Line arrangements with the maximal number of triple points
Marcin Dumnicki, Lucja Farnik, Agata Glowka, Magdalena Lampa-Baczynska, Grzegorz Malara, Tomasz Szemberg, Justyna Szpond, Halszka Tutaj-Gasinska
aa r X i v : . [ m a t h . C O ] J un Line arrangements with the maximal number of triplepoints
M. Dumnicki, L. Farnik, A. G l´owka, M. Lampa-Baczy´nska, G. Malara,T. Szemberg ∗ , J. Szpond, H. Tutaj-Gasi´nskaJune 6, 2018 Abstract
The purpose of this note is to study configurations of lines in projectiveplanes over arbitrary fields having the maximal number of intersection pointswhere three lines meet. We give precise conditions on ground fields F over whichsuch extremal configurations exist. We show that there does not exist a fieldadmitting a configuration of 11 lines with 17 triple points, even though such aconfiguration is allowed combinatorially. Finally, we present an infinite seriesof configurations which have a high number of triple intersection points. Keywords arrangements of lines, combinatorial arrangements, Sylvester-Gallai problem
Mathematics Subject Classification (2000)
Configurations of points and lines have been the classical object of study in geometry.They come up constantly in various branches of contemporary mathematics, amongothers serving as a rich source of interesting examples and counter-examples. By wayof example, in algebraic geometry, arrangements of lines have been studied recentlyby Teitler in [21] in the context of multiplier ideals, in [10] as counter-examplesto the containment problem for symbolic powers of ideals of points in the complexprojective plane and in [5] in the setup of the Bounded Negativity Conjecture andHarbourne constants.In combinatorics point line arrangements are subject of classical interest andcurrent research. Notably, in the last year we have witnessed a spectacular proof ofa long standing conjecture motivated by the Sylvester-Gallai theorem on the numberof ordinary lines. It has been proved by Green and Tao [13] with methods closelyrelated to the real algebraic geometry. Symmetric ( n k ) configurations (mostly inthe real Euclidean plane) are another classical topic of study in combinatorics. Suchconfigurations with triple points (i.e. k = 3) are well understood, see the beautifulmonograph by Gr¨unbaum [14]. The classification of ( n ) configurations has wit-nessed much progress in recent years and is almost completed, mainly due to worksof Bokowski and his coauthors, see e.g. [7], [6] for an up to date account on thatpath of research. In the present note we take up a slightly different point of view and ∗ TS was partially supported by NCN grant UMO-2011/01/B/ST1/04875 investigate what the maximal number of triple points in a configuration of s linesover an arbitrary ground field is. In Theorem 2.1, which is our main result, we givea full classification for s
11. The interest in this particular bound is explained bythe observation that it is the first value of s for which a configuration with maximalcombinatorially possible number of triple points cannot be realized over any field.To the best of our knowledge this is the first example of this kind. More precisely,non-realizable configurations were known previously, probably the first example wasfound by Lauffer [17], see also [20] and [12]. However, the known examples concernconfigurations whose numerical invariants (for example Lauffer’s example is numeri-cally the Desargues (10 ) configuration) allow several combinatorial realizations and some of them are not realizable over any field. In the case studied here, there aretwo combinatorial realizations possible but none of them has a geometrical realiza-tion. It is then natural to ask to what extend the combinatorial upper bound onthe number of triple points found by Sch¨onhein [19] (see also equation (3) below)can be improved in general. Our result is rendered by a series of configurations witha high number of triple points presented in Section 3. This series of examples setssome limits to possible improvements in (3).Given a positive integer s and a projective plane over a field with sufficientlymany elements, it is easy to find s lines intersecting in exactly (cid:0) s (cid:1) distinct points.In fact this is the number of intersection points of a general arrangement of s lines(such arrangements in algebraic geometry are called star configurations , see [11])and this is also the maximal possible number of points in which at least two out ofgiven s lines intersect.Given a configuration of s mutually distinct lines, let t k denote the number ofpoints where exactly k > (cid:18) s (cid:19) = X k > t k (cid:18) k (cid:19) . (1)Let T k ( s ) denote the maximal number of k -fold points in an arrangement of s distinctlines in the projective plane over an arbitrary field F . The discussion above showsthat T ( s ) = (cid:18) s (cid:19) . The aim of this note is to investigate the numbers T ( s ). Deciding the existence or non-existence of a configuration with certain propertiesis a problem which can, in principle, be always solved by combinatorial and semi-algebraic methods. The combinatorial part evaluates the collinearity conditions andchecks whether the resulting incidence table can be filled in or not. The restrictionsare imposed by the number of lines intersecting in configuration points and thecondition that two lines cannot intersect in more than one point. Table 2 on page 6is an example of what we call an incidence table.If a configuration is combinatorially possible, then we assign coordinates to theequations of configuration lines and check if the system of polynomial equationsresulting from evaluating combinatorial data has solutions. Typically this is thecase and this is where the semi-algebraic part comes into the picture, as one hasto exclude various degenerations, for example points or lines falling together. Thiskind of conditions is given by inequalities .This note, in a sense, is a field case study of the effective applicability of theapproach described above. Evaluating all conditions carefully, one can actuallystudy moduli spaces of configurations in the spirit of [2] and [1]. However, since weare interested in the existence of configurations over arbitrary fields, we do not dwellon this aspect of the story.Some of the computations were supported by the symbolic algebra programSingular [9].The equality (1) yields the following upper bound on the number t of triplepoints in an arrangement of s lines: t $ (cid:0) s (cid:1) % . (2)This naive bound has been improved by Kirkman in 1847, [15], with a correctionof Sch¨onheim in 1966, [19]. Theorem 3.1 shows that this bound is close to beattained, on the other hand Theorem 2.1 shows that there is place for some furtherimprovements. Let U ( s ) := (cid:22)(cid:22) s − (cid:23) · s (cid:23) − ε ( s ) , (3)where ε ( s ) = 1 if s ≡ ε ( s ) = 0 otherwise. Then T ( s ) U ( s ) . (4)We refer to Section 5 in [8] for a nice discussion of historical backgrounds. In thenext table we present a few first numbers resulting from (4) s U ( s ) 0 0 1 1 2 4 7 8 12 13 17 20It is natural to ask to which extend the numbers appearing in the above table aresharp. Our main result is the following classification Theorem. Theorem 2.1. a) For s , there are configurations of lines with U ( s ) triple points in projective planes P ( F ) over arbitrary fields.b) A configuration of lines with triple points exists only in characteristic (the smallest such configuration is the Fano plane P ( F ) ).c) A configuration of lines with triple points exists over any field containing anon-trivial third degree root of . Moreover such a configuration always arisesby taking out one line from a configuration of lines with triple points.d) A configuration of lines with triple points exists over any field containinga non-trivial third degree root of .e) A configuration of lines with triple points exists onlye1) over a field F of characteristic containing a non-trivial third root ofunity. In this case one of the points has in fact multiplicity ;e2) over any field F of characteristic . f ) There is no configuration of lines with triple points. There exist config-urations of lines with triple points.Proof. The first part of the Theorem is well known for s
9. We go briefly throughall the cases for the sake of the completeness and discuss s = 10 in more detail asthis configuration seems to be new.For s = 1 , s = 3 , s = 5 is easy as well, see Figure 1. A C E F
Figure 1 : s = 5 A C E F
Figure 2 : s = 6We pass to the case s = 6 adding the line through both double points, seeFigure 2.For s = 7 we obtain the famous Fano plane P ( F ). It is well known thatthis configuration is possible only in characteristic 2. The picture below (Figure 3)indicates collinear points as lying on the segments or on the circle.Figure 3 : s = 7 Figure 4 : s = 8For s = 8 there is the M¨obius-Kantor (8 ) configuration. This configuration can-not be drawn in the real plane. Collinearity is indicated by segments and the circlearch, see Figure 4. This configuration can be obtained from the next configurationby removing one line.For s = 9 there is the dual Hesse configuration. It is easier to describe theoriginal Hesse configuration. It arises taking the nine order 3 torsion points of asmooth complex cubic curve (which carries the structure of an abelian group andthe torsion is understood with respect to this group structure). There are 12 linespassing through the nine points in such a way that each line contains exactly 3torsion points and there are 4 lines passing through each of the points. See [4] fordetails. This configuration cannot be drawn in the real plane.Beside the geometrical realization over the complex numbers, the dual Hesse config-uration can be also easily obtained in characteristic 3, more precisely, in the plane P ( F ) taking all 13 lines and removing from this set all 4 lines passing through afixed point. We leave the details to the reader.Before moving on, we record for further reference the following simple but usefulfact. Lemma 2.2.
Let L = { L , . . . , L s } be a configuration of lines. Let L ∈ L be a fixedline. Let P ( L ) , . . . , P r ( L ) be intersection points of L with other configuration lineswith corresponding multiplicities m ( L ) , . . . , m r ( L ) . Then s − r X i =1 ( m i − . In particular, if there are only triple points on a line L , then s is an odd number. lines Now we come to the case s = 10. We work over an arbitrary field F . The upperbound (4) implies that in this case there can be at most 13 points of multiplicity atleast 3. We first deal with the case when points of higher multiplicity might appear. The combinatorial equality (1) implies that there are no points with multiplicity m > t = 2, t = 11, t = 0.(ii) t = 1, t = 12, t = 3.Case (i) is excluded since there exists a configuration line L which does not passthrough any of the 4-fold points. Hence this line contains only 3-fold points. Since s = 10, this contradicts Lemma 2.2.Passing to (ii) we start with the 4-fold point W . We denote the lines passingthrough W by M , . . . , M as indicated in the picture below. W P M P M P M P M P P P P P P P P Figure 5The remaining 6 lines L , . . . , L (not visible in the figure above) intersect pair-wise in 15 mutually distinct points — 12 of these points are the 12 configurationtriple points and the remaining 3 points (not visible in the figure above) contributeto t . Note that 6 general lines intersect in 15 mutually distinct points, but in oursituation there are additional collinearities which are reflected in the picture aboveand in the table below. Passing to the details let P ij = L i ∩ L j for 1 i < j
6. Upto renumbering of points we may assume that they are distributed in the followingway: line points on the line M P , P , P M P , P , P M P , P , P M P , P , P Table 1Moreover, we may assume that the lines L , . . . , L have the following equations(we omit “= 0” in the equations) L : x, L : y, L : z,L : x + y + z, L : ax + by + z, L : cx + dy + z, (5)with a, b, c, d ∈ F ∗ and det a b c d = 0 . (6)Indeed, the first four equations are obvious. The coefficients at z in the lines L and L can be normalized to 1 since otherwise the star configuration condition would fail.Similarly the conditions in (6) are necessary in order to guarantee that L , . . . , L form a star configuration.Evaluating collinearity conditions in Table 1 above we obtain the following sys-tem of linear and quadratic equations: a − b − c + d = 0 − ad + a − c + d = 0 a − bc = 0 bc − d = 0 (7)Additionally, the condition that the lines M , . . . , M belong to the same pencilgives − ab + a + bc − . (8)A solution to the above system of equations (7) and (8) satisfying additionallythe non-equality condition (6) exists only in characteristic 2. This has been verifiedwith the aid of Singular. Moreover, in that case a satisfies a + a + 1 = 0 . and then, consequently, b = a , c = a and d = a .It follows that the configuration (ii) exists in P ( F q ), for all q > q = 1 is excluded as there are evidently not enough points in the Fano plane. Theconfiguration lines are then given by equations L : x, L : y, L : z,L : x + y + z, L : ax + a y + z, L : a x + ay + z,M : x + y, M : ax + z, M : a x + y + z, M : x + a y + z Then the configuration points have coordinates: W = (1 : 1 : a ) , P = (0 : 0 : 1) , P = (0 : 1 : 0) , P = (0 : 1 : 1) ,P = (0 : 1 : a ) , P = (1 : 0 : 1) , P = (1 : 0 : a ) , P = (1 : 0 : a ) ,P = (1 : 1 : 0) , P = ( a : 1 : 0) , P = (1 : a : 0) , P = ( a : a : 1) ,P = (1 : 1 : 1) . The incidence table in this case reads:
W P P P P P P P P P P P P L + + + + L + + + + L + + + + L + + + + L + + + + L + + + + M + + + + M + + + + M + + + + M + + + +Table 2 t = 13 andconsequently t = 6. There is an odd number of 2-fold points on each configurationline. This implies that there is a configuration line M containing exactly three2-fold points D , D , D . We have again two cases:(A) The lines M , M , M passing through the points D , D , D meet in a singlepoint W . D D M D WM M M Figure 6 : Case (A)(B) The lines M , M , M form a triangle with vertices Z , Z , Z . M M M M Z Z Z Figure 7 : Case (B)The first case is impossible. This follows similarly to the case (e1) with a 4-fold point. Indeed, the six lines L , . . . , L not visible in the Figure 6 form a starconfiguration, i.e. there are 15 mutually distinct intersection points P ij = L i ∩ L j .Up to renumbering incidences between the lines M i and the points P jk are as in theTable 3.Moreover the condition that M , M and M meet at one point W is the sameas in equation (8). This gives the same solution as in the previous case (e1). It iseasy to check that this implies that the line M goes through W , a contradiction.Now we consider the remaining case (B). There are three 3-fold points on theline M . The points Z , Z , Z must be also 3-fold points of the configurationand on each of the lines M , M , M there are two more 3-fold points. This givesaltogether twelve 3-fold points. Hence the six remaining lines L , . . . , L have a 3-fold intersection point. We call this point D , and assume that L , L and L passthrough D . We can assume that D = (1 : 1 : 1) and then the equations of the lines L i are L : x, L : y, L : z,L : ax − ( a + 1) y + z, L : bx − ( b + 1) y + z, L : cx − ( c + 1) y + z, (9)with a , b , c mutually distinct and different from zero. The combinatorics impliesthat each of Z , Z , Z lie on one of the lines L , L , L . Up to renumbering wecan assume Z ∈ L , Z ∈ L , Z ∈ L . Then the incidence table is determined asfollows: line points on the line M P , P , P M Z , Z , P , P M Z , Z , P , P M Z , Z , P , P Table 3See the text after Table 5 for hints how to fill in such a table.Using equations as in equation (9) and evaluating incidences we obtain the fol-lowing conditions ab + ac + a − bc = 0 ac + a − b + c = 0 ab + c = 0 a + bc = 0 (10)This implies that b = 1. If b = − a = −
1, a contradiction. If b = 1 then a = 3 and a + 1 = 0. It follows that char F = 2 or char F = 5. In the first case a = c , a contradiction. In the second case we obtain a = 3, b = 1, c = 2, thus thelines are L : x, L : y, L : z,L : 3 x + y + z, L : x + 3 y + z, L : 2 x + 2 y + z,M : x + y + z, M : 2 x + 4 y, M : 3 y + z, M : 2 x + z The points have coordinates D = (1 : 1 : 1) , Z = (2 : 3 : 1) , Z = (4 : 3 : 2) , Z = (4 : 3 : 1) ,P = (0 : 0 : 1) , P = (0 : 1 : 0) , P = (0 : 4 : 1) , P = (0 : 4 : 3) ,P = (1 : 0 : 0) , P = (1 : 0 : 4) , P = (1 : 0 : 3) , P = (4 : 3 : 0) ,P = (3 : 2 : 0) . The incidence table is
D Z Z Z P P P P P P P P P L + + + + L + + + + L + + + + L + + + + L + + + + L + + + + M + + + M + + + + M + + + + M + + + +Table 4Passing to the last assertion f) of Theorem 2.1 we assume that a configurationof 11 lines with 17 triple points exists. Then (1) implies that there are 4 doublepoints in the configuration. Hence each line meets 10 other lines, there is an evennumber of double points on each line. If there are 4 double points on a line, thenthis condition fails on the 4 lines meeting the given one in the double points. Hence,there must be 2 pairs of lines in the configuration with double points situated in theintersection points of lines from different pairs as indicated in the figure below0 N W M M W N Now, there are two cases(I) the line W W belongs to the configuration,(II) the line W W is not a configuration line.We begin with the case (I) N W M M W N L Figure 8 : Case (I)Let L be the line W W . In the figure above there are 5 configuration lines.The remaining lines L , . . . , L must form a star configuration and their intersectionpoints P ij = L i ∩ L j have to distribute in five collinear triples lying on the lines L , M , M , N and N . Up to renumbering the points the collinear triples are P , P , P P , P , P P , P , P P , P , P P , P , P (11)Indeed, the first column contains the points lying on the line L . The first row isthen completed just by assigning numbers. The index 2 must appear somewhere inthe second row. Since the pairs 3 , , P . These labeling determines the rest of the table.Without loss of generality as in (5) and (6) we may assume that L : x, L : y, L : z,L : x + y + z, L : ax + by + z, L : cx + dy + z, a, b, c, d ∈ F ∗ and det a b c d = 0 . We can now compute coordinates of all points P ij for 1 i < j a − b − c + d = 0 − ad + a − c + d = 0 a − bc = 0 bc − d = 0 ad − a + b − d = 0 (12)This system has a solution satisfying conditions (6) only if the ground field F hascharacteristic 2. In that case we have a = d = ε, b = c = ε , with ε a solution of the equation x + x + 1 = 0, i.e. a primitive root of unity oforder 3. This implies that the equations of the lines L , N , N , M , M are (up toordering) x + y, εx + z, ε x + y + z, x + ε y + z, εy + z. These lines belong all to the pencil of lines passing through the point (1 : 1 : ε ), acontradiction.Now we pass to the second case (II). In this situation we start with the followingfigure N W M M W N Z M Z N Z Z Z Figure 9 : Case (II)There are now 5 remaining configuration lines, which we call as usual L , . . . , L .They form a star configuration, i.e. there are 10 mutually distinct intersection points P ij = L i ∩ L j , for 1 i < j
5. Up to renumbering these points are distributed asfollows2line points on the line M P , P M P , P N P N P , P N P , P M P Table 5The Table 5 is filled as follows. The first two rows are filled just by assigninglabels. They imply immediately that Z ∈ L and Z ∈ L . There is another triplepoint of the configuration not depicted in Figure 9. The line L cannot pass throughthis point so that it must be P (note that L and L intersect N already in Z and Z respectively). The points P and P must then lie one on N and theother on N . We have selected the labeling in such a way, that Z i ∈ L i holds for all i = 1 , . . . , L i are L : x, L : y, L : z,L : x + y + z, L : ax + by + z. Evaluating the conditions Z i ∈ L i for i = 1 , . . . , Z ∈ L , is satisfied automatically) − a + ab + ab − b = 0 a − ab + ab − a = 0 − ab + ab − a + b = 0 a − ab − a + b = 0 (13)This system is equivalent to the system a − b ) = 0 ab − b + a = 0 a − ab + b − a = 0 (14)The latter system has to be treated differently in case of characteristic of F equaleither 2 or 3 but all cases lead to the same solution a = b = 1, we omit the details.This solution means L = L , a contradiction.We conclude the proof of Theorem 2.1 with an example of a configuration of11 lines with 16 triple points. Our example is constructed over an arbitrary field F which contains the golden section ratio. Our example is dual to the example in [8,page 398, figure (i)].Turning to details, let b ∈ F satisfy b + b − L : x, L : y, L : z, L : x + y + z,L : − bx + z, L : bx + y + bz, L : y + z, L : b x + by + z,L : bx − by − z, L : − b x − y − bz, L : − b x + (1 − b ) y. P = (0 : − , P = (1 : 0 : 0) , P = (0 : 1 : 0) ,P = (1 : 0 : − , P = ( − b + 1 : − b ) , P = ( − b : 0) ,P = (1 : 0 : b ) , P = (0 : − b : 1) , P = (0 : − b ) ,P = (1 : 0 : − b ) , P = (1 − b : − b : b ) , P = ( − b : − b ) ,P = (1 : 1 : − P = (0 : 0 : 1) , P = (1 : 1 : 0) ,P = (1 : 1 : − . The incidence table in this case reads: P P P P P P P P P P P P P P P P L + + + + + L + + + + + L + + + + L + + + + L + + + + L + + + + L + + + + + L + + + + + L + + + + L + + + + L + + + +And finally, the configuration is visualized in Figure 10. In this figure the dashedcircle indicates the line at the infinity on which parallel lines intersect. So that forexample parallel lines L and L intersect in point P . The line at infinity is aconfiguration line. L L L L L L L L L L L P P P P P P P P P P P P P P P P Figure 104
In complex algebraic geometry a point where exactly two lines meet is a node . Thisis the simplest singularity one encounters and is denoted by A in the A - D - E –classification of simple singularities of curves, see for example [3]. Plane curves(not necessarily splitting in lines) with A singularities are well understood, see forexample [18]. When exactly three lines meet in a point, then there is a D singularityin that point. Apart from A , this is the only simple singularity which can appearin an arrangement of lines. Plane curves containing D singularities are way lessunderstood, see for example [16, Section 11]. Results of this note can be consideredas a step towards completing this picture.The construction we present here is directly motivated by the passage from theHesse configuration to its dual.Let E be an elliptic curve embedded as a smooth plane cubic. The group lawon E is related to the embedding by the following equivalent conditionsa) the points P , Q and R on the curve E are collinear;b) P + Q + R = 0 in the group E .Let p be a prime number >
3. There are exactly p mutually distinct solutionsto the equation pX = 0 on E . These solutions form a subgroup E ( p ) of p –torsionpoints. Since they form a subgroup and by the above equivalence, any line joiningtwo distinct points in E ( p ) intersects E in another point which is also an elementof E ( p ). The tangent line to E at 0 is tangent there to order 3, in particular theequation 2 X = 0 on E has no non-trivial solution in E ( p ). The tangent lines to E at every other point X ∈ E ( p ) \ { } intersects E ( p ) in some other point Y . This isbecause the equation 2 X + Y = 0 has a unique solution in E ( p ) for all Y = 0. Inparticular the point X also lies on a line tangent to E at some point Z ∈ E ( p ).Hence, there are altogether ( p +4)( p − lines determined by pairs of points in E ( p ). There are p − configuration lines passing through 0 and p +12 lines passingthrough every other point in E ( p ).Passing to the dual configuration, we obtain thus p lines with t ( p ) := ( p − p − triple points (and p − X ∈ E ( p ) \ { } ). The equality in (1) guarantees that there are no other intersectionpoints between the lines. Since p is a prime, there is no rounding in (3) and it cannotbe p ≡ U ( p ) − t ( p ) = p − .Hence we have proved our final result. Theorem 3.1.
For any prime number p > , there exists a configuration of p linesintersecting in ( p − p − triple points (and p − double points). Acknowledgement.
This notes originated in a workshop on Arrangements ofLines held in Lanckorona in April 2014. We thank the Jagiellonian University inCracow for financial support. We thank also Brian Harbourne and Witold Jarnickifor helpful discussions.
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[email protected] Agata G l´owka, Magdalena Lampa-Baczy´nska, Grzegorz Malara, Tomasz Szemberg, JustynaSzpond, Instytut Matematyki UP, Podchor¸a˙zych 2, PL-30-084 Krak´ow, Poland
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