138 research outputs found
Moduli Spaces of Arrangements of 10 Projective Lines with Quadruple Points
We classify moduli spaces of arrangements of 10 lines with quadruple points.
We show that moduli spaces of arrangements of 10 lines with quadruple points
may consist of more than 2 disconnected components, namely 3 or 4 distinct
points. We also present defining equations to those arrangements whose moduli
spaces are still reducible after taking quotients of complex conjugations.Comment: Changed notations in the definition of moduli space to improve
clarity. Results unchange
Classification of Moduli Spaces of Arrangements of 9 Projective Lines
In this paper, we present a proof that the list of the classification of
arrangements of 9 lines by Nazir and Yoshinaga is complete.Comment: Changed notations in the definition of moduli space to improve
clarity. Results unchange
Moduli spaces of arrangements of 12 projective lines with a sextic point
*This paper is from 2018*
In this paper, we try to classify moduli spaces of arrangements of lines
with sextic points. We show that moduli spaces of arrangements of lines
with sextic points can consist of more than two connected components. We also
present defining equations of the arrangements whose moduli spaces are not
irreducible taking quotients by the complex conjugation by supply some
potential Zariski pairs. Through complex conjugation we take quotients and
supply some potential Zariski pairs.Comment: *A paper from 2018*. arXiv admin note: text overlap with
arXiv:1206.248
Numerical invariants and moduli spaces for line arrangements
Using several numerical invariants, we study a partition of the space of line
arrangements in the complex projective plane, given by the intersection lattice
types. We offer also a new characterization of the free plane curves using the
Castelnuovo-Mumford regularity of the associated Milnor/Jacobian algebra.Comment: v3: A new proof of a result due to Tohaneanu, giving the
classification of line arrangements with a Jacobian syzygy of minimal degree
2 is given in Theorem 4.11. Some other minor change
On the Sylvester-Gallai and the orchard problem for pseudoline arrangements
We study a non-trivial extreme case of the orchard problem for
pseudolines and we provide a complete classification of pseudoline arrangements
having triple points and double points. We have also classified those
that can be realized with straight lines. They include new examples different
from the known example of B\"or\"oczky. 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 Gr\"unbaum's problems. We formulate some
open problems which involve our new examples of line arrangements.Comment: 5 figures, 11 pages, to appear in Periodica Mathematica Hungaric
On the containment problem
The purpose of this note is to provide an overview of the containment problem
for symbolic and ordinary powers of homogeneous ideals, related conjectures and
examples. We focus here on ideals with zero dimensional support. This is an
area of ongoing active research. We conclude the note with a list of potential
promising paths of further research.Comment: 13 pages, 1 figur
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