The typical genius is of:
1. Very high intelligence, and
2. This very high intelligence is channelled into a specific ability, and
3. Motivation is also channelled into that specific ability.
Genius is rare because this situation is unusual. These three things must occur together.
High ability - but not all-round ability but instead that ability channelled narrowly, and motivation also channelled into that ability - rather than being spread across a range of activities.
In particular, most humans' interests for most of the time are social and reproductive - but this is not the case for geniuses
Most humans - in ancestral conditions - aim to be esteemed by other humans; to have status and power; in other words to optimise their reproductive potential - typically this is achieved indirectly and implicitly via sexual instincts, and so on.
But geniuses generally do not behave in ways that optimise their reproductive potential, and they do not want to. Geniuses want to do what they excel-at-doing even when this is low status and leads to problems. In fact, geniuses will do what they excel at doing, unless they are actively prevented.
Consequently the average reproductive success of a genius is very low.
In other words, considered at the level of the specific person: genius is pathological, an illness, an impairment.
More exactly, what makes geniuses work so hard and for so long at that thing in which they excel is that they have relatively impaired social and reproductive motivations.
The channelling of abilities and motivations away from reproduction and into a specific ability is a result of genetic impairment caused by mutations which weaken or close-off the usual social and reproductive lines of development.
Because genius is reliant upon mutations, there is no specific pattern to genius - it is extremely varied what a specific genius is good-at and interested-in. Rather the specificity comes from what the genius is not good-at, and not interested-in.
So, genius is a pathology which occurs occurs in a person of very high intelligence. (Anything which damages average reproductive success is a pathology.)
Yet the occurrence of geniuses seems non-random in the sense that the rate of occurrence of geniuses varies widely between places and at different times.
Furthermore, although genius is a pathology at the level of the individual; geniuses are sometimes massively adaptive in terms of enhancing the reproductive of their group.
Even a single genius can change the world. For example, a genius who invents a new tool or weapon may allow his 'tribe' of relatives to expand greatly. If that tool or weapon spreads widely, it can change the world. This probably happened many times in history - although usually the inventor's name is forgotten as with the stone axe, the spade, the wheel, the arch, the stirrup, and many other breakthrough innovations.
Therefore, it is possible that some populations may be selected (by their specific circumstances) to facilitate genius by allowing the kind of genetic pathology which leads to an increased frequency of geniuses.
(Mutations will happen spontaneously, this is merely a matter of passively allowing them to happen - it is a matter of mutations removing or impairing some of the mechanisms that regulate and repair certain genes or types of genes.)
Of course, most individuals who experience mutations in the genes which support abilities or motivations in the social and reproductive domains will not be geniuses; they will just be pathological, ill, impaired. These will die or leave behind no offspring.
(Most historical human populations were under very strong selection
against mutations by extremely high child mortality rates - so the wrong
kind of pathology or pathologies of excessive severity would be
filtered out by this mechanism.)
Only among some of the most highly intelligent individuals will the pathology have the desired effect of channelling developmental resources into useful but non-reproductive abilities and motivations.