The Mishna on our daf discusses a case where a person decided to divorce his wife and went so far as to have a geṭ written. At that point, however, he changed his mind and decided that he did not want to divorce her. In such a case, Beit Shammai rules that the act of having the geṭ written will create a situation that would no longer allow her to marry a kohen (since kohanim cannot marry divorced women). Beit Hillel disagrees, arguing that even if the husband actually gives a divorce to his wife, if the divorce is predicated on conditions that are not fulfilled and the divorce does not take effect, we are not concerned with “re’ah ha-geṭ“, and the woman can marry a kohen (if, for example, her husband dies and she becomes a widow).
In explanation of Beit Shammai’s position, the Me’iri points to the concept of re’ah ha-geṭ (literally, the “smell” of a divorce), that the Gemara raises in another context, suggesting that such a concept may be the underlying principle here, as well. Others suggest that this is a type of gezera, or Rabbinic enactment, that treats a geṭ that has been written as if it were actually delivered with regard to certain halakhot.
The Ḥatam Sofer suggests that the disagreement between Beit Shammai and Beit Hillel may be dependent on another argument between them. According to Beit Shammai, a man is not allowed to divorce his wife unless he has strong reason to suspect that she has been involved in sexual promiscuity of some sort. Thus, if a man reaches the point of actually writing a geṭ, even if it is never delivered it would appear that there was a basis for suspicion with regard to her behavior. This itself is enough to keep her from marrying a kohen, according to Beit Shammai. According to Beit Hillel, who allows a man to divorce his wife for any number of reasons, the decision to write a geṭ gives us no information beyond the fact that there is a difficulty in their relationship, something that would not affect her status regarding a kohen in any way.