### Bipartite Algebraic Graphs

Joint work with Boris Bukh

### Extremal graph problem

$\operatorname{ex}(n, F)$ – largest # of edges in an $F$-free graph

Erdős–Stone–Simonovits Theorem

$$\operatorname{ex}(n, F) = \left(1 - \frac{1}{\chi - 1}\right){n \choose 2} + o(n^2)$$

Degenerate case $\chi = 2$: complete bipartite $K_{s,t}$

### Complete bipartite

Kövari–Sós–Turán Theorem

$$\operatorname{ex}(n, K_{s,t}) \le c_{s,t}\cdot n^{2-1/s}+o(n^{2-1/s})$$

Constructions that match the upper bounds:

• A bipartite graph whose partite sets are $\mathbb{F}_p^s$,
• and $\bar{u}\sim\bar{v}$ iff $(\bar{u},\bar{v})$ is on an algebraic hypersurface.
 For $K_{2,2}$, $x_1y_1+x_2y_2=1$; For $K_{3,3}$, $(x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2=1$.

### Questions

What does an extremal $K_{s,t}$-free graph look like?

Füredi's result

Every $K_{2,2}$-free graph with $q$ vertices and $\frac{1}{2}q(q+1)^2$ edges is obtained from a projective plane via a polarity with $q+1$ absolute elements.

What does a $K_{s,t}$-free algebraic graph look like?

• Finite field (number theory) is hard.
• Projective algebraic geometry is hard but easier.

### Algebraic graphs

Given hypersurface $H$ in $\mathbb{P}^s\times \mathbb{P}^s$.

Example

$$(x_1y_0-x_0y_1)^2+(x_2y_0-x_0y_2)^2$$ $$+(x_3y_0-x_0y_3)^2=x_0^2y_0^2.$$

$H$ is said to contain an $(s,t)$-grid if there exist $S, T\subset \mathbb{P}^s$ such that $s = |S|, t = |T|$ and $S\times T \subset H$.

$H$ is almost-$(s,t)$-grid-free if there are "big" sets $X,Y\subset \mathbb{P}^s$ such that $H\cap(X\times Y)$ is $(s,t)$-grid-free.

### Meta conjecture

Every almost-$(s,t)$-grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in $\bar{y}$ is low.

### Heuristics

Suppose deg of $F(\bar{x},\bar{y})$ in $\bar{y}$ is $d$, $\bar{u}_1, \dots, \bar{u}_s$ are points in $\mathbb{P}^s$.

Generically, $F(\bar{u}_1,\bar{y})=\dots=F(\bar{u}_1,\bar{y})=0$ has $d^s$ solutions.

### Simplest case

Given hypersurface $H \subset \mathbb{P}^1\times \mathbb{P}^1$.

The $s=1$ case

If $H$ is almost-$(1,t)$-grid-free, then there exists $F(\bar{x}, \bar{y})$ of degree $< t$ in $\bar{y}$ such that $H$ is almost equal to $\{F = 0\}$.

### Partial results

Given hypersurface $H \subset \mathbb{P}^2\times \mathbb{P}^2$ and a "big" $X \subset \mathbb{P}^2$

$s=t=2$ and $Y = \mathbb{P}^2$

If $H\cap(X\times \mathbb{P}^2)$ is $(2,2)$-grid-free, then $\exists F(\bar{x},\bar{y})$ of degree $\le 2$ in $\bar{y}$ such that $H$ is almost equal to $\{F = 0\}$.

$s=t=2$ and $\mathbb{P}^2\setminus Y = \{\bar{v}_1, \dots, \bar{v}_n\}$

If $H\cap (X\times Y)$ is $(2,2)$-grid-free, then either we are 😄 or $\mathbb{P}^2\times \{\bar{v}_i\}\subset H$ for some $i$.

### Proof idea

Think of $H\cap (X\times\mathbb{P}^2)$ as a family of algebraic curves in $\mathbb{P}^2$:

$$C(\bar{u}) := \left\{\bar{v}\in\mathbb{P}^2:(\bar{u},\bar{v})\in H\right\}.$$

Hypersurface $H$ is $(2,2)$-grid-free if and only if $C(\bar{u})$ and $C(\bar{u}')$ intersect at $\le 1$ point for distinct $\bar{u},\bar{u}'\in X$.

A corollary of Moura's result

For a generic point $\bar{v}$ on an algebraic curve $C$ in $\mathbb{P}^2$, any algebraic curve $C'$ with $\bar{v}\in C'$ intersects with $C$ at another point unless $C$ is irreducible of degree $\le 2$.

### Meta conjecture

Every almost-$(s,t)$-grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in $\bar{y}$ is low.

Examples

$H_0 := \left\{x_0y_0+x_1y_1+x_2y_2=0\right\}$

$H_1 := \left\{x_0y_0^d+x_1y_0^{d-1}y_1+x_2\left(y_0^{d-1}y_2+y_1^d\right)=0\right\}$

### Birational automorphism

Birational automorphism $\sigma\colon$ defined by

$$\sigma(y_0:y_1:y_2):=(y_0^d:y_0^{d-1}y_1:y_0^{d-1}y_2+y_1^d)$$

is a biregular map from $\mathbb{P}^2\setminus\{y_0=0\}$ onto itself.

Conjecture

If $H$ is almost-$(s,t)$-grid-free, then there exists a birational automorphism $\sigma$ such that $H$ is equal, up to $\mathrm{id}\times \sigma$, to a hypersurface of low degree in $\bar{y}$.