  We classify the solutions to a system of equations, introduced by Bondal,
which encode numerical constraints on full exceptional collections of length 4
on surfaces. The corresponding result for length 3 is well-known and states
that there is essentially one solution, namely the one corresponding to the
standard exceptional collection on the surface $\mathbb{P}^2$. This was
essentially proven by Markov in 1879. It turns out that in the length 4 case,
there is one special solution which corresponds to
$\mathbb{P}^1\times\mathbb{P}^1$ whereas the other solutions are obtained from
$\mathbb{P}^2$ by a procedure we call numerical blowup. Among these solutions,
three are of geometric origin ($\mathbb{P}^2\cup \{\bullet\}$,
$\mathbb{P}^1\times\mathbb{P}^1$ and the ordinary blowup of $\mathbb{P}^2$ at a
point). The other solutions are parametrized by $\mathbb{N}$ and very likely do
not correspond to commutative surfaces. However they can be realized as
noncommutative surfaces, as was recently shown by Dennis Presotto and the first
author.
