Individual differences in educational attainment are linked to differences in intelligence, and predict important social, economic, and health outcomes. Previous studies have found common genetic factors that influence educational achievement, cognitive performance and total brain volume (i.e., brain size). Here, in a large sample of participants from the UK Biobank, we investigate the shared genetic basis between educational attainment and fine-grained cerebral cortical morphological features, and associate this genetic variation with a related aspect of cognitive ability. Importantly, we execute novel statistical methods that enable high-dimensional genetic correlation analysis, and compute high-resolution surface maps for the genetic correlations between educational attainment and vertex-wise morphological measurements. We conduct secondary analyses, using the UK Biobank verbal–numerical reasoning score, to confirm that variation in educational attainment that is genetically correlated with cortical morphology is related to differences in cognitive performance. Our analyses relate the genetic overlap between cognitive ability and cortical thickness measurements to bilateral primary motor cortex as well as predominantly left superior temporal cortex and proximal regions. These findings extend our understanding of the neurobiology that connects genetic variation to individual differences in educational attainment and cognitive performance.