Learn Russian: У директора тоже нет документа. - The director also does not have a document.

Because of the word нет (does not have / there is no). In Russian, when you use нет to say something is missing, the missing thing must take the Genitive case. Since документ is a masculine noun ending in a consonant, we add the ending -а to make it документа.

Why does the sentence start with У директора to mean "The director"?

Russian doesn't typically use a direct verb for "to have." Instead, to say someone has or lacks something, you use the preposition у (meaning "at" or "by") plus the person in the Genitive case. Директор becomes директора, so У директора literally means "At the director."

Wait, are both директора and документа in the Genitive case here?

Exactly! But they take the Genitive case for two completely different reasons. Директора is Genitive because it follows the preposition у (showing who we are talking about). Документа is Genitive because it follows нет (showing what is missing).

Why is тоже (also) placed in the middle of the sentence?

In Russian, тоже usually goes right before the idea it is emphasizing. Here, it emphasizes the lack of the document (нет), so it sits right before it. The literal flow of the sentence is "At the director also there-is-no document."