2-Hereditary algebras and almost Fano weighted surfaces

Tilting bundles T on a weighted projective line X have been intensively studied by representation theorists since they give rise to a derived equivalence between X and the finite dimensional algebra EndT. A classical result states that if EndT is hereditary, then X is Fano and conversely, for every Fano weighted projective line, there exists a tilting bundle T with EndT hereditary. In this paper, we examine the question of when a weighted projective surface has a tilting bundle whose endomorphism ring is 2-hereditary in the sense of Herschend-Iyama-Oppermann. It is natural to conjecture that they are the almost Fano weighted surfaces, weighted only on rational curves, and we give evidence to support this.