On an analogue of the Markov equation for exceptional collections of length 4

Abstract: 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.