Commentarii Mathematici Helvetici

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Volume 78, Issue 3, 2003, pp. 447–462

DOI: 10.1007/s00014-003-0762-0

Elementary modifications and line configurations in P^2

Hal Schenck (1)

(1) Department of Mathematics, University of Illinois, 1409 W Green Street, IL 61801, URBANA, UNITED STATES

Associated to a projective arrangement of hyperplanes ${\mathcal A}$ in P^n is the module D$({\mathcal A})$, which consists of derivations tangent to ${\mathcal A}$. We study D$({\mathcal A})$ when ${\mathcal A}$ is a configuration of lines in P^2. 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$({\mathcal A})$. 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 Teraos conjecture.

Keywords: Hyperplane arrangement - Vector bundle - Castelnuovo-Mumford regularity - Stability, jump locus