doug woods

I just bought a 2004 cavalier for my daughter. It's her first car. It has 130,000 miles. When I first start it, it starts perfect. Its runs good but the only thing is after I run it for a while maybe drive it for about 10 or 20 min, if I shut it off, sometimes it wont start back right away. It takes about maybe 5 or 6 min wait. The engine turns over fine but it wont start. Any ideas what could that be? Trying to fix all of the bugs before I hand the keys over.