The Generalized Sylvester's And Orchard Problems Via Discriminantal arrangement

TL;DR

This paper explores the connection between discriminantal arrangements and classical geometric problems, providing new insights into their combinatorics and potential to solve longstanding open problems.

Contribution

It establishes a link between discriminantal arrangements and generalized Sylvester's and orchard problems, highlighting how changes in combinatorics relate to special point configurations.

Findings

Discriminantal arrangements can classify special point configurations.

The connection offers new approaches to classical geometric problems.

Changing combinatorics of arrangements relates to open problems in geometry.

Abstract

In 1989 Manin and Schechtman defined the discriminantal arrangement $\mathcal{B}(n, k,\mathcal{A})$ associated to a generic arrangement $\mathcal{A}$ of $n$ hyperplanes in a $k$-dimensional space. An equivalent notion was already introduced by Crapo in 1985 with the name of geometry of circuits. While both those papers were mainly focused on the case in which $\mathcal{B}(n, k,\mathcal{A})$ has a constant combinatorics when $\mathcal{A}$ changes, it turns out that the case in which the combinatorics of $\mathcal{B}(n, k,\mathcal{A})$ changes is quite interesting as it classifies special configurations of points in the $k$-dimensional space. In this paper we provide an example of this fact elucidating the connection between the well known generalized Sylvester's and orchard problems and the combinatorics of $\mathcal{B}(n, k,\mathcal{A})$. In particular we point out how this connection could be helpful to address those old but still open problems.