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Line and rational curve arrangements, and Walther's inequality | Alexandru Dimca
; Gabriel Sticlaru
; | Date: |
14 Mar 2018 | Abstract: | There are two invariants associated to any line arrangement: the freeness
defect $
u(C)$ and an upper bound for it, denoted by $
u’(C)$, coming from a
recent result by Uli Walther. We show that $
u’(C)$ is combinatorially
determined, at least when the number of lines in $C$ is odd, while the same
property is conjectural for $
u(C)$. In addition, we conjecture that the
equality $
u(C)=
u’(C)$ holds if and only if the essential arrangement $C$ of
$d$ lines has either a point of multiplicity $d-1$, or has only double and
triple points. We prove this conjecture in some cases, in particular when the
number of lines is at most 10. We also extend a result by H. Schenck on the
Castenuovo-Mumford regularity of line arrangements to arrangements of possibly
singular rational curves. | Source: | arXiv, 1803.5386 | Services: | Forum | Review | PDF | Favorites |
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