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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.
Complete moduli of cubic threefolds and their intermediate Jacobians
The intermediate Jacobian map, which associates to a smooth cubic threefold
its intermediate Jacobian, does not extend to the GIT compactification of the
space of cubic threefolds, not even as a map to the Satake compactification of
the moduli space of principally polarized abelian fivefolds. A much better
"wonderful" compactification of the space of cubic threefolds was constructed
by the first and fourth authors --- it has a modular interpretation, and
divisorial normal crossing boundary. We prove that the intermediate Jacobian
map extends to a morphism from the wonderful compactification to the second
Voronoi toroidal compactification of the moduli of principally polarized
abelian fivefolds --- the first and fourth author previously showed that it
extends to the Satake compactification. Since the second Voronoi
compactification has a modular interpretation, our extended intermediate
Jacobian map encodes all of the geometric information about the degenerations
of intermediate Jacobians, and allows for the study of the geometry of cubic
threefolds via degeneration techniques. As one application we give a complete
classification of all degenerations of intermediate Jacobians of cubic
threefolds of torus rank 1 and 2.Comment: 56 pages; v2: multiple updates and clarification in response to
detailed referee's comment
Algebraic functions and closed braids
This article was originally published in Topology 22 (1983). The present
hyperTeXed redaction includes references to post-1983 results as Addenda, and
corrects a few typographical errors. (See math.GT/0411115 for a more
comprehensive overview of the subject as it appears 21 years later.)Comment: 12 pages, 2 figure
Classification of the line-soliton solutions of KPII
In the previous papers (notably, Y. Kodama, J. Phys. A 37, 11169-11190
(2004), and G. Biondini and S. Chakravarty, J. Math. Phys. 47 033514 (2006)),
we found a large variety of line-soliton solutions of the
Kadomtsev-Petviashvili II (KPII) equation. The line-soliton solutions are
solitary waves which decay exponentially in -plane except along certain
rays. In this paper, we show that those solutions are classified by asymptotic
information of the solution as . Our study then unravels some
interesting relations between the line-soliton classification scheme and
classical results in the theory of permutations.Comment: 30 page
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