Bipartite algebraic graphs without quadrilaterals

TL;DR

This paper investigates hypersurfaces in complex projective spaces that avoid certain bipartite subgraphs, establishing bounds on their degrees and exploring their structural properties related to the Turán problem.

Contribution

It proves that (2,2)-grid-free hypersurfaces in complex projective space can be described by polynomials of degree at most 2 in one variable, advancing understanding of bipartite graph restrictions.

Findings

(2,2)-grid-free hypersurfaces are defined by quadratic polynomials.

The degree bound in the variable y is at most 2 for such hypersurfaces.

Results extend to algebraically closed fields of large characteristic.

Abstract

Let $\mathbb{P}^s$ be the $s$-dimensional complex projective space, and let $X, Y$ be two non-empty open subsets of $\mathbb{P}^s$ in the Zariski topology. A hypersurface $H$ in $\mathbb{P}^s\times\mathbb{P}^s$ induces a bipartite graph $G$ as follows: the partite sets of $G$ are $X$ and $Y$, and the edge set is defined by $\overline{u}\sim\overline{v}$ if and only if $(\overline{u},\overline{v})\in H$. Motivated by the Tur\'an problem for bipartite graphs, we say that $H\cap (X\times Y)$ is $(s,t)$-grid-free provided that $G$ contains no complete bipartite subgraph that has $s$ vertices in $X$ and $t$ vertices in $Y$. We conjecture that every $(s,t)$-grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in $\overline{y}$ is bounded by a constant $d = d(s,t)$, and we discuss possible notions of the equivalence. We establish the result that if $H\cap(X\times \mathbb{P}^2)$ is $(2,2)$-grid-free, then there exists $F\in \mathbb{C}[\overline{x},\overline{y}]$ of degree $\le 2$ in $\overline{y}$ such that $H\cap(X\times \mathbb{P}^2) = \{F = 0\}\cap (X\times \mathbb{P}^2)$. Finally, we transfer the result to algebraically closed fields of large characteristic.