On the algebraic properties of the B\"or\"oczky configuration

The B\"or\"oczky configuration of lines and (multiple) points exhibits extremal behavior in commutative algebra and combinatorics. Examples of this appear in the context of the containment problem for ordinary and symbolic powers and the proof of the Dirac-Motzkin conjecture by Green and Tao. This paper studies the algebraic properties of B\"or\"oczky configurations for arbitrary values of $n$. Our results compute the Waldschmit constant of the defining ideal of these configurations. Moreover, we use the weighted projective plane $\mathbb{P}(1,2,3)$ to give an upper bound for the degree of the minimal generators of their ideal. Finally, this construction is applied to an elliptic curve in $\mathbb{P}^2$ to give a new counterexample to the containment $I^{(3)}\subseteq I^2$.