Bound the degree of configuration varieties for expanding incidence graphs

Prove that for the point-line incidence relation in the complex plane, an ε-expander bipartite graph with n vertices on each side and n^α edges, where α>1, has configuration variety of degree at most a fixed power of log n.

Background

When the configuration variety has dimension 8, its degree measures, up to the projective symmetry, the number of distinct configuration families or projective orbits. Number-field constructions suggest that this degree can grow at least logarithmically with n.

The conjecture asserts that expansion and a superlinear number of incidences impose a polylogarithmic upper bound on this degree, which would further restrict the complexity of near-extremal point-line configurations.

References

\begin{conj} \label{degbound} Let $Z \subset C2 \times C2$ be the point-line incidence relation in $C2$ given by eqZpointline. Suppose that $G$ is an $\epsilon$-expander bipartite graph with $n$ vertices on each side and $n\alpha$ edges for $\alpha > 1$. Then $\deg V(G,Z) \le (\log n){O(1)}$. \end{conj}

— Perspectives on the unit distance problem  (2609.10791 - Guth, 9 Sep 2026) in Conjecture \ref{degbound}, subsection “Incidence graphs and configuration varieties”