Fact-checking requires retrieving evidence related to a claim under investigation. The task can be formulated as question generation based on a claim, followed by question answering.However, recent question generation approaches assume that the answer is known and typically contained in a passage given as input,whereas such passages are what is being sought when verifying a claim.In this paper, we present Varifocal, a method that generates questions based on different focal points within a given claim, i.e. different spans of the claim and its metadata, such as its source and date.Our method outperforms previous work on a fact-checking question generation dataset on a wide range of automatic evaluation metrics.These results are corroborated by our manual evaluation, which indicates that our method generates more relevant and informative questions.We further demonstrate the potential of focal points in generating sets of clarification questions for product descriptions.