Interpolation matrices and jumping lines of logarithmic bundles

TL;DR

This paper investigates the loci of jumping lines of logarithmic bundles associated with finite point sets in the projective plane, using interpolation matrices to describe these loci as explicit determinants and exploring their geometric properties.

Contribution

It introduces a new geometric interpretation of jumping lines via interpolation matrices, linking classical results to the framework of unexpected curves and hypersurfaces.

Findings

Determinants define irreducible curves of expected degree for general points.

Special point configurations lead to fixed components in the determinants.

Provides a new geometric perspective on classical jumping lines.

Abstract

We study jumping lines loci of logarithmic bundles associated with finite sets of points in the projective plane. Using the interpolation matrix introduced in [DMTG25], we describe these loci as the zero sets of explicit determinants depending on parameters $(d,m)$ determined by the number of points. We show that for points in general position the determinant defines an irreducible curve of the expected degree, while for special configurations it acquires fixed components related to the combinatorics of the arrangement. The approach provides a new geometric interpretation of the classical jumping lines of Dolgachev--Kapranov and Barth, and connects them to the framework of unexpected curves and hypersurfaces.