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I have always interpreted the Investiture of Flame spell as having a flamethrower effect, similar to Aganazzar's Scorcher, but one of my players pointed out a different interpretation.

Here is the relevant segment of the spell:

You can use your action to create a line of fire 15 feet long and 5 feet wide extending from you in a direction you choose. Each creature in the line must make a Dexterity saving throw. A creature takes 4d8 fire damage on a failed save, or half as much damage on a successful one.

My player pointed out that the spell as a whole is Concentration, up to 10 minutes, while no duration has been specified for the line of fire. That has led them to believe the lines of fire stick around.

I believe this is just poor wording by the writers, who were assuming the reader would interpret it as an instantaneous effect, rather than a hazard-creating spell because such spells almost always have the clause "deal n damage when a creature begins its turn there or enters it for the first time on its turn." Nevertheless, I wanted to double-check to make sure I wasn't severely underpowering this 6th-level spell.

How long does the line of fire last? Is it instantaneous, or does it remain there until the spell ends?

Your assumption is correct and is backed up by what you pointed out, specifically that there are no rules that match continuous effects that govern what happens when a creature starts its turn in the fire. If this was to be anything aside an instantaneous effect it would have rules governing what happens to creatures that start their turn in the fire like all other continuous effects that have a "damaging" area.

In fact, right in this spell you have an example of the 'what happens on the creature's turn' text and it would have the same type of wording on the line of fire effect we are discussing if it was meant to be continuous.

Any creature that moves within 5 feet of you for the first time on a turn or ends its turn there takes 1d10 fire damage.

To further back this up, please see Create Bonfire, Incendiary Cloud, and Wall of Fire as each one makes a continuous effect of fire and each one you can see the rules that govern what happens when a creature ends or begins their turn near it.

The individual lines of fire don't have a duration, they only exist as an immediate effect of your action. The other effects of the spell remain: the immunity to fire and resistance to cold, the automatic damage to creatures near you, and the ability to continue slinging those lines of fire for 100 combat rounds.

(Side note: since the line of fire is just an action, you can ready it, if you're looking to wait for someone to move so you can catch multiples in the same line. Nothing like a line of flame to interrupt an enemy's turn!)