Home /  Workshop /  Schedules /  Modules of Derivations and Applications to Vector Fields

Modules of Derivations and Applications to Vector Fields

Connections Workshop: Commutative Algebra January 18, 2024 - January 19, 2024

January 18, 2024 (09:30 AM PST - 10:30 AM PST)
Speaker(s): Claudia Polini (University of Notre Dame)
Location: SLMath: Eisenbud Auditorium, Online/Virtual
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC

Modules of Derivations and Applications to Vector Fields


The talk is concerned with the structure of the module of derivations and its connections with singularities and vector fields of varieties. Modules of derivations are not well understood -- despite great advances on the Zariski-Lipman conjecture, there is still no complete characterization for when they are free. Our work is partially motivated by a question of Poincaré, who asked how to decide whether a polynomial vector field in the complex plane leaves some algebraic curve invariant. We reformulate this problem in terms of bounding from below the initial degree of the module consisting of all vector fields that leave a fixed curve invariant. This module is a quotient of the module of derivations. We also treat the case of smooth varieties of higher dimension. For plane curves of low genus we establish a correspondence between finer invariants of the module of derivations, such as its graded betti numbers, and properties of the singularities of the curve. This is joint work with Chardin, Hassanzadeh, Simis, and Ulrich.

Supplements No Notes/Supplements Uploaded
Video/Audio Files

Modules of Derivations and Applications to Vector Fields

Troubles with video?

Please report video problems to itsupport@slmath.org.

See more of our Streaming videos on our main VMath Videos page.