2016/07/14 by Louis de Thanhoffer de Völcsey, Michel Van den Bergh, de Volcsey, Louis de Thanhoffer +1 · 3 citations
Mathematics · #11D99 #14A22 #14C40 #Advanced Operator Algebra Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1607.04246
openalex publication_date 2016/07/14 · openalex created_date 2016/08/23 · openalex updated_date 2026/07/28
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 ℙ2. 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 ℙ1×ℙ1 whereas the other solutions are obtained from ℙ2 by a procedure we call numerical blowup. Among these solutions, three are of geometric origin (ℙ2∪ \\bullet\, ℙ1×ℙ1 and the ordinary blowup of ℙ2 at a point). The other solutions are parametrized by 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.