Unexpected curves arising from special line arrangements

Marca, Michela Di; Malara, Grzegorz; Oneto, Alessandro

doi:10.1007/s10801-019-00871-0

Cited by 23 publications

(23 citation statements)

References 14 publications

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“…A line arrangement in is supersolvable if it has a so-called modular point , that is, a point with the property that if and if is the intersection of and then the line joining and is a line of . (See [DMO18] for examples of supersolvable line arrangements giving rise to unexpected curves.) A standard fact is that if is a supersolvable line arrangement consisting of lines, of which pass through the modular point , then is free, and the splitting type is .…”

Section: Examplesmentioning

confidence: 99%

Line arrangements and configurations of points with an unexpected geometric property

Cook¹,

Harbourne²,

Migliore

et al. 2018

Compositio Math.

133

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The SHGH conjecture proposes a solution to the question of how many conditions a general union of fat points imposes on the complete linear system of curves in P 2 of fixed degree d, and it is known to be true in many cases. We propose a new problem, namely to understand the number of conditions imposed by a general union of fat points on the incomplete linear system defined by the condition of passing through a given finite set of points Z (not general). Motivated by work of Di Gennaro-Ilardi-Vallès and Faenzi-Vallès, we give a careful analysis for the case where there is a single general fat point, which has multiplicity d − 1. There is an expected number of conditions imposed by this fat point, and we study those Z for which this expected value is not achieved. We show, for instance, that if Z is in linear general position then such unexpected curves do not exist. We give criteria for the occurrence of such unexpected curves and describe the range of values of d for which they occur. Unexpected curves have a very particular structure, which we describe, and they are often unique for a given set of points. In particular, we give a criterion for when they are irreducible, and we exhibit examples both where they are reducible and where they are irreducible. Furthermore, we relate properties of Z to properties of the arrangement of lines dual to the points of Z. In particular, we obtain a new interpretation of the splitting type of a line arrangement. Finally, we use our results to establish a Lefschetz-like criterion for Terao's conjecture on the freeness of line arrangements. 2

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Section: Examplesmentioning

confidence: 99%

Line arrangements and configurations of points with an unexpected geometric property

Cook¹,

Harbourne²,

Migliore

et al. 2018

Compositio Math.

133

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“…Recently, unexpected curves and hypersurfaces have been intensively studied. In the paper [6], Di Marca, Malara and Oneto present a way to produce families of unexpected curves using supersolvable arrangements of lines. In [2], Bauer, Malara, Szemberg and Szpond consider the existence of special linear systems in P 3 and exhibit there a quartic surface with unexpected postulation properties.…”

Section: Mathematicians Have Been Working On Interpolation Problems Fmentioning

confidence: 99%

On the unique unexpected quartic in $${\mathbb {P}}^2$$

et al. 2019

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The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in algebraic geometry. In general, it is still an open problem to understand when the points fail to impose independent conditions. Despite many partial results, a complete solution is not known, even if the fixed points are in general position. The answer in the case of general points in the projective plane is predicted by the famous Segre-Harbourne-Gimigliano-Hirschowitz conjecture. When we consider fixed points in special position, even more interesting situations may occur. Recently, Di Gennaro, Ilardi and Vallès discovered a special configuration Z of nine points with a remarkable property: A general triple point always fails to impose independent conditions on the ideal of Z in degree four. The peculiar structure and properties of this kind of unexpected curves were studied by Cook II, Harbourne, Migliore and Nagel. By using both explicit geometric constructions and more abstract algebraic arguments, we classify low-degree unexpected curves. In particular, we prove that the aforementioned configuration Z is the unique one giving rise to an unexpected quartic.

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“…, a n ] is a form (possibly a constant) describing the codimension 1 part of the locus of points B such that F (B, ·) ≡ 0. In the case of the unexpected plane quartics discussed above, we have n = 2, d = 4 and m = 3 so F has bidegree ( m+n−1 n (d−m+1), d) = (12, 4), but H = ab 3 c 3 (a+b−c)(a−b+c) (see Example 13), so in this case H has bidegree (9, 0) and G has bidegree (3,4) which gives the bidegree of F to be (12, 4), as asserted. While we do not have an explicit expression for the bidegree of H (or for G), our results show that BMSS duality holds for F (and hence for G, whatever its bidegree is).…”

Section: Introductionmentioning

confidence: 99%

“…The form G has bidegree (3,4) and defines a hypersurface in P 2 × P 2 . The fibers with respect to the first projection are quartics G(B, ·) = Q B (·) = 0 with a triple point at B, while the fibers with respect to the second projection are cubics G(·, S) = Q S (·) = 0 with a triple point at S. The connection between G(·, S) and G(B, ·) is that they have the same tangent cone at B = S.…”

Section: Introductionmentioning

confidence: 99%

A matrixwise approach to unexpected hypersurfaces

Dumnicki

Farnik

Harbourne

et al. 2020

Linear Algebra and its Applications

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The aim of this note is to give a generalization of some results concerning unexpected hypersurfaces. Unexpected hypersurfaces occur when the actual dimension of the space of forms satisfying certain vanishing data is positive and the imposed vanishing conditions are not independent. The first instance studied were unexpected curves in the paper by Cook II, Harbourne, Migliore, Nagel. Unexpected hypersurfaces were then investigated by Bauer, Malara, Szpond and Szemberg, followed by Harbourne, Migliore, Nagel and Teitler who introduced the notion of BMSS duality and showed it holds in some cases (such as certain plane curves and, in higher dimensions, for certain cones).They ask to what extent such a duality holds in general. In this paper, working over a field of characteristic zero, we study hypersurfaces in P n × P n defined by determinants. We apply our results to unexpected hypersurfaces in the case that the actual dimension is 1 (i.e., there is a unique unexpected hypersurface). In this case, we show that a version of BMSS duality always holds, as a consequence of fundamental properties of determinants.2010 Mathematics Subject Classification. 14C20, 15A06.

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Unexpected curves arising from special line arrangements

Cited by 23 publications

References 14 publications

Line arrangements and configurations of points with an unexpected geometric property

Line arrangements and configurations of points with an unexpected geometric property

On the unique unexpected quartic in $${\mathbb {P}}^2$$

A matrixwise approach to unexpected hypersurfaces

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