# Fortnightly links (152)

Giulia Gugiatti, Franco Rota: Full exceptional collections for anticanonical log del Pezzo surfaces takes the reader on a beautiful voyage through the lands of homological mirror symmetry for stacky (or singular) del Pezzo surfaces, the McKay correspondence for $\mathrm{GL}_2$, colorful pictures showing weighted blowups, the geometry of Eckardt points on $\operatorname{Bl}_7\mathbb{P}^2$, the structure of exceptional collections on the minimal resolution and canonical stack.

Fabian Reede: Picard schemes of noncommutative bielliptic surfaces discusses aspects of last remaining case of Picard schemes for Azumaya algebras on Kodaira dimension 0 surfaces, the bielliptic surfaces. Recommended if you like some interesting noncommutative algebraic geometry involving orders.

Riemann-Roch for $\overline{\operatorname{Spec}\mathbb{Z}}$ reminds me of the $\mathbb{F}_1$-seminar Lieven ran in Antwerp in 2011 (or 2012?), where if memory serves me right we disussed how a good theory for $\operatorname{Spec}\mathbb{Z}$ (or some compactified version) over $\mathbb{F}_1$ would yield an approach to the $abc$-conjecture. One can read more about this in this article by Lieven. Browsing through these notes, it seems that the essential ingredient is not quite Riemann–Roch, but rather Riemann–Hurwitz. In Hartshorne these are only separated by 5 pages, so surely the world must be getting close to a non-IUTT "proof" of the $abc$-conjecture?!