Triangle singularities, ADE-chains, and weighted projective lines

We study the singularity category of the triangle singularity f=xa+yb+zcf=xa+yb+zc, or S=k[x,y,z]/(f)S=k[x,y,z]/(f), attached to the weighted projective line XX, given by the weight triple (a,b,c)(a,b,c), by investigating the stable category vect¯-X of vector bundles on XX, obtained from the category of vector bundles by factoring out all the line bundles.We show that vect¯-X is fractional Calabi–Yau whose CY-dimension is a function of the Euler characteristic of XX and establish the existence of a tilting object, having the shape of an (a−1)×(b−1)×(c−1)(a−1)×(b−1)×(c−1)-cuboid. The weight types (2,a,b)(2,a,b) yield an explanation of Happel–Seidel symmetry for a class of important Nakayama algebras. Further, the weight sequence (2,3,p)(2,3,p) corresponds to an ADE-chain, the EnEn-chain, extrapolating the exceptional Dynkin cases E6,E7E6,E7 and E8E8 to a whole sequence of triangulated categories.