On the algebraic properties of the Böröczky configuration
Jake Kettinger, Shahriyar Roshan-Zamir
TL;DR
This work analyzes the algebraic properties of Böröczky configurations $B_n$ of lines and triple points across arbitrary $n$. It computes the Waldschmidt constant $\\widehat{\alpha}(I_n)$, bounds the minimal generator degree via the weighted projective plane ${\mathbb P}(1,2,3)$, and shows the product of the configuration lines generates the unique degree-$n$ element of $I_n^{(3)}$ not in $I_n^2$, with the action of $D_6$ yielding an alternating representation. It also produces a novel counterexample to the containment $I^{(3)}\subseteq I^2$ by applying the construction to an elliptic curve, and discusses an extended elliptic-curve version (Elliptified Böröczky). The paper closes with open problems and computational data guiding further exploration of containment phenomena in symmetric point-line configurations.
Abstract
The Böröczky 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 Böröczky 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 $I^{(3)}\subseteq I^2$.