Logarithmic vector fields for quasihomogeneous curve configurations in P^2

TL;DR

This paper develops an inductive method using elementary modifications of rank two bundles to analyze the splitting of vector fields tangent to quasihomogeneous curve configurations in the projective plane, aiding geometric understanding.

Contribution

It introduces a new approach using elementary bundle modifications to relate vector fields tangent to quasihomogeneous curve arrangements in P^2.

Findings

Provides a method to study bundle splitting based on divisor geometry

Establishes a relation between tangent vector bundles before and after adding a curve

Offers tools for analyzing quasihomogeneous curve configurations in algebraic geometry

Abstract

Let A be a union of smooth plane curves C_i, such that each singular point of A is quasihomogeneous. We prove that if C is a smooth curve such that each singular point of A U C is also quasihomogeneous, then there is an elementary modification of rank two bundles, which relates the O_{P^2} module Der(log A) of vector fields on P^2 tangent to A to the module Der(log A U C). This yields an inductive tool for studying the splitting of the bundles Der(log A) and Der(log A U C), depending on the geometry of the divisor A|_C on C.