On an analogue of the Markov equation for exceptional collections of   length 4
  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. 
Louis de Thanhoffer de Volcsey and Michel Van den Bergh
math.AG