Alexei confronts his wife who gave birth to an illegitimate girl with her lover Count Vronsky:
Alexei: You would have no position. And worse, if we divorce, you would be the guilty party. That means you cannot legally remarry. Your union with Count Vronsky would be illegitimate, as would your daughter, who now has the protection of my name. And this is what you want?
What does "protection of my name" mean?
It's about the newborn girl's standing in society. For now, the girl is seen as Alexei's daughter, as a member of his family. If somebody were to mess with the girl, Alexei would be obliged to come to her defense - and any potential aggressor knows that, because the girl bears Alexei's last name. She has the protection of his name.
If Alexei and his wife were to get a divorce, the girl wouldn't be a member of his family anymore, and no longer enjoy Alexei's protection. The girl wouldn't be a member of Count Vronsky's family either, she would be illegitimate and without protection.
