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Interpolation matrices and jumping lines of logarithmic bundles

Elena Guardo, Graham Keiper, Grzegorz Malara

TL;DR

The paper develops a determinantal framework to study jumping lines of logarithmic (syzygy) bundles associated with finite point sets in the plane. By using interpolation matrices, the jumping locus in the dual plane is captured as the zero set of a determinant $F_{d,m;Z}$, with $(d,m)$ dictated by the parity of $|Z|$. For general-position point sets, this yields irreducible curves of degree $d(d-1)$ (or a secant-locus description in the even-parity case), linking jumping lines to the second Chern class and to unexpected hypersurfaces in a dual setting. Special configurations produce fixed components (e.g., lines or conics) in the degeneracy locus, reflecting the combinatorics of $Z$, and revealing a richer geometry that combines classical bundle theory with contemporary ideas on unexpected curves. These results provide a new, computable bridge between the theory of logarithmic bundles, interpolation determinants, and the geometry of point configurations.

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.

Interpolation matrices and jumping lines of logarithmic bundles

TL;DR

The paper develops a determinantal framework to study jumping lines of logarithmic (syzygy) bundles associated with finite point sets in the plane. By using interpolation matrices, the jumping locus in the dual plane is captured as the zero set of a determinant , with dictated by the parity of . For general-position point sets, this yields irreducible curves of degree (or a secant-locus description in the even-parity case), linking jumping lines to the second Chern class and to unexpected hypersurfaces in a dual setting. Special configurations produce fixed components (e.g., lines or conics) in the degeneracy locus, reflecting the combinatorics of , and revealing a richer geometry that combines classical bundle theory with contemporary ideas on unexpected curves. These results provide a new, computable bridge between the theory of logarithmic bundles, interpolation determinants, and the geometry of point configurations.

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 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.
Paper Structure (5 sections, 4 theorems, 77 equations)

This paper contains 5 sections, 4 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jumping lines and the interpolation approach
  4. Points in general position
  5. Special configurations of points and forced components of the degeneracy locus

Key Result

Proposition 3.1

With the above notation, assume that $\binom{d+2}{2}=|Z|+\binom{m+1}{2}$. Then that is, a line $L$ is a jumping line of the first kind if and only if the corresponding dual point $B=P_L$ satisfies

Theorems & Definitions (19)

  • Definition 2.1
  • Example 2.2
  • Proposition 3.1
  • proof
  • Remark 3.2: Parity condition and choice of parameters
  • Remark 3.3
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Proposition 4.3
  • ...and 9 more