Planar quadratic differential systems with invariant straight lines of the total multiplicity 4

Abstract

In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with four invariant lines, including the line at infinity and including multiplicities. For each orbit we exhibit its configuration. We characterize in terms of algebraic invariants and comitants and also geometrically, using divisors of the complex projective plane, the class of quadratic differential systems with four invariant lines. These conditions are such that no matter how a system may be presented, one can verify by using them whether the system has exactly four invariant lines including multiplicities, and if it is so, to check to which orbit (or family of orbits) it belongs.