Nakayama Algebras and Fuchsian Singularities

This present paper is devoted to the study of a class of Nakayama algebras \(N_n(r)\) given by the path algebra of the equioriented quiver \(\mathbb {A}_n\) subject to the nilpotency degree r for each sequence of r consecutive arrows. We show that the Nakayama algebras \(N_n(r)\) for certain pairs (n, r) can be realized as endomorphism algebras of tilting objects in the bounded derived category of coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras \(N_n(r)\) of Fuchsian type, that is, derived equivalent to the bounded derived categories of extended canonical algebras. We also provide a new way to prove the classification result on Nakayama algebras of piecewise hereditary type, which have been done by Happel–Seidel before.