# Fortnightly links (23)

It seems that John Calabrese has stopped using the arXiv to post his preprints, but luckily they are available on his website. So the first three links are just papers of his.

Relative singular twisted Bondal–Orlov is a generalisation of result that there are no non-trivial Fourier–Mukai partners for a smooth projective variety provided that the (anti)canonical bundle is ample. John proves the analogous result for the

*relative situation*$X,Y\to S$ where things are allowed to be algebraic stacks, and the ampleness condition is now taken on all the fibers, which are only required to be*Gorenstein*, and it is also possible to incorporate*Brauer classes*. Awesome!I do remember Orlov claiming in a talk that there might be some issues with earlier work of Ballard in this direction (who proved a version of

*singular Bondal–Orlov*), but I never figured out what it might be. Do you know?A note on derived equivalences and birational geometry proves a nice birationality criterion for fully faithful functors: if the functor maps one skyscraper to another, then they are birational. This is a cool explanation for the dichotomy between chopping up derived categories in the minimal model program, and chopping up derived categories for more noncommutative reasons, e.g. Kuznetsov's K3 category for a cubic fourfold.

A remark on generators of $\mathbf{D}(X)$ and flags gives an explicit generator for a smooth projective variety with ample (anti)canonical bundle, by summing together the structure sheaves of a complete flag. Another obvious choice would of course have been the direct sum of tensor powers of the ample line bundle you are given, but the situation at hand appears in the moduli of marked curves and the study of $\mathrm{A}_\infty$-structures as started by Polishchuk. This paper gives the geometric motivation behind some observations in his work, and is (like all the others) a very nice read.

Now for the non-John stuff.

Karim El Haloui, Ample tangent bundles on smooth projective stacks shows that the tangent bundle of a weighted projective stack is an ample vector bundle. Hartshorne conjectured for smooth projective varieties that the only such variety is projective space, which was proved by Mori, and the author addresses one direction of the generalised question for algebraic stacks.

hodge_numbers.sage is a tool to compute Hodge numbers for hypersurfaces in weighted projective spaces, somewhat similar to my tool for arbitrary complete intersections in (non-weighted) projective space.

The next topic for ANAGRAMS is weighted projective spaces, so maybe we will use it at some point. Feel free to come!

Also, the musical discovery of the day is Bully. To say it in the words of a YouTube commenter: If this came out in '94, it'd be an absolute classic.