The cards are exiled face up.

The cards in the pile that wasn’t chosen remain exiled. Likewise, any cards in the chosen pile that you can’t play or you choose not to play remain exiled.

If you cast a card “without paying its mana cost,” you can’t pay any alternative costs. You can pay additional costs, such as conspire costs and kicker costs.

You can play a land card from the chosen pile only if it’s your turn (which it probably is, since Brilliant Ultimatum is a sorcery) and you haven’t yet played a land this turn. That means that if there are two lands in the chosen pile, you’ll be able to play a maximum of one of them.

You play cards from the chosen pile as part of the resolution of Brilliant Ultimatum. You may play them in any order. Timing restrictions based on the card’s type (such as creature or sorcery) are ignored. Other play restrictions are not (such as “Cast [this card] only during combat”). You play all of the cards you like, putting land onto the battlefield and spells on the stack, then Brilliant Ultimatum finishes resolving and is put into your graveyard. The spells you cast this way will then resolve as normal, one at a time, in the opposite order that they were put on the stack.

One of the piles may have zero cards in it if the opponent wishes.