GAAG seminar - On Mori Dreamness of blowups along space curves

Mori Dream Spaces are a special kind of varieties introduced by Hu and Keel that enjoy very good properties with respect to the Minimal Model Program. While Mori Dreamness is a very desirable property, it is not very well behaved with respect to even the simplest birational maps: blowups. In this talk we study Mori Dreamness of blowups along space curves: we provide various sufficient criteria as well obstructions to the blowup being a Mori Dream Space. We also study how this property behaves while varying the curve in the corresponding Hilbert scheme and show that it is neither an open nor a closed condition. Furthermore we exhibit examples of Hilbert schemes whose general element does not give rise to a Mori Dream Space, while special elements do, and vice versa. This is joint work in progress with Tiago Duarte Guerreiro.