Because the verb being changed is 'does' - 'to do'. It's followed by an infinitive 'to have' I do not have He/She/It does not have We do not have They do not have

becasue in this sentence the "have" is an infinitive, not a conjugated verb She doesn't swim ( do is the main verb (the conjugated one) and "swim" is an infinitive) She doesn't eat well (eat is infinitve ) He is doesn't have his books with him (have is infinitive) or with another help verb/main verb he wants to eat icecream (eat is infinitive) she loves to write (write is infintive)

She has a pen. (Correct) She hasn't a pen. (Not incorrect, but to an American it sounds old-fashioned and awkward, perhaps less so to British speakers) She doesn't have a pen. (Correct, natural sounding, and you have to conjugate "to do")