Elementary modifications and line configurations in P²

2003/07/02 by Henry K. Schenck · 3 citations
Mathematics · #math.AG #math.AC #msc:14N20 #msc:14J60 #msc:52C30 #msc:14F05

paper · pdf

published as Comment. Math. Helv. 78 (2003) 447-462 · 16 pages, 3 figures

arxiv created 2003/07/02 · arxiv updated 2009/11/30

Abstract

Associated to an arrangement of projective hyperplanes A is the module D(A), which consists of derivations tangent to A. We study D(A) when A is a configuration of lines in the projective plane. In this setting, we relate the deletion/restriction construction used in the study of hyperplane arrangements to elementary modifications of bundles. This allows us to obtain bounds on the Castelnuovo-Mumford regularity of D(A). We also give simple combinatorial conditions for the associated bundle to be stable, and describe its jump lines. These regularity bounds and stability considerations impose constraints on Terao's conjecture.

Cited by

• Addition theorems for Ziegler pairs of hyperplane arrangements
• On the Jacobian syzygies for generic toric models
• Bourbaki modules and the module of Jacobian derivations of projective hypersurfaces

Related

• Projective dimensions of hyperplane arrangements
• On the Castelnuovo-Mumford regularity of curve arrangements
• On the module of derivations of a line arrangement
• Numerical invariants and moduli spaces for line arrangements
• Hyperplane sections of projective bundle associated to the tangent bundle of ℙ².