Martin Herschend, Osamu Iyama, Hiroyuki Minamoto, Steffen Oppermann: Representation theory of Geigle-Lenzing complete intersections is a preprint from 2014, but it just got an update. They introduce a higher-dimensional analogue of weighted projective lines: objects which appear in the intersection of representation theory of finite-dimensional algebras, the algebraic geometry of Deligne–Mumford stacks, the noncommutative algebraic geometry of hereditary orders.
Example 3.1.15 made me chuckle a bit.
Lev Borisov, JongHae Keum, Research announcement: equations of a fake projective plane really made me chuckle. They give 84 (yes, eighty-four) equations in 10 variables, whose vanishing locus is a fake projective plane. For me, these are interesting objects because their derived category can contain a (quasi-)phantom, but numerically they look completely like an actual projective plane. But as they are surfaces of general type, they cannot deform noncommutatively, whereas the usual projective plane does.