# Fortnightly links (55)

Andrei Okounkov, Takagi lectures on Donaldson-Thomas theory are lecture notes by one of the leading figures in enumerative geometry. Even if you don't care about these things, you should check out the amazing pictures in these lecture notes. I witnessed him giving very visual lectures at the 2015 Salt Lake City monster conference, and these notes are again really nice.

Adeel Khan, Virtual Cartier divisors and blow-ups shows how in derived algebraic geometry blowups satisfy a universal property for

*all*morphisms, provided one replaces the condition that the schematic fibre is an effective Cartier divisor by the condition that the schematic fibre is a virtual effective Cartier divisor.Ciaran Meachan, Giovanni Mongardi, Kota Yoshioka: Derived equivalent Hilbert schemes of points on K3 surfaces which are not birational and Shinnosuke Okawa: An example of birationally inequivalent projective symplectic varieties which are D-equivalent and L-equivalent are two preprints giving examples of derived equivalent but non-birational hyperkähler varieties. Bondal–Orlov predicts that the opposite implication is true: birational varieties with trivial canonical bundle (which are all built up using abelian varieties, Calabi–Yau varieties and hyperkähler varieties) are derived equivalent. In the abelian and Calabi–Yau case there were plenty of examples of the other direction failing, now there are also hyperkähler varieties using $\operatorname{Hilb}^n\mathrm{K}3$'s.