in Q thenℓ∈ L orℓ∈ M.

(4) Given x∈Q there exist unique lines L_p(x)∈ L and M_q(x)∈ M such that{x}=L_p(x)∩M_q(x).

One other simple but important fact which will help us to prove our main result (Theorem 1) is the next Lemma.

Lemma 2. Given the linesℓ₁,ℓ₂andℓ₃inP3_{such thatℓi}∩_ℓj=;_for1≤_{i< j}≤_{3, there exists}

a nonsingular quadric surface Q inP3_{containingℓ1,ℓ2andℓ3.}

Proof. Take 3 points p1i,p2i,p3i on each lineℓi (i=1, 2, 3) and note that W={G∈C_[X]2|_{G(pi j) =0 for all 1}≤_i,j≤3}_(heren=3)

is a subspace ofC_{[X]2 of dimension greater than or equal to 1. So we can chooseG} 6_{= 0 in}

W and takeQ =Z(G). Now follows from Proposition 2 that each line ℓ_i is contained inQ. Finally, sinceQcontains three pairwise disjoint lines, thenQmust be a nonsingular surface in P3_.

3. The Plücker’s Quadric

Q

in

P

The Plücker embeddingP :G_k(P(V))−→P(Λk+1_{V)is the map that enable us to identify} the grassmannianG_k(P_{(V))with a projective variety in}P_(Λk+1_V)(Λk+1_{V denotes the(k+1)} -th exterior power ofV). It is defined byP_(W)7→_[u0∧_{. . .}∧_{uk], ifW = [u0, . . . , uk].}

The Plücker’s quadricQinP5_{. In what follows we will considerV =}C4_{. Now, fix the base} {e_i∧e_j}1≤i<j≤4 ofΛ2C4 where{ei}4i=1is the canonical basis ofC

4_{. Let us consider}

W = [u, v]∈G2(C4)with u= (u0,u1,u2,u3)and v= (v0,v1,v2,v3), then we see that

u∧v= X 1≤k<l≤4

w_k−1,l−1e_k∧e_l wherew_{i j}=u_iv_j−u_jv_i for 0≤i< j≤3.

Thus the Plücker embeddingP above in coordinates is given by P :G₁(P3) −→P5

P_{([u, v])} 7−→_{[w01:w02:w03:w12:w13:w23].} (1) Proposition 3. The Plücker map P : G₁₍P3₎ −→ P5 _{defined in (1) is an embedding of the}

grassmannianG₁₍P3_{)over the nonsingular quadric hypersurface}Q₌Z_(F)⊂P5_where

Proof. See Theorem 11 at p. 409 in[2]or p. 209-211 in[4].

We will refer toQ in Proposition 3 as the Plücker’s quadricQ(inP5_).

Next, we will make a remark that tells us who is the image of the families of linesL and M of a nonsingular quadric surface inP3_{, under the Plücker embedding}P _{(cf. Lemma 1).}

Remark 1. It can be shown thatP(L) =Z(X₀+X₅,X₁−X₄,X₂+X₃,F)and

P(M) =Z(X0−X5,X1+X4,X2−X3,F), so the image of the families of linesL andM, under

the Plücker embeddingP (in(1)) are nonsingular conics lying in complementary planes (also see p. 196 in[6]).

3.1. Linear Subspaces in the Plücker’s quadric

Q

We begin by proving a simple result, which allows us to conclude that the Plücker’s quadric Q does not contain 3-planes or hyperplanes. In fact, it is verified thatQ only contains lines and planes, and this will be described in Propositions 5 and 6, respectively (see[1]).

Proposition 4. Let Z ⊂Pn _{be a nonsingular quadric hypersurface defined by G}∈C[X]₂ andΛ a r-linear subspaces ofPn _{. IfΛ}⊂_{Z, then we have2r<n.}

Proof. SetΛ =P_{(W)whereW = [w0, . . . , wr]. Assume thatG=}Pn

i=0X2i (otherwise make

a linear change of coordinates). SoG induz the bilinear formB:Cn+1×Cn+1→C _{given by}

B(v, w) =v0w0+· · ·+vnwn, if v= (v0, . . . ,vn)and w= (w0, . . . ,wn). Let

W⊥={v∈Cn+1|_{B(v, w) =0 for all w}∈_W}

be the the orthogonal subspace associated toW.

Note thatW⊥ =ker(T0)∩ · · · ∩ker(Tr)where Ti is the linear functional overCn+1 given

by Ti(v) =B(v, wi),i∈ {0, . . . ,r}. Since these linear functionals are linearly independent we

conclude that dimW⊥=n+1−dimW =n−r.

On the other hand, Λ⊂ Z if and only if G(w) =B(w, w) =0 for all w∈W. ThusΛ⊂ Z

if and only if W ⊆ W⊥. So, if Λ ⊂ Z then dimW ≤ dimW⊥ = n+1−dimW. Therefore, 2 dimW≤n+1 which implies 2r<n.

In what follows, we denoted by ℓ₁, . . . ,ℓ_kthe smallest linear subspace ofPn _containing the linesℓ1, . . . ,ℓk ofPn. For example, ifℓ1 andℓ2 are two distinct lines inPn having a com-mon point, thenℓ₁,ℓ₂is a plane, elseℓ₁,ℓ₂is a 3-plane.

Next we give the description of lines and planes inQ.

Lines in the Plücker’s quadricQ. The lines in the Plücker’s quadricQare parametrized by the incidence varietyΓ ={(p,Π)| p∈Π} ⊂P3×G₂₍P3_{). In fact, let p}∈P3 _andΠ⊂P3 _{be a} plane throughp. Set

Proposition 5. Let P :G₁₍P3₎ → P5 _{be the Plücker embedding in (1). If L is a line in} P5

contained inQ, then we have:

(i) P(Ω_p(Π))is a line inP5_{contained in}Q_{for all(p,Π)}∈_Γ.

(ii) If P0,P∈L are two different points such that P0=P(ℓ0)and P=P(ℓ), thenℓ0∩ℓ={p}

and they determine the planeΠ =ℓ₀,ℓ. In fact, for every point Q=P(m)∈L holds that m∈Ω_p(Π).

Planes in the Plücker’s quadric Q. There are two families of planes in the Plücker’s quadricQparametrized by P3 _{and its dual}

∨

P3₌G₂₍P3_{), respectively. In fact, let p}∈P3 _and

Π⊂P3 _{be a plane. Set}

Ω_p=¦ℓ∈G₁₍P3₎|_p∈_ℓ©_{andΩ(Π) =}¦_ℓ∈G₁₍P3₎|_ℓ⊂_Π©_.

Proposition 6. LetP :G₁₍P3₎ → P5 _{be the Plücker embedding in (1). If Λ is a plane in}P5

contained inQ, then we have:

(i) P(Ω_p)andP(Ω(Π))are planes inP5_{contained in}Q_.

(ii) The pre-image ofΛunder the Plücker embeddingP is either: Ω_pfor some p∈P3 _orΩ(Π)

for someΠ∈G₂₍P3_{). In fact, if Pi=}P_{(ℓi), i=0, 1, 2are three points in}Q_{whose linear}

span it is equal toΛ, then Λ = Ω_p if p ∈ ∩2_i=0ℓ_i (and these lines are non-coplanar)else Λ = Ω(Π)withΠ =〈ℓ0,ℓ1,ℓ2〉.

4. Description of

Λ

∩ Q

According to the Relative Position of the Four Given

Lines in

P

We begin this section by introducing some more notation. As we did for lines we denote by 〈p1, . . . ,pk〉the linear spanofp1,. . . ,pk∈Pn(i.e. the smallest linear subspace ofPncontaining

these points). Moreover, if pis a point andℓis a line inPn_{, then}〈_p,ℓ〉_{also denote the linear} span ofpandℓinPn_{. So}〈_p,ℓ〉_{=ℓ, if p}∈_ℓelse〈_p,ℓ〉_{is a 3-plane.}

Letℓ₁,ℓ₂,ℓ₃ andℓ₄ be four distinct lines inP3_{. LetPi} ∈ Q _{(i=1, . . . , 4) be the image of}

ℓ_i under the Plücker embeddingP (in (1)) andΛ =〈P1,P2,P3,P4〉 ⊂P5. SetS be the set of solutions of the 4-lines problem in Schubert calculus, i.e.

S =ℓ∈G₁₍P3₎|_ℓ∩_ℓi6₌;_{for 1}≤_i≤_{4 .}

We have three major cases to be considered (cf. subsections 4.1, 4.2 and 4.3). In each of these cases, the linear spaceΛ andΛ∩ Q are reported in tables, whose rows are labeled according to the position that the lines can have inP3_{. Next, we determineΛandΛ}∩ Q_in some of these cases, which we believe are sufficient to illustrate the technique used in this computations, we left to the reader the other cases

4.1. The Four Lines are Concurrent, Say

p

∈ ∩

ℓ

.

Table 1: Four Lines are Concurrent.

Subcase Position of the lines S Λ Λ∩ Q

1.1 ℓ1,ℓ2,ℓ3andℓ4are contained

in the planeΠ Ωp∪Ω(Π) Line Ωp(Π)

1.2 ℓ1,ℓ2,ℓ3 andℓ4 are non

coplanar lines Ωp Plane Ωp

1.1 Naturally any line passing trough p is a solution. On the other hand, ifℓ∈ S is a solution such that p6∈ℓ, it certainly meets the linesℓ1 andℓ2 at two different points, which implies that ℓ ⊂ Π. Therefore S = Ω_p∪Ω(Π). On the other hand, since P_i ∈ Ω_p(Π) for

i=1, . . . , 4, thenΛ≡Ω_p(Π)is a line contained inQ.

Figure 1: Case 1.1

4.2. The Four Lines have no Common Point and at Least Two are Coplanar.

Of course in all the subcases listed below we consider an index reordering if necessary.

Table 2: Four Lines have no Common Point and at Least Two are Coplanar.

Subcase Position of the lines S Λ Λ∩ Q

2.1 ℓ1,ℓ2,ℓ3andℓ4are contained

in the planeΠ Ω(Π) Plane Ω(Π)

Table 3: 2.2 Exactly Three Lines are Coplanar. Assume thatℓ1,ℓ2andℓ3are contained in the planeΠand

Π∩ℓ4={q}.

Subcase Position of the lines S Λ Λ∩ Q

2.2.1 T3_i=1ℓi={p} Ωp(〈p,ℓ4〉)∪Ωq(Π) Plane

Union of two lines

2.2.2 T3_i=1ℓ_i =; Ω_q(Π) 3-plane Union of

two planes

2.2.1Letℓ∈ S be a solution which meets the linesℓ1andℓ2 at two different points, then ℓ∈Ω_q(Π) (sinceΠ∩ℓ₄ = {q}). Else ℓmeetsℓ₁ andℓ₂ at p andℓ₄ at some point different ofp, thenℓ⊂ 〈p,ℓ₄〉. Soℓ∈Ω_p(〈p,ℓ₄〉). Therefore, S = Ω_q(Π)∪Ω_p(〈p,ℓ₄〉)and it can be identified with two projective lines having a common point (just a cross!). On the other hand, sinceP₁,P₂andP_ilies on the lineΩ_p(Π)andP₄6∈Ω_p(Π)(p6∈ℓ₄) we conclude thatΛis a plane inP5_{not contained in}Q. So_Λ∩ Q _{is a conic. Note that the line L1,2=}〈_P1,P2〉 ⊂_Λ∩ Q _and

P46∈L1,2. Therefore,Λ∩ Q is the union two lines. In fact,Λ∩ Q=L1,2∪LwhereL=〈P4,M〉 withM=P(〈p,q〉).

Table 4: 2.3 Any Three Lines are non Coplanar and at Least Two Pair of Lines are Coplanar with

ℓi6= Π1∩Π2∀i. Assume thatℓ1∩ℓ2={p}andℓ3∩ℓ4={q}. SetΠ1=〈ℓ1,ℓ2〉andΠ2=〈ℓ3,ℓ4〉.

Subcase Position of the lines S Λ Λ∩ Q

2.3.1 p,q∈Π₁∩Π₂ Ω_p(Π₂)∪Ω_q(Π₁) Plane Union of two lines 2.3.2 p∈Π₁∩Π₂6∋q Ω_p(Π₂) 3-plane Union of two planes 2.3.3 p6∈Π₁∩Π₂6∋q ¦Π₁∩Π₂,ℓ_p,q© 3-plane Nonsingular

quadric

Figure 2: Case 2.3.3

2.3.3Letℓbe a solution. Next we consider the following two cases.

(ii) Assume that p 6∈ℓ andq 6∈ℓ. Thenℓ meets the linesℓ₁ andℓ₂ (respectively, ℓ₃ and ℓ₄) at two different points which implies thatℓ⊂Π₁ (respectively, ℓ⊂Π₂). Therefore, ℓ= Π₁∩Π₂.

Now, note that the line L_1,2 = 〈P₁,P₂〉 ⊂ Λ∩ Q and P_i 6∈ L_1,2, i = 3, 4. So 〈P₁,P₂,P₃〉is a plane not contained inQ (sincep6∈ℓ_i andℓ_i 6⊂Π₁ fori=3, 4). On the other hand, the line

L3,4=〈P3,P4〉is also contained inΛ∩ Qand we observe that L1,2and L3,4are disjoint (since

L_1,2= Ω_p(Π₁)and L_3,4= Ω_q(Π₂)), soP₄ does not belong to the plane〈P₁,P₂,P₃〉. Therefore, Λis a 3-plane. Keeping in mind thatΛ∩ Q is a quadric surface containing four non coplanar points and two disjoint lines, we conclude thatΛ∩Qis a union of two planes or a nonsingular quadric. Suppose thatΛ∩ Q is a union of two planes, say Λ∩ Q = Λ₁∪Λ₂. Thus we can assume that L_1,2 ⊂ Λ₁ and L_3,4 ⊂ Λ₂. Then necessarilyΛ₁ it is either Ω_p or Ω(Π₁), and in the same formΛ₂it is eitherΩ_q orΩ(Π₂). But in any case,Λ₁∩Λ₂ will be empty or a point. Therefore,Λ∩ Qis a nonsingular quadric surface.

Table 5: 2.4 Any Three Lines are non Coplanar and at Least Two Pair of Lines are Coplanar withΠ₁∩Π₂=ℓ1. Assume thatℓ1∩ℓ2={p}andℓ1∩ℓ3={q}. SetΠ1=〈ℓ1,ℓ2〉andΠ2=〈ℓ1,ℓ3〉. Let{ri}= Πi∩ℓ4fori=1, 2.

Subcase Position of the lines S Λ Λ∩ Q

2.4.1 p=q Ω_p(〈p,ℓ₄〉) 3-plane Union of

two planes 2.4.2 p6=qandp∈ℓ4

(orp6=qandq∈ℓ₄)

Ω_p(〈p,ℓ3〉)

(orΩ_q(〈q,ℓ₂〉)) 3-plane

Union of two planes 2.4.3 r1=r2 and

#{p,q,r1}=3

¦

ℓ1

©

3-plane Quadric cone 2.4.4 p6=qandr₁6=r₂ ¦ℓ_q,r

1,ℓp,r2

©

3-plane Nonsingular quadric

2.4.4Letℓ∈ S. Here we have two possibilities:

(i) p6∈ℓ. Thenℓ∩ℓ₁ 6=ℓ∩ℓ₂ and we have thatℓ⊂Π₁. Thusℓ∩ℓ₃⊂Π₁∩ℓ₃ ={q}and ℓ∩ℓ4⊂Π1∩ℓ4={r1}. Therefore,ℓ=ℓq,r1(q6=r1 sinceℓ46⊂Π1).

(ii) p ∈ ℓ. Note that ℓ meets the lineℓ₃ at a point p₁ (p₁ ∈ Π₂) different from p. Thus ℓ⊂Π₂. On the other handℓ∩ℓ4⊂Π2∩ℓ4={r2}. Therefore,ℓ=ℓp,r2.

As in case2.3.3we have that〈P₁,P₂,P₃〉is a plane not contained inQ. Since the lines

L_1,2=〈P₁,P₂〉and L_1,3=〈P₁,P₃〉are contained in〈P₁,P₂,P₃〉 ∩ Q we conclude that

〈P1,P2,P3〉 ∩ Q= L1,2∪L1,3. So P46∈ 〈P1,P2,P3〉. Therefore,Λis a 3-plane. Keeping in mind that Λ∩ Q is a quadric surface containing four non coplanar points, {P₁} = L_1,2∩L_1,3 and

L_1,4=〈P₁,P₄〉 6⊂Λ∩ Q(sinceL_1,46⊂ Q), we conclude thatΛ∩ Qis a union of two planes or a nonsingular quadric. Suppose thatΛ∩ Qis a union of two planes, sayΛ∩ Q= Λ₁∪Λ₂. Thus we can assume that L_1,2 ⊂ Λ₁. Now, note that L_2,3 6⊂Λ∩ Q (since L_2,3 6⊂ Q). So P₃ 6∈Λ₁, which implies L_1,3⊂Λ₂. Finally, observe that L_2,4 andL_3,4 are not contained inΛ∩ Q. Thus

Figure 3: Case 2.4.4

Table 6: 2.5 Exactly Two Lines are Coplanar. Assume that ℓ1 and ℓ2 are contained in the plane Π, ℓ1∩ℓ2={p},Π∩ℓi={pi}for i=3, 4andp36=p4. LetC={p,p3,p4}.

Subcase Position of the lines S Λ Λ∩ Q

2.5.1 p=p3

(orp=p₄)

Ω_p(〈p,ℓ₄〉)

(orΩ_p(〈p,ℓ₃〉)) 3-plane

Union of two planes 2.5.2 #C=3 and

p∈ℓ_p

3,p4

¦

ℓ_p3,p4

©

3-plane Quadric cone 2.5.3 p6∈ℓp3,p4

(so #C =3)

¦

〈p,ℓ3〉 ∩ 〈p,ℓ4〉,ℓp3,p4

©

3-plane Nonsingular quadric

2.5.3Letℓbe a solution and consider the following two possibilities.

(i) ℓ⊂Π. Sinceℓ∩ℓ_i ⊂Π∩ℓ_i={p_i}thenp_i∈ℓfori=3, 4. Thereforeℓ=ℓ_p

3,p4.

(ii) ℓ6⊂Π. Now having in mind thatℓ∩ℓ_i 6=;fori=1, 2 andℓ6⊂Πwe concluded thatp∈ℓ. On the other handℓ∩ℓ_i ={q_i} ⊂ℓ_i ⊂ 〈p,ℓ_i〉fori=3, 4. Note thatq_i6=p(sincep6∈ℓ_i fori=3, 4) which impliesℓ=ℓ_p,q

i ⊂ 〈p,ℓi〉fori=3, 4. Therefore,ℓ=〈p,ℓ3〉 ∩ 〈p,ℓ4〉. Note that P_i 6∈ L_1,2 =〈P₁,P₂〉 ⊂ Q for i=3, 4. Thus 〈P₁,P₂,P₃〉 is a plane not contained in Q since p6∈ℓ3 6⊂Π. Since L1,2∪ {P3} ⊂ 〈P1,P2,P3〉 ∩ Q, ifP4∈ 〈P1,P2,P3〉thenP4∈ L1,2or

P₄belong to the componentLpassing throughP₃in the conic〈P₁,P₂,P₃〉 ∩ Q =L_1,2∪L. But

P₄6∈ L_1,2 becauseℓ₄ 6⊂Π. Moreover, P₄ 6∈L sinceℓ₄∩ℓ₃=;. Thus Λis a 3-plane. Keeping in mind thatΛ∩ Q is a quadric surface containing four non coplanar points and L1,3 is not contained inΛ∩ Q, we conclude thatΛ∩ Q is a union of two planes or a nonsingular quadric. Suppose thatΛ∩ Q is a union of two planes, sayΛ∩ Q= Λ₁∪Λ₂. Thus we can assume that

Figure 4: Case 2.5.3

4.3. No Pair of Lines is Coplanar

LetQbe a nonsingular quadric surface inP3_{containingℓ1,ℓ2andℓ3(as stated in Lemma 2).} Since ℓ_i∩ℓ_j = ;for 1 ≤ i < j ≤ 3 then they belong to the same family of lines in Q (cf. Lemma 1). So we will assume thatℓ1,ℓ2 andℓ3 belong to the familyL.

Note that any solutionℓ∈ S meets the lineℓ_i at a point p_i ∈ℓ_i ⊂Qfori=1, 2, 3. Since this three points p₁,p₂ and p₃ are different and belong toℓ∩Q. Then follows from Proposi-tion 2 thatℓ⊂Q, which implies thatℓ∈ M. Therefore any solution belong to the familyM.

In order to determine the set of solutionsS and the linear spaceΛ, we will consider the following two subcases.

3.1 ℓ₄⊂Q . In this caseℓ₁,ℓ₂,ℓ₃ andℓ₄ belong to the same familyL (sinceℓ₄if disjoint to the others three). ThereforeS =M.

Now, since the lineL_i,j =P_i,P_j6⊂ Q for 1≤i< j≤3, we conclude from Proposition 6 that P1,P2,P3

is a plane not contained in Q. But, from Remark 1 we conclude that

P₁,P₂,P₃∩ Q=P(L). Consequently, Λ =P₁,P₂,P₃(keep in mind that P₄∈ P(L)

too) andΛ∩ Q is a nonsingular conic.

3.2 ℓ₄6⊂Q . In this case follows from Proposition 2 thatℓ₄∩Qis non empty and

ℓ₄∩Q={x,y}(withx and y do not necessarily distinct). LetM_x andM_y be the unique lines in the familyM passing throughx and y, respectively (cf. (4) in Lemma 1). Indeed

Mx and My are solutions (in fact, if p∈ {x,y}thenMp∩ℓi is non empty fori=1, 2, 3

becauseℓ_i∈ L andM_p∩ℓ₄ ={p}). Now, we will show that M_x andM_y are the unique solutions. Letℓ∈ M be a solution, thenℓ∩ℓ₄⊂Q∩ℓ₄={x,y}which implies thatx ∈ℓ or y∈ℓ. Therefore follows from (4) in the Lemma 1 thatℓ=Mx orℓ=My.

Again, since the line Li,j =

Pi,Pj

6⊂ Q for 1≤ i< j ≤3, we conclude thatP1,P2,P3

is a plane not contained inQandP₁,P₂,P₃∩ Q is a nonsingular conic. On the other hand, the conditionℓ₄ 6⊂Q implies thatℓ₄ 6∈ L. So, we are left to conclude that P4 6∈

P1,P2,P3

Assume that ℓ₄ is tangent to Q. Let x ∈ℓ₄∩Q be the tangent point. Let M_x ∈ M and

L_x ∈ L be the (only) lines inQpassing through x. It is verified thatℓ₄ ⊂ T_{xQ =}〈_Mx,Lx〉. Therefore, ℓ4 ∈ Ωx(TxQ). Set M = P(Mx),L = P(Lx) ∈ Q. Thus L belong to the conic

P₁,P₂,P₃∩ Q=P(L). Moreover M ∈ 〈L,P₄〉 ⊂Λ∩ Q(since, x ∈ℓ₄ ⊂ 〈M_x,L_x〉). On the

other hand the linesL_i=〈M,P_i〉 ⊂Λ∩ Q,i=1, . . . , 4. Therefore,Λ∩ Q is a quadric cone.

Thus we have proved the following Theorem. Theorem 1. Given four linesℓ1,ℓ2,ℓ3,ℓ4inP3. Set

S =ℓ∈G₁₍P3₎|_ℓ∩_ℓi 6₌;_for1≤_i≤_{4 .}

Let P_i (i = 1, . . . , 4)be the image ofℓ_i in the Plücker’s quadricQ ⊂P5 _{under the Plücker}

em-beddingP (in(1)). SetΛ =P1, . . . ,P4

be the linear span of those four points inP5_{. Then the}

cardinal of the setS is either 1, 2 or infinite. Moreover we have.

(i) #S=2if and only ifΛis a 3-plane andΛ∩ Q is a smooth quadric. (ii) #S=1if and only ifΛis a 3-plane andΛ∩ Q is a quadric cone. (iii) If#S=∞then

(a) S is a line if and only ifΛis a 3-plane andΛ∩ Q is a union of two planes.

(b) S is a conic if and only ifΛis a plane andΛ∩ Qis a nonsingular conic.

(c) S is a reduced and reducible conic if and only ifΛis a plane andΛ∩ Q is a reduced and reducible conic.

(d) S is a plane if and only ifΛis a plane inQ.

(e) S is a union of two distinct planes if and only ifΛis a line inQ.

In terms of the lines position Theorem 1 tell us:

#S=2⇐⇒

                                  

• ∩4

i=1ℓi =;and any three lines are non coplanar.

(i) ℓ1∩ℓ2={p},ℓ3∩ℓ4={q}withp6=qandp6∈Π1∩Π26∋q, ifΠ₁=〈ℓ₁,ℓ₂〉,Π₂=〈ℓ₃,ℓ₄〉.

(ii) ℓ₁∩ℓ₂={p},ℓ₁∩ℓ₃={q}withp6=q. IfΠ_i=〈ℓ₁,ℓ_i+1〉 fori=1, 2, then it is verified thatΠ_i∩ℓ4={ri}i=1, 2

andr₁6=r₂.

(iii) ℓ₁∩ℓ₂={p}. IfΠ =〈ℓ₁,ℓ₂〉then it is verified fori=3, 4 thatΠ∩ℓi={ri},r36=r4 andp6∈ 〈r3,r4〉.

• No pair of lines is coplanar.

Note that (i), (ii) and (iii) above correspond to subcases2.3.3,2.4.4and2.5.3in subsec-tion 4.2. Already (iv) corresponds to subcase3.2in subsection 4.3.

#S=1⇐⇒

              

             

• ∩4

i=1ℓi=;and any three lines are non coplanar. (α) ℓ₁∩ℓ₂={p},ℓ₁∩ℓ₃={q}withp6=q. IfΠ_i =〈ℓ₁,ℓ_i+1〉

fori=1, 2, then it is verified thatΠ_i∩ℓ4={ri}i=1, 2.

withr₁=r₂and #{p,q,r₁}=3.

(β) ℓ₁∩ℓ₂={p}. IfΠ =〈ℓ₁,ℓ₂〉then it is verified fori=3, 4 thatΠ∩ℓi={ri}, r36=r4, p∈ 〈r3,r4〉andp6=ri.

• No pair of lines is coplanar.

(γ) The non singular quadric surfaceQ⊂P3_{containingℓ1,ℓ2} andℓ₃, meetsℓ₄exactly in one point.

Note that (α) and (β) above correspond to subcases2.4.3 and2.5.2 in subsection 4.2. Already (γ) correspond to subcase3.2in subsection 4.3.

ACKNOWLEDGEMENTS The first author wishes to express her gratitude to I. Vainsencher (DM-UFMG) for a helpful conversation about the subject of this paper.

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