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We study the problem of selecting the top-k candidates from a pool of applicants, where each candidate is associated with a score indicating his/her aptitude. Depending on the specific scenario, such as job search or college admissions, these scores may be the results of standardized tests or other predictors of future performance and utility. We consider a situation in which some groups of candidates experience historical and present disadvantage that makes their chances of being accepted much lower than other groups. In these circumstances, we wish to apply an affirmative action policy to reduce acceptance rate disparities, while avoiding any large decrease in the aptitude of the candidates that are eventually selected. Our algorithmic design is motivated by the frequently observed phenomenon that discrimination disproportionately affects individuals who simultaneously belong to multiple disadvantaged groups, defined along intersecting dimensions such as gender, race, sexual orientation, socio-economic status, and disability. In short, our algorithm's objective is to simultaneously: select candidates with high utility, and level up the representation of disadvantaged intersectional classes. This naturally involves trade-offs and is computationally challenging due to the the combinatorial explosion of potential subgroups as more attributes are considered. We propose two algorithms to solve this problem, analyze them, and evaluate them experimentally using a dataset of university application scores and admissions to bachelor degrees in an OECD country. Our conclusion is that it is possible to significantly reduce disparities in admission rates affecting intersectional classes with a small loss in terms of selected candidate aptitude. To the best of our knowledge, we are the first to study fairness constraints with regards to intersectional classes in the context of top-k selection.