Learn Icelandic: Hún er með ofnæmi fyrir hnetum.

While hefur is used for owning objects, Icelandic uses the phrase vera með (literally "to be with") for physical characteristics, conditions, and illnesses. Since an allergy is a medical condition, you say hún er með (she is with).

Why is the preposition fyrir used here?

English says you have an allergy "to" something, but Icelandic says you have an allergy "for" it. Prepositions rarely translate one-to-one between languages, so it is best to memorize the combination ofnæmi fyrir as a fixed phrase.

What case is hnetum, and what triggers it?

It is in the dative case. The preposition fyrir can govern either the accusative or dative case depending on its meaning, but in the phrase ofnæmi fyrir, it always requires the dative. The nominative plural for nuts is hnetur, which changes to hnetum in the dative plural.

The current lesson is about prepositions governing the genitive case. Is there a genitive in this sentence?

Actually, no! This sentence is a great refresher on older rules. It uses með (which takes the accusative here for ofnæmi) and fyrir (which takes the dative here for hnetum). It serves as a reminder that even when you are learning new genitive phrases, everyday prepositions keep their normal accusative or dative rules.