A counterexample to the containment I<sup>(3)</sup> ⊂ I<sup>2</sup> over the reals

The purpose of this note is to show a new series of examples of homogeneous ideals I in K[x, y, z, w] for which the containment I (3) ⊂ I 2 fails. These ideals are supported on certain arrangements of lines in P 3 , which resemble Fermat configurations of points in P 2 , see [14]. All examples exhibiting the failure of the containment I (3) ⊆ I 2 constructed so far have been supported on points or cones over configurations of points. Apart from providing new counterexamples, these ideals seem quite interesting on their own.

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“…Taking this for granted for a while, this shows that the coefficient of the monomial y n−2 w n−2 in h 4,4 , and hence also in h 4,4 is −1.…”

Section: The Non-containment Resultsmentioning

confidence: 93%

Fermat-type configurations of lines in P3 and the containment problem

Malara

Szpond

2018

Journal of Pure and Applied Algebra

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“…Over C, one can also take Z to be the points of the Fermat for any n ≥ 3 [36], or the Klein or Wiman [8,49] (again see Remark 1.1.4). Additional failures of containment for N = r = 2 are given in [17,23]. A recent paper [1] leverages these examples, by obtaining others by pulling them back by a finite cover of P 2 .…”

Section: 3mentioning

confidence: 99%

“…Counterexamples also occur over the reals [17] and one of them can be made to work over the rationals [23]. This one is displayed in Figure 10.…”

Section: 3mentioning

confidence: 99%

Asymptotics of linear systems, with connections to line arrangements

Harbourne

2018

Banach Center Publ.

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The main focus of these notes is recent work on linear systems in which line arrangements play a role, including problems such as semi-effectivity, containment problems of symbolic powers of homogeneous ideals in their powers, bounded negativity, and a new perspective on the SHGH Conjecture. Along the way we will be concerned with asymptotic invariants such as Waldschmidt constants, resurgences and H-constants. Acknowledgements: My participation at the Spring, 2016 miniPAGES was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. Some of the revisions to these notes were done at the Banach Center in Warsaw in the weeks after this material was presented at the workshop at the Pedagogical University in Krakow. I also thank J. Szpond for her helpful comments, and the referee for generously working though these notes and providing copious, helpful feedback.

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“…It turned out that in general (1) does not hold and most counter-examples are based on radical ideals of points which are given by intersection points of line arrangements. In particular, in [7] the very first counter-example to (1) in the real projective plane is provided and it is given by the radical ideal of triple points of Böröczky's arrangement of 12 lines. In order to understand better the containment problem, one can formulate the following question.…”

Section: Problems and Applicationsmentioning

confidence: 99%

On the Sylvester–Gallai and the orchard problem for pseudoline arrangements

Bokowski

Pokora

2017

Period Math Hung

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We study a non-trivial extreme case of the orchard problem for 12 pseudolines and we provide a complete classification of pseudoline arrangements having 19 triple points and 9 double points. We have also classified those that can be realized with straight lines. They include new examples different from the known example of Böröczky. Since Melchior's inequality also holds for arrangements of pseudolines, we are able to deduce that some combinatorial point-line configurations cannot be realized using pseudolines. In particular, this gives a negative answer to one of Grünbaum's problems. We formulate some open problems which involve our new examples of line arrangements.Keywords line arrangements, pseudoline arrangements, orchard problem, Sylvester's problem1 The Sylvester-Gallai problemWe begin with a few definitions. A line arrangement in the real projective plane P 2 R is a finite set of lines in P 2 R . A pseudoline in P 2 R is a simple closed curve such that its removal does not cut P 2 R in two connected components. A pseudoline arrangement is a set of pseudolines in P 2 R such that every pair of pseudolines has precisely one point in common where the two curves intersect each other. A first book about pseudoline arrangements was written by Grünbaum [13]. It has turned out later that pseudoline arrangements are isomorphic to reorientation classes of oriented matroids in rank 3. This implies a close connection of our investigation to the theory of oriented matroids. The interested reader can find more about this relation in [2] and [3]. For a given line arrangement, or for a given pseudoline arrangement, we count the number of points that are incident with precisely k lines or k pseudolines, respectively, with k 2, and we denote this number of the arrangement by t k . We call a point incident with precisely r lines or with precisely r pseudolines an r-point. We use also the notion double point for r = 2, triple point for r = 3, and quadruple point for r = 4. We speak of an essential line (pseudoline) arrangement when all lines (pseudolines) do not intersect at one point. Our article can also be seen in the spirit of Grünbaum's book about point-line configurations, see [14]. The reader will get some benefit for understanding our paper when she/he has a look at this book. For a point-line configuration we have not only a set of lines but also a set of points together with an incidence relation between the set of points and the set of lines. It is clear that the lines can be replaced with pseudolines and we arrive at a point-pseudoline arrangement. Moreover, when we forget about any underlying geometric set of points, lines, or pseudolines, we arrive at an abstract point-line configuration. Point-line configurations with n lines and n points in which (•) the lines are incident with precisely k points and (••) the points are incident with precisely k lines have been

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A counterexample to the containment I⁽³⁾ ⊂ I² over the reals

Abstract: 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.

Cited by 29 publications

References 13 publications

Fermat-type configurations of lines in P3 and the containment problem

Fermat-type configurations of lines in P3 and the containment problem

Asymptotics of linear systems, with connections to line arrangements

On the Sylvester–Gallai and the orchard problem for pseudoline arrangements

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A counterexample to the containment I(3) ⊂ I2 over the reals

Abstract: 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.

Cited by 29 publications

References 13 publications

Fermat-type configurations of lines in P3 and the containment problem

Fermat-type configurations of lines in P3 and the containment problem

Asymptotics of linear systems, with connections to line arrangements

On the Sylvester–Gallai and the orchard problem for pseudoline arrangements

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A counterexample to the containment I⁽³⁾ ⊂ I² over the reals