On the derived category of the classical Godeaux surface

Abstract: We construct an exceptional sequence of length 11 on the classical Godeaux surface X which is the Z/5-quotient of the Fermat quintic surface in P³. This is the maximal possible length of such a sequence on this surface which has Grothendieck group Z¹¹+Z/5. In particular, the result answers Kuznetsov's Nonvanishing Conjecture, which concerns Hochschild homology of an admissible subcategory, in the negative. The sequence carries a symmetry when interpreted in terms of the root lattice of the simple Lie algebra of type E₈. We also produce explicit nonzero objects in the (right) orthogonal to the exceptional sequence.