If you look back carefully at Chapter Five of this amazing novel, you can see that the animosity that exists between Dede and Minerva has its roots in the conflict they have over a man. When Virgilio Morales first enters the lives of the sisters, both Minerva and Dede are attracted to him. However, it is Minerva who makes the first move and Dede is left with her cousin, who has already been making overtures towards her. It is Dede's sense of jealousy and also her confusion about the realities of living in a police state that cause conflict between her and Minerva. However, what finally cements their animosity is when Virgilio is forced to leave the country and he delivers a letter to Minerva asking her to flee the country with him. Although Dede is entrusted with this letter, she burns it, convincing herself that she is acting in her sister's best interest. A careful reading of the text, however, would indicate otherwise:
The paper lit up. Ashes fluttered like moths, and Dede ground them to dust on the floor. She had taken care of the problem, and that was that. Looking up at the mirror, she was surprised by the wild look on her face. The ring on her finger flashed a feverish reminder. She brushed her hair up into a tight ponytail and put on her nightgown. Having blown out the light, she slept fitfully, holding her pillow like a man in her arms.
We can see by the "wild look" in her face and the way she cuddles the pillow to her in her sleep that this is an act borne out of jealousy. Not having the opportunity to be with Virgilio herself, she cannot bear to let her sister have that opportunity.
Thus the seeds of the conflict are sown, and throughout the rest of the novel Dede becomes increasingly exasperated about Minerva's willingness to become involved in the rebel movement.