This is a common problem for acrylic, especially on the lathe. Of course, it can be a problem with wood too.
Here's a few things to try if you are not already doing them:
-ensure the blank is very well secured in the chuck. If you can round it first and secure it in a collet chuck, even better.
-Try punching a small hole in the centre of the end of the blank with...well, with whatever. I use the centre (sorry, 'center' :biggrin
tip of a Brad-point drill bit. Ensure your mark is centred well.
-Start the hole with a very small size drill bit, and work up to the size you want using regular tip drill bits. This allows for less debris to exit around the bit.
-Drill only 5-10mm at a time, withdraw, and cool the bit with a wet rag. This avoid excessive heat buildup, which melts the acrylic, and causes the drill to start 'bouncing around', enlarging the hole.
OK, there's my 2 cents worth of experience.
Cheers,
Ken