Genius Breeding Act

In any world with a basic (and we talk about very basic here) understanding of genetics, and of the advantages that intelligence can bestow upon an individual, somebody is going to get the idea to selectively breed the best and the brightest to each other to produce smarter and smarter people. When a leader, government, or other higher-up starts to command, pressure or incentivize the smartest people to pair off for breeding purposes, you have this trope.

Note the frequent pairing with plots about eugenics.

In Real Life, humans always breed back towards the mean, so expect eugenics only to decrease genetic diversity if they do anything at all. Genetic engineering won't get you far, too, as the many forms of intelligence (be it logical or emotional intelligence) don't appear to be hereditary. No encoding genes = no magic Einstein potion.

Compare Stalker with a Test Tube, Super Breeding Program, Shipper on Deck, Designer Babies. For genius cast members getting together in general, see Pair the Smart Ones.

No real life examples, please; As mentioned above, Real Life genetics don't work this way.

Examples of Genius Breeding Act include:

Fan Works

In the Relationships Series, The "Higher Ups" actually were pushing Yuuno to get with Nanoha and trying to keep Fate from being near Nanoha to push this through.

Film

The premise behind the 1994 romantic comedy I.Q. Albert Einstein does some Playing Cyrano when his niece is attracted to a garage mechanic but insists on marrying an intelligent stuck-up Jerkass so their children will have a high IQ. Einstein and his fellow scientists make the mechanic look smarter than he is—she eventually sees through the ruse but marries him anyway.

Literature

In Larry Niven's Known Space stories, the Earth is so overpopulated that in order to have children at all, one has to be extraordinarily talented (high intelligence, good teeth, superior eyesight, cancer resistance, etc.). A very few Einstein-level geniuses get Unlimited Breeding Licenses that basically allow them to have all the kids they want.
The Arisians of Lensman have been running massive breeding programs for millions of years to develop humans with enough mental ability (both in intelligence and Psychic Powers) to finally defeat the Eddorians. To do this, they have used infiltrators in human society, control over the Amplifier Artifacts that allow Lensmen to use their Psychic Powers, and outright Mind Manipulation to make sure that the right people breed with each other (and, even more importantly, that people who aren't supposed to breed before it's time don't). There's even a mention that the fact that humans breed back towards the mean isn't a concern, because a side-effect of the breeding program is that the mean (very slowly) approaches the result that the Arisians need with every generation. When it comes time for Kimball and Clarissa to be married and produce the Children of the Lens, virtually the entire Galactic Patrol and the universe itself seem to start shipping them together.
In The Number of the Beast, it's briefly suggested that the four members of the Gay Deceiver crew ought to have babies together, as they're all extremely intelligent and would presumably pass that on to their offspring. By the time we meet them in the sequel, The Cat Who Walks Through Walls, they have, and the kids are indeed geniuses.
Brave New World has a very developed version of this trope. Embryos are created in labs, and people are born into different classes: Alpha, Beta, Delta, Gamma, and Epsilon. These groups are engineered to have different intelligence levels both through genetic selection and differences in their artificial fetal environment; for example, an Alpha is made from Alpha gametes and incubated in an optimal fetal environment.
In the first Dune book, the Bene Gesserit had a breeding program and required specific members of the Sisterhood to participate. The goal wasn't generic smarts, but specifically creating a super-prescient Kwisatz Haderach. It worked.
In David Brin's Uplift series, humans use both genetic engineering and selective breeding to improve the intelligence of their uplifted dolphins and chimpanzees. Most chimps and 'fins have to apply for a license to reproduce. The ones with unlimited breeding licenses are the smartest and most talented of their generation. It's stated that most alien clans have similar or stricter breeding programs for their client races.
Mack Reynolds' Section G series novel Brain World. All of the initial settlers of the planet Einstein were required to have a minimum IQ of 130. They all bred with each other, and the result was a planet of geniuses with intellectual abilities far exceeding the rest of humanity.
In Ender's Game, this is mentioned as one reason the government allowed Ender to be born, despite being a usually illegal third child. His parents and siblings were so intelligent, they figured any future kids would have to be around the same intelligence, and they still needed their genius military commander.

Live-Action TV

In Doctor Who, Rattigan explains his master plan for a new world to the other Teen Geniuses he'd collected, and mentions that he's written up a breeding program. They are appropriately appalled.
In the first scene of the first episode of The Big Bang Theory Sheldon & Leonard go to a sperm bank that only accepts donations from people with high IQs.
One episode of Law and Order SVU dealt with smart and/or successful men getting sperm-jacked by an unscrupulous woman and her mother for use at a geniuses-only sperm bank.

Video Games

Sequence The entire plot of the game is the main leads are in a Batman Gambit arranged to make them fall in love and reproduce.

Western Animation

Futurama mentions a Genius Breeding Act from a time when aliens landed on Earth and forced the smartest members to mate continuously. Farnsworth was disappointed that the latest alien invasion wasn't going to involve this.