On the derived category of the classical Godeaux surface (Q392962)

Let \(S\) be a complex surface with \(p_g=q=0\). Then any line bundle \(L\) on \(S\) is exceptional, that is \(\mathrm{Ext}^i(L,L) = 0\) for \(i >0\), and \(\mathrm{End}(L)= {\mathbb C}\). Since this definition is purely homological, one should naturally consider exceptional objects in the derived categroy \(D^b(S)\), where line bundles (in general, coherent sheaves) identify to one-term complexes concentrated in degree 0. A sequence \((E_1, \ldots, E_l)\) of exceptional objects is exceptional if \({\Hom}_{D^b(S)}(E_j,E_k[i])=0\) for all \(i\) whenever \(j > k\). Such a sequence is full if \({\Hom}_{D^b(S)}(A,E_i)=0\) for all \(i\) implies that \(A=0\). If \(S\) has a full exceptional collection of length \(n\), then \(K_0(S)\) is free of rank \(n\), and hence \(n=\rho+2\): it follows that the maximal length of an exceptional collection is \(\rho+2\). If \(S\) is a rational surface, then it has a full exceptional collection of line bundles. On the other hand, it is natural to wonder about the exsitence of an exceptional collection of maximal length on a surface with \(p_q=q=0\) and to question about its fullness or about its orthogonal complement.NEWLINENEWLINEIn this paper, the authors provide the first example of an exceptional collection of maximal length on a surface of general type, the classical Godeaux surface \(S\) obtained as a quotient of the Fermat quintic surface. Since \(K_0(S)\) has a torsion subgroup, the collection cannot be full. Hence the orthogonal complement \(\mathbb{A}\) to the collection is nontrivial, but \(K_0(\mathbb{A})\) is torsion. This also provide a first example of quasi-phantom triangulated category, and a counterexample to Kuznetsov's Nonvanishing Conjecture.NEWLINENEWLINEThe proof is constructive: the authors first describe the Picard lattice, based on the explicit description of \(S\) as a quotient of a Fermat quintic, and on the \(E_8\)-symmetry of the Picard lattice. They can provide a sequence of length 11, which is maximal since \(\rho(S)=9\). Some calculations are obtained using Macaulay. Finally they draw some interesting consequences on the structure of \(D^b(S)\).