I started with a type of yarn that is recommended - sock yarn - and an appropriate crochet hook - a 2.5mm. Waaayy too small to work this pattern with. Went to a bigger 8-ply yarn with a 4mm hook. This is the result.
I have put the star aside. The pattern was too compact for this yarn and the hook, just like the thinner yarn. It is curling up at the edges, and looks lumpy. Instead of getting overly frustrated with it, I just put it aside. I'll go back to it, but sometimes projects stumble, and it's better to have a res and work on something else for a while. It gives you time to think about how to tackle the stumling block. In this case, I'm going to continue with this yarn, but go to a size 5mm hook. The yarn might be the issue: it might be too big a ply, but with a bigger hook it should work in the star's favour. I could also go back to the thinner 2-ply sock yarn and try a 3.25/3.5/3.75mm hook.