I'm assuming that you didn't file a separation agreement with court that included this? The deduction goes to whatever parent provided the most support during the year. If you provided more than 50% of her support, you are entitled to claim her. Because she lives with you - and assuming that you don't have some other complex arrangement going on where he's paying a substantial part of your living expenses or your parents are supporting your daughter, etc. - the deduction is yours. The link below lists all kinds of scenarios and possible outcomes:
http://www.irs.gov/publications/p17/ch03.html#en_US_2013_...#
As to how this is enforced...they may or may not catch it initially, or ever, but he really should file an amended return to keep himself out of trouble:
https://turbotax.intuit.com/tax-tools/tax-tips/Family/Wha...
Basically, you go ahead and claim your daughter as your dependent. If it gets flagged, you'll provide whatever documentation they request and be recognized as having filed correctly and your husband will have a few years to file an amended return and pay back anything he owes.
Do make sure that this gets spelled out clearly in your separation and divorce agreements.
In response to some other inaccurate info you've been given...you should both be filing as married filing separately - that's how separated couples file and it's perfectly correct to do so. As to whether or not your child lived with you for more than 50%, of the year, the time that you lived together doesn't count. Of the remainder of the year, when you were living apart, your child lived with you and therefore, you provided more of the support for your child than your husband did and are entitled to claim her as your dependent.