Yes, most cases of Adams-Oliver syndrome (AOS) are hereditary. There are currently six known genes - defects in any one of these genes can cause AOS. However, there are many patients that do not have identified mutations in any of these genes, which suggests that there are more genes that have not yet been identified. There are three ways in which a mutation can occur in these genes:
1) Autosomal dominant inheritance: mutations in the ARHGAP31, RBPJ, NOTCH1 and DLL4 genes have all been reported to cause autosomal dominant AOS. Autosomal dominant refers to mutations that have been inherited from one parent. A person who carries a defect in an autosomal dominant gene has a 50% chance of passing this on to their children.
2) de novo (new mutation): this is when the mutation has occurred spontaneously during formation of the egg/sperm, or at an early stage during development of the baby in the womb. In this case, the mutation is not inherited from a parent. However, an affected person then has a 50% chance of passing it on to their own children.
3) Autosomal recessive inheritance: mutations in the DOCK6 and EOGT genes have been reported to cause autosomal recessive AOS. Autosomal recessive refers to mutations that have been inherited from both parents. In this case, both copies of the gene must be defective to show signs of the condition. The parents of an affected individual each carry one copy of the mutated gene but they are typically healthy. Mutation 'carriers' with one copy of the defective gene carry a 25% chance of having children affected with the condition.
See https://ghr.nlm.nih.gov/condition/adams-oliver-syndrome for more information