The "enough to investigate her" - if you can call it that - was that some people, prosecutor included, thought she seemed weird.
Really, she was arrested because the prosecutor was a disgraced, ultra conservative religious fanatic, who had been ridiculed over a ludicrous and baseless claims of Satanism in another case.
Desperate to salvage his reputation, and having noted the murder took place on All Saint's Day, he instantly leaped to a theory of Satanic panic.
Then, his misogyny told him that the woman having premarital sex probably had a vagina embued with satanic powers.
After using illegal and extreme methods to extract a confession and the implication of Labamba from Knox, he thought he had his vindication, in front of the world's media - only for Rudy Gued to be definitively implicated.
Unable to accept that he was a deluded clown, he ignored real evidence in favor of the false, and used the media to spread fabtastical claims about Knox, which were consumed by the jury.
wUltimately, through Supreme-Court ordered independent tests and reviews, it was determined not a single peice of reliable physical evidence linking Knox or Sollecito to the crime and, beyond that, the physical evidence demonstrated it was not possible she could have materially participated in the crime.