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

Abstract

Abstract Any collection of $n$ compact convex planar sets $K_1,\dots , K_n$ defines a vector of ${n\choose 2}$ mixed areas ${\operatorname {V}}(K_i,K_j)$ for $1\leq i<j\leq n$. We show that for $n\geq 4$ 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 $({\operatorname {V}}(K_i,K_j)\,:\,1\leq i<j\leq 4)$. For arbitrary $n\geq 4$, 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.