We met her in Star Trek 2: The Wrath of Khan. She's a brilliant, powerful, influential, respected, super-genius Federation scientist. What kind of scientist is she? She's a molecular biologist.
Dr. Marcus concieved of and directed the ambitious Project Genesis, which developed a miraculous process to rapidly terraform uninhabitable planets into worlds suitable for human life. Her son, Dr. David Marcus also became a noted scientist, and worked closely with his famous mother on the Genesis Project.
Carol Marcus is a very strange woman (a domineering ball-breaker, for one thing) who apparently was once intimate with young James T. Kirk. How intimate is harder to precisely say than you might imagine. True, her handsome son, Dr. David Marcus, is revealed to be Kirk's offspring. But Jim Kirk was totally uninvolved in David's upbringing, in accordance with Carol's request/demand.
It's not strictly speaking correct to say that Kirk is David's "father": Kirk never had a chance to be any kind of a father to little David. In fact, David wasn't even informed that Jim Kirk was his biological sire until adulthood, during the events depicted in ST2. Jim had no relationship with the child at all, which he had obviously agreed to, but it did seem to hurt him that Carol hadn't even told the boy who his biological sire was... as if he were someone she was ashamed of.
Carol may be a respectable scientist, but she apparently is a man-hater. She cared less than nothing for both Jim's and David's feelings and rights in the domestic arena. The only man in her life all those years, it seems, was her son - whom she kept tied tightly to herself with apron-strings of iron. Then she erected an iron wall between father and son, through which no communication whatsoever ever passed.
It's nothing personal. It's not as if she hates Jim personally. Carol tells David that Jim was "never a boy scout" (which seems to imply that she might disapprove of him), but later she defends him when Jim is (unjustly) accused by David of doing nogoodnik things. She knows that Jim is a good guy, and generally, she obviously trusts him.
Many people like to imagine that Carol Marcus was one of Kirk's "great loves" but in fact there is no love evident between them. They are at best polite to one another, and no mention is ever made of a love affair, or even of a one-night stand. If we stick closely only to the facts that are presented onscreen, it's logical to assume that David was the product of artificial insemination, which would neatly explain why Jim had no parental rights or responsibilities whatsoever.
She is a brilliant molecular biologist. She might have taken Jim's DNA from a fallen hair, and used it to impregnate herself. Think about it: someone like her would know how to do that.
Of course, if she and Jim actually did have an affair it wouldn't be surprising. She's exactly the kind of woman that we have seen that he likes. She has the one quality that Kirk finds irresistable in women: She's smarter than Stephen Hawkings.
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