A long-standing problem in evolutionary biology has been determining whether and how gradual, incremental changes at the gene level can account for rapid speciation and bursts of adaptive radiation. Using genome-scale computer simulations, we extend previous theory showing how gradual adaptive change can generate nonlinear population transitions, resulting in the rapid formation of new, reproductively isolated species. We show that these transitions occur via a mechanism rooted in a basic property of biological heredity: the organization of genes in genomes. Genomic organization of genes facilitates two processes: (i) the build-up of statistical associations among large numbers of genes and (ii) the action of divergent selection on persistent combinations of alleles. When a population has accumulated a critical amount of standing, divergently selected variation, the combination of these two processes allows many mutations of small effect to act synergistically and precipitously split one population into two discontinuous, reproductively isolated groups. Periods of allopatry, chromosomal linkage among loci, and large-effect alleles can facilitate this process under some conditions, but are not required for it. Our results complement and extend existing theory on alternative stable states during population divergence, distinct phases of speciation and the rapid emergence of multilocus barriers to gene flow. The results are thus a step towards aligning population genomic theory with modern empirical studies.