On Mori dreamness of blowups along space curves

We study the problem of determining when the blowup X → P³ along a smooth space curve C is a Mori Dream Space. We obtain sufficient conditions, as well obstructions to the Mori dreamness of X based on the external geometry of C. We furthermore find infinitely many pairs $(g,d)$ such that the corresponding Hilbert schemes Hg,dS admit components whose general element has these obstructions. As a consequence we show that Mori dreamness is not an open property in flat families and exhibit various degenerational pathologies.