1,029 research outputs found
On supersolvable and nearly supersolvable line arrangements
We introduce a new class of line arrangements in the projective plane, called
nearly supersolvable, and show that any arrangement in this class is either
free or nearly free. More precisely, we show that the minimal degree of a
Jacobian syzygy for the defining equation of the line arrangement, which is a
subtle algebraic invariant, is determined in this case by the combinatorics.
When such a line arrangement is nearly free, we discuss the splitting types and
the jumping lines of the associated rank two vector bundle, as well as the
corresponding jumping points, introduced recently by S. Marchesi and J.
Vall\`es. As a by-product of our results, we get a version of the Slope
Problem, looking for lower bounds on the number of slopes of the lines
determined by points in the affine plane over the real or the complex
numbers as well.Comment: v.3, a version of the Slope Problem, valid over the real and the
complex numbers as well, is obtained, see Thm. 1.1 and Thm. 4.
Singular hypersurfaces characterizing the Lefschetz properties
In the paper untitled "Laplace equations and the Weak Lefschetz Property" the
authors highlight the link between rational varieties satisfying a Laplace
equation and artinian ideals that fail the Weak Lefschetz property. Continuing
their work we extend this link to the more general situation of artinian ideals
failing the Strong Lefschetz Property. We characterize the failure of SLP (that
includes WLP) by the existence of special singular hypersurfaces (cones for
WLP). This characterization allows us to solve three problems posed by Migliore
and Nagel and to give new examples of ideals failing the SLP. Finally, line
arrangements are related to artinian ideals and the unstability of the
associated derivation bundle is linked with the failure of SLP. Moreover we
reformulate the so-called Terao's conjecture for free line arrangements in
terms of artinian ideals failing the SLP
Logarithmic Bundles Of Hypersurface Arrangements In P^n
Let D = {D_{1},...,D_{l}} be an arrangement of smooth hypersurfaces with
normal crossings on the complex projective space P^n and let
\Omega^{1}_{P^n}(log D) be the logarithmic bundle attached to it. Following
[1], we show that \Omega^{1}_{P^n}(log D) admits a resolution of lenght 1 which
explicitly depends on the degrees and on the equations of D_{1},...,D_{l}. Then
we prove a Torelli type theorem when all the D_{i}'s have the same degree d and
l >= {{n+d} \choose {d}}+3: indeed, we recover the components of D as unstable
smooth hypersurfaces of \Omega^{1}_{P^n}(log D). Finally we analyze the cases
of one quadric and a pair of quadrics, which yield examples of non-Torelli
arrangements. In particular, through a duality argument, we prove that two
pairs of quadrics have isomorphic logarithmic bundles if and only if they have
the same tangent hyperplanes.Comment: 21 pages, 2 figures, Collectanea Mathematica 201
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