### Saturating the Jacobian ideal of a line arrangement via rigidity theory

- Series
- Algebra Seminar
- Time
- Monday, March 6, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
- Location
- Skiles 005
- Speaker
- Michael DiPasquale – University of South Alabama – mdipasquale@southalabama.edu

A line arrangement is a collection of lines in the projective plane. The intersection lattice of the line arrangement is the set of all lines and their intersections, ordered with respect to reverse inclusion. A line arrangement is called free if the Jacobian ideal of the line arrangement is saturated. The underlying motivation for this talk is a conjecture of Terao which says that whether a line arrangement is free can be detected from its intersection lattice. This raises a question - in what ways does the saturation of the Jacobian ideal depend on the geometry of the lines and not just the intersection lattice? A main objective of the talk is to introduce planar rigidity theory and show that 'infinitesimal rigidity' is a property of line arrangements which is not detected by the intersection lattice, but contributes in a very precise way to the saturation of the Jacobian ideal. This connection builds a theory around a well-known example of Ziegler. This is joint work with Jessica Sidman (Mt. Holyoke College) and Will Traves (Naval Academy).