# Fortnightly links (43)

Katrina Honigs, Max Lieblich, Sofia Tirabassi: Fourier-Mukai partners of Enriques and bielliptic surfaces in positive characteristic is a nice and short paper which proves that there are

*no*Fourier-Mukai partners for Enriques (resp. bielliptic) surfaces in characteristic at least 3 (resp. 5). The condition on the characteristic comes from the required liftability to the Witt vectors. Whether there might be Fourier-Mukai partners in small characteristic therefore remains open.Mgbar is a Stacks project blogpost concerning commit a5d4ac0. In this commit the proof that $\overline{\mathcal{M}}_g$ is a smooth and proper Deligne-Mumford stack over $\mathop{\rm Spec}\mathbb{Z}$. This is theorem 5.2 in the seminal Deligne-Mumford paper on the irreducibility of $\overline{M}_g$, which introduced the notion of an algebraic stack to the world (although Mumford had some preliminary work).

An interesting quote from this paper is the following:

Full details on the basic properties and theorems for algebraic stacks will be given elsewhere.

It turns out they never gave those details themselves, but they are now (almost 50 years after they were promised) at an EGA-level of completeness in the Stacks project!