Plücker-Type Inequalities for Mixed Areas and Intersection Numbers of Curve Arrangements

Any collection of n compact convex planar sets K1, ⋯, Kn defines a vector of n2 mixed areas V(Ki, Kj) for 1≤i < j≤n. We show that for n ≥ 4 these numbers satisfy certain Plücker-type inequalities. Moreover, we prove that for n = 4, these inequalities completely describe the space of all mixed area vectors (V(Ki, Kj): 1≤i < j≤4). For arbitrary n ≥ 4, we show that this space has a semialgebraic closure of full dimension. As an application, we show that the pairwise intersection numbers of any collection of n tropical curves satisfy the Plücker-type inequalities. Moreover, in the case of four tropical curves, any homogeneous polynomial relation between their six intersection numbers follows from the corresponding Plücker-type inequalities.