Questions & Answers about Gætirðu spilað tónverkið?
I thought the verb geta takes a genitive object. Why isn't tónverkið in the genitive case?
You are right that geta governs the genitive case when used as a main verb meaning to mention or to guess. However, in this sentence, geta acts as an auxiliary (helper) verb meaning to be able to. When used this way, it takes a second verb instead of a noun, and it is that second verb that decides the case of the noun object.
Since geta doesn't decide the case here, what case is tónverkið in?
It is in the accusative case. The action happening to the musical piece is playing, and the verb spila governs the accusative case. Because tónverk is a neuter noun, its accusative form with the definite article attached looks exactly the same as its nominative form. If it were genitive, it would be tónverksins.
Why does the sentence use gætirðu instead of the present tense geturðu?
Gætir is the past subjunctive form of geta. Just like in English, where we soften can you to could you to make a polite request, Icelandic uses the past subjunctive to sound much more polite. The -ðu on the end is just the pronoun þú merged with the verb, which happens very often in questions.
Why is the verb spilað in this form instead of the infinitive að spila?
The verb geta is unique in Icelandic because it requires the following verb to be in the supine form (sagnbót), rather than the infinitive. The supine looks identical to the neuter singular past participle, which is why we must use spilað here instead of spila.
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