A COUNTEREXAMPLE TO THE CONTAINMENT I (3) ⊂ I 2 OVER
THE REALS
ADAM CZAPLIŃSKI, AGATA GŁÓWKA, GRZEGORZ MALARA, MAGDALENA
LAMPA–BACZYŃSKA, PATRYCJA ŁUSZCZ–ŚWIDECKA, PIOTR POKORA,

arXiv:1310.0904v2 [math.AG] 4 May 2014

AND JUSTYNA SZPOND
Abstract. The purpose of this note is to give deﬁned over the real numbers
counterexamples to a question relevant in the commutative algebra, concerning
a containment relation between algebraic and symbolic powers of homogeneous
ideal.

1. The main result
In algebraic geometry and in commutative algebra there has been recently a lot
of interest in comparing usual (algebraic) and symbolic powers of homogeneous
ideals, see for example [2], [10], [3]. If I ⊂ K[x0 , x1 , . . . , xn ] is a homogeneous ideal,
then its algebraic r–th power I r is deﬁned as the ideal generated by r–th powers
f r of all elements f in I. This is a purely algebraic concept. On the other hand,
homogeneous ideals are deﬁned geometrically as sets of all polynomials vanishing
along a given set, a subvariety V ⊂ Pn (K). For example, the ideal determined by
simultaneous vanishing in points P = (1 : 0 : 0), Q = (0 : 1 : 0) and R = (0 : 0 : 1) in
the projective plane P2 (K) is generated by the monomials yz, xz and xy. Studying
the geometry of algebraic varieties one is often interested in polynomials vanishing
to certain order m along the subvariety. For example the monomial xy vanishes to
order 2 in point R and only to order 1 in points P and Q, whereas the monomial
xyz vanishes to order 2 in all these points. All polynomials vanishing to order m
along the subvariety deﬁned by an ideal I form again an ideal. This ideal is denoted
by I (m) and is called the m–th symbolic power of I. This is a geometric concept.
It follows from the deﬁnition that I m ⊂ I (m) holds for all homogeneous ideals. The
reverse inclusion however might fail already in the simplest situations. For example
this happens for the ideal I of the three points P, Q and R deﬁned above. Indeed,
the least degree of a polynomial in I 2 is 4 (because generators of I have degree 2),
whereas xyz is a polynomial of degree 3 contained in I (2) . Hence there is surely no
containment I (2) ⊂ I 2 in this case.
It came as a big surprise that nevertheless there is also a uniform containment
relation in the reverse direction, when one takes into account the dimension of the
ambient space (or equivalently the number of variables in the polynomial ring).
More precisely, it has been discovered independently by Ein, Lazarsfeld and Smith
1991 Mathematics Subject Classification. 14C20; 52C30; 13A15; 52A20.
Keywords: symbolic powers, fat points, conﬁgurations.
1

[7] in characteristic 0 and Hochster and Huneke [12] in ﬁnite characteristic, that
there is always the containment
I (m) ⊂ I r ,
provided m ≥ n · r, where n is the dimension of the ambient space. Whereas the
lower bound n · r cannot be improved in general, it has been expected that it can
be improved under certain assumptions on varieties deﬁned by I. In particular, for
points in the projective plane P2 (K) Huneke asked if there is always the containment
(1)

I (3) ⊂ I 2 .

It is the ﬁrst instance of a more general statement predicting for ideals deﬁned by
vanishing along points the containment
(2)

I (nr−(n−1)) ⊂ I r

for all r ≥ 1.
A ﬁrst counterexample to the containment in (1) was announced in 2013 by
Dumnicki, Szemberg and Tutaj-Gasińska in [6]. In that counterexample the set
of relevant points is taken as all the intersection points of 9 lines whose union is
deﬁned by the polynomial equation
(x3 − y 3 )(y 3 − z 3 )(z 3 − x3 ) = 0.
These lines form the so called dual Hesse configuration, see [1] for a beautiful account on that, interesting in its own right, subject. The 9 lines in this conﬁguration
intersect by 3 in 12 points. Since there are no points where only two of these lines intersect, such a conﬁguration cannot be realized over the reals as it would contradict
the following well-know theorem of Sylvester [13] and Gallai.
Theorem 1 (Sylvester–Gallai). Given a finite set of points in the real plane, either
all points are collinear or there exists a pair of points not collinear with any other
point in the set.
The contradiction arises if one passes to the dual statement.
Theorem 2 (dual Sylvester–Gallai Theorem). Given a finite number of lines in the
real plane, either all these lines belong to the same pencil of lines passing through
a fixed point, or there exists a point in the plane, where only two of these lines
intersect.
Shortly after the counterexample in [6] was announced, Harbourne and Seceleanu
[11] constructed a series of counterexamples to the containment in (2) for various
values of n and r. Their counterexamples however are deﬁned over ﬁnite ﬁelds.
Since conﬁgurations of real lines are subject to stronger combinatorial constrains
than sets of lines either in the complex plane or in planes deﬁned over ﬁnite ﬁelds,
it is not immediately clear that counterexamples to (1) can be constructed in the
real plane. This issue has motivated our research. The main result we want to
announce is the following.

Theorem 3. There exists a series of counterexamples to the containment in (1)
defined in the real plane.
In fact our counterexamples are well known in the combinatorics. They were
introduced by Füredi and Palásti in [8] in connection with the following SylvesterGallai problem motivated by Theorem 2.
Problem 4. Given a conﬁguration of s lines in the real (projective) plane, not all
belonging to the same pencil of lines through a ﬁxed point, what is the minimal
number of points where only two lines meet?
It has been long conjectured, see for example [4], that there are at most Rs =

1+⌊ s(s−3)
⌋ points where three lines meet, so that there are at least 2s −3−3⌊ s(s−3)
⌋
6
6
points where the lines meet in pairs only. This conjecture, at least for large s, was
recently proved by Green and Tao [9]. A series of examples with number of triple
points equal or close to Rs was constructed in [8].
In the recent preprint [14] Terence Tao pointed out that many combinatorial
problems have been solved, or substantial progress has been obtained, when applying methods from algebra or algebraic geometry. In this note the directions are
reversed, we present a solution to an algebraic problem based on a combinatorial
construction.
2. Proof of the main result
In order to prove Theorem 3, we are interested in conﬁgurations of lines with a
high number of points where three of them meet. The motivation for this approach
is the following. Taking V as the set of all triple points in the conﬁguration and
denoting by I = I(V ) the ideal of V , it is immediate from the description of
symbolic powers given above, that the product
F = L1 · . . . · Ls
of equations of the lines in the conﬁguration is contained in I (3) .
If the number of triple points is high when related to the number of lines, then
F will be the only element of degree s in I (3) . On the other hand, the generators
of I should be of relatively high degree. In the most optimistic situation, the least
degree d of a generator of I would satisfy 2d > s. In that case, it would be clear
that F cannot be contained in I 2 . This scenario never happens in the reality, so
that the actual argument requires somewhat more eﬀort.
We recall now how the examples of Füredi and Palásti work. To this end it
is convenient to identify the real plane R2 with the set of complex numbers C in
the usual way. Let n be an even and positive integer. Let ξ = exp(2πi/n) be a
primitive n–th root of unity and let Pi = ξ i for i = 0, . . . , n − 1. Denote by Lm,k
the (real) line passing through the points Pm and Pk if m 6= k and the tangent line
to the unit circle at the point Pm if m = k.
Then we construct the conﬁguration of points and lines in two steps. First we
deﬁne the conﬁguration of lines
n−1
,
Ln = {Li, n2 −2i }i=0

where the indices are understood mod n. We will denote the linear form deﬁning
Qn−1
the line Li, n2 −2i by Li for i = 0, . . . , n − 1. Let Fn = i=0
Li . Then deg(Fn ) = n.
Let Zn be the set of all triple points in the conﬁguration Ln .
Note that the lines Li , Lj , Lk are concurrent if and only if n divides i + j + k, see
[8, Lemma]. It follows from [8, Property 4] that there are exactly 1 + ⌊n(n − 3)/6⌋
points in the set Zn .
By the way of an example we focus now on the case n = 12. This is the minimal
number of lines, which leads to a conﬁguration of points giving a counterexample
to the containment in (1).
The conﬁguration of lines arising in this case is illustrated on the following picture
prepared with the aid of Geogebra.

P2

P6

P10

The points in Z12 are marked by dots. There are 19 of them. There are 3 lines,
each containing 4 conﬁguration points, these lines are the tangents to the unit circle
at points P2 , P6 and P10 . On each of the remaining 9 conﬁguration lines there are
exactly 5 conﬁguration points.
Thus the conﬁguration is in fact the union of two conﬁgurations. One depicted in
the picture below using solid lines is a real analogue of the dual Hesse conﬁguration
mentioned before.

In the real case there are only 10 triple points. The remaining 6 double points
(distinguished in the picture by larger dots) determine in pairs, 3 additional lines
(dotted lines in the picture), which intersect in pairs on the lines of the solid conﬁguration. These 3 lines form the second conﬁguration, and altogether there are
12 lines and 19 triple points.
In order to determine generators of the ideal J = I(Z12 ), we can intersect the
ideals of single points in the set Z12 , i.e.
J = I(Q1 ) ∩ . . . ∩ I(Q19 ).
Note that the ideals I(Qj ) are generated by any two lines passing through the point
Qj , so they can be easily written explicitly down. In fact, these calculations can
be done by hand, but this is a tedious procedure. Instead, we prefer to revoke a
Singular [5] script in this place.
In fact, our computational strategy in the script attached at the end of this note
is slightly diﬀerent (and more eﬃcient). We compute ﬁrst the points P0 , . . . , P11
on the unit circle, the vertices of the regular 12–gon with P0 = (1, 0) (in the script
we use projective coordinates, so that P0 = (1 : 0 : 1)). Then we compute the
equations of lines L0 , . . . , L11 . In the next step, we don’t compute the triple points
in the conﬁguration L12 one by one. Instead, we compute the ideal J of all of
them simultaneously taking into account that they are the worst singularities of
the conﬁguration (in algebraic geometry a triple point is considered to be a worse
singularity than a double point). Finally, we check the containment of F12 (which
is called F in the script) in J 2 . Again, this step could be carried out by hand in the

theory, but this is a lengthy and dull computation, which can be safely relegated
to a machine.
Checking any given conﬁguration with an even number n ≥ 12 goes along the
same lines.
This ﬁnishes the proof of Theorem 3.
Remark 5. There is a similar construction for an odd number of lines n in [8]. We
have checked also these examples and found out that they give counterexamples to
the containment in (1) for all n ≥ 13. Thus 12 is the minimal number of real lines
such that triple points of their conﬁguration provide a counterexample to (1).
Remark 6. All counterexamples to the containment (2) found up to now are based
on conﬁgurations of points coming from some extremal, from the combinatorial
point of view, conﬁgurations of lines. It would be very interesting to understand if
there are some deeper connections justifying these phenomena lurking behind the
scenes.
3. A Singular script
LIB "elim.lib";
ring R=(0,a),(x,y,z),dp; option(redSB);
minpoly=a2-3;
proc gline(ideal I1, ideal I2) {
ideal I=intersect(I1,I2);
I=std(I);
return(I[1]);
}
ideal P0=x-z,y; ideal P1=x-a/2*z,y-1/2*z; ideal P2=x-1/2*z,y-a/2*z;
ideal P3=x,y-z; ideal P4=x+1/2*z,y-a/2*z; ideal P5=x+a/2*z,y-1/2*z;
ideal P6=x+z,y; ideal P7=x+a/2*z,y+1/2*z; ideal P8=x+1/2z,y+a/2*z;
ideal P9=x,y+z; ideal P10=x-1/2*z,y+a/2*z; ideal P11=x-a/2*z,y+1/2*z;
poly
poly
poly
poly

L0=gline(P0,P6); poly L1=gline(P1,P4); poly L2=1/2*x+a/2*y-z;
L3=gline(P3,P0); poly L4=gline(P4,P10); poly L5=gline(P5,P8);
L6=-x-z; poly L7=gline(P7,P4); poly L8=gline(P8,P2);
L9=gline(P9,P0); poly L10=1/2*x-a/2*y-z; poly L11=gline(P11,P8);

poly F=L0*L1*L2*L3*L4*L5*L6*L7*L8*L9*L10*L11;
ideal I=F,diff(F,x),diff(F,y),diff(F,z);
ideal J=I,diff(I,x),diff(I,y),diff(I,z);
ideal M=x,y,z; J=std(J);
J=sat(J,M)[1]; J=std(J);
reduce(F,std(J^2));
quit;
Acknowledgement. This note resulted from the second workshop on linear series held
in Lanckorona in September 2013. We thank Pedagogical University of Cracow for

ﬁnancial support. Last but not least we thank Marcin Dumnicki, Tomasz Szemberg
and Halszka Tutaj-Gasińska for valuable comments.
References
[1] M. Artebani, I. Dolgachev, The Hesse pencil of plane cubic curves, L’Enseignement
Mathématique. Revue Internationale. 2e Série 55 (2009), 235–273.
[2] C. Bocci, B. Harbourne, Comparing Powers and Symbolic Powers of Ideals, J. Algebraic
Geometry 19 (2010), 399–417.
[3] C. Bocci, S. M. Cooper, B. Harbourne, Containment results for ideals of various conﬁgurations
of points in PN . J. Pure Appl. Algebra 218 (2014), 65–75
[4] S. A. Burr, B. Grünbaum, N. J. A. Sloane, The orchard problem, Geometriae Dedicata 2
(1974), 397-424.
[5] W. Decker, G.-M. Greuel, G. Pﬁster, H. Schönemann, Singular 3-1-3 A computer algebra
system for polynomial computations. http://www.singular.uni-kl.de (2011).
[6] M. Dumnicki, T. Szemberg, H. Tutaj-Gasińska, A counter-example to a question by Huneke
and Harbourne, J. Algebra 393 (2013), 24–29.
[7] L. Ein, R. Lazarsfeld, K.E. Smith, Uniform bounds and symbolic powers on smooth varieties.
Invent. Math. 144 (2001), 241–252.
[8] Z. Füredi, I. Palasti, Arrangments of lines with a large number of triangles, Proc. Amer.
Math. Soc. 92 (1984), 561–566.
[9] B. Green, T. Tao, On sets deﬁning few ordinary lines, Discrete Comput. Geom. 50 (2013),
409–468.
[10] B. Harbourne, C. Huneke, Are symbolic powers highly evolved ?, J. Ramanujan Math. Soc.
28 (2013), 311–330.
[11] B. Harbourne, A. Seceleanu, Containment counterexamples for ideals of various conﬁgurations
of points in Pn , arXiv:1306.3668.
[12] M. Hochster, C. Huneke, Comparison of symbolic and ordinary powers of ideals. Invent.
Math. 147 (2002), 349-369.
[13] J. J. Sylvester, Problem 11851, Math. Questions from the Educational Times 59 (1893),
98–99.
[14] T. Tao, Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory, arXiv:1310.6482

Agata Główka–Habura, Grzegorz Malara, Magdalena Lampa–Baczyńska, Patrycja Łuszcz–
Świdecka, Piotr Pokora, Justyna Szpond, Pedagogical University of Cracow, Institute of
Mathematics, Podchora̧żych 2, PL-30-084 Kraków, Poland
Adam Czapliński, Johannes Gutenberg-Universität Mainz, Institute of Mathematics,
Staudingerweg 9, 55099 Mainz

