Deuteronomy 24:1-4 outlines some guidance for divorce but this was open to significant variations in rabbinic interpretation. The rabbinical school of Hillel allowed for a wide range of acceptable reasons to end a marriage – it was more liberal than the rabbinical school of Shammai. Jesus’ restrictive rules on divorce echo those of the rabbinical school of Shammai.
Jesus made it clear that God hates divorce and it should only be granted on the grounds of adultery:
“You have heard the law that says, ‘A man can divorce his wife by merely giving her a written notice of divorce.’ But I say that a man who divorces his wife, unless she has been unfaithful, causes her to commit adultery. And anyone who marries a divorced woman also commits adultery” (Matthew 5:31-32). See also Matthew 19:1-8.
Moses permitted divorce only as a concession to the hard hearts of the people. Divorce is never God’s desire (Malachi 2:14-16); it is an expression of human sin. Marriage was designed to be permanent (see Mark 10:11-12). Source: New Living Translation Bible notes
Paul allows for separation in the case of abandonment by an unbeliever (1 Corinthians 7: 10-15) but urges reconciliation. As for 1 Corinthians 7:27, the NLT says “If you have a wife, do not seek to end the marriage.” That would suggest divorce, not separation.