"During the next semester we will study authors." Is this quote a prepositional or a gerund phrase?
Let us just briefly remind ourselves of what a gerund phrase is. A gerund phrase is a phrase that will begin with a gerund, or an -ing word (which is a verb), and will also include modifiers and/or objects. What is important to note about Gerund phrases is that they always count as nouns. Don't mistake a present participle for a gerund phrase, which will function as a verb.
If we have a look at the example you have given, "during" is definitely not a gerund. Although it is a -ing word, it is not a gerund, and is actually a preposition as it refers to time. Therefore we can see that the phrase you have quoted is a prepositional phrase rather than a gerund phrase.
A prepositional phrase is a phrase that shows the relationships between subjects and other nouns. In this case, it is talking about the time involved with the coming semester (the object of the preposition).
To have this act as a gerund (or act as the subject of the sentence in verb form), it has to act as a noun in the sentence: Playing soccer is something that Lucy enjoys.
'Playing soccer' is acting as a noun and in this instance, the subject. You can check that by asking 'what is something that lucy enjoys?' since the gerund also acts as the subject.