Genes define
capacity for
intelligence.
My genes differ from that of a hamster. I am (generally) more intelligent than a hamster. The genetic difference between me and the small, furry mammal is responsible for this difference in intelligence. I like to think that even if the hamster had the same advantages in education and stimulation, I would still come out tops in the IQ stakes.
The real question is whether or not the natural variation in the human population is significant, or whether cultural effects dominate.
I believe the latter.
Nobody is surprised if I say I share cognitive characteristics or aptitudes with my immediate family. I take after my parents. I was lucky, and had more opportunity and encouragement to stretch my mind and pursue my interests. For me and my genes, life is good and my genes get to express themselves. Many other people don't get the opportunity to develop their mind. I reckon this latent capacity completely outweighs any genetically-linked variation of performance in simplistic IQ tests.