Turns out blowing up asteroids might not be such a great idea because, like the Terminator 2, they rebuild themselves:
As time went on, the gravitational pull of the asteroid’s resilient core was able to pull back ejected shards. It seems that asteroids don’t just absorb mind-boggling amounts of damage, but, as previous work has hinted, they also are able to rebuild themselves.If We Blow Up an Asteroid, It Might Put Itself Back Together
The best strategy seems to be deflection. Isn’t that a pretty good general principle?