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So, I had this idea for an albino orc, who was thrown out of his clan for being different, and ended up being raised by some elves. These elves taught him the way of magic, and thus, there was Vajor Galailo, the albino orc wizard.

I usually mess around with HeroForge to create a visual for my character, and came across the option to embed a gem in his forehead.

So I came up with this idea: Vajor, still being an orc with a low wisdom score, keeps losing his arcane focus, a gem. After encoutering this problem multiple times, he decides to make sure he never loses it again, and embeds the gem in his own forehead.

Would this be allowed?

This would have some mechanical benefits for your character

If your arcane focus is physically attached to your body, this gives several potential advantages, the most obvious of which is that you cannot be forced to drop it. This would mean that, for example, a fighter's Disarming Attack would not be able to deprive you of your focus. Perhaps more importantly, if your party is captured and has all their items confiscated, it won't be possible for them to confiscate your focus (either that, or they will confiscate it very painfully).

In addition, you will be able to "drop" your arcane focus in order to grab something else without fear of losing it. For example, normally you wouldn't be able to stow your focus, pull out a weapon, and attack all on the same turn. Any character can do this by dropping their focus on the ground (a free action) instead of stowing it, but then they risk losing the focus if they aren't able to return to pick it back up later. With your focus embedded in your forehead, you will not run this risk, which gives you a little more flexibility in battle. Essentially, the hand with which you manipulate your focus will always be free to do other things in between casting spells. (I'm assuming you would still need to touch the focus with your hand in order to cast spells with it.)

All in all, these are minor advantages that probably won't come up very often, unless you specifically design your character around them by intentionally taking advantage of the ability to quickly switch what you're holding. If your DM finds these minor effecs acceptable, they could certainly allow you do run with this character concept. However, be prepared for them to say no if they feel it would be unfair.

In official Adventurer's League play, I'm pretty sure this would not be allowed.

No and yes

No, if you want it to be permanently affixed so that it grants mechanical advantages such that it couldn't be removed.

Yes, if this is just how you imagine and describe your character but that the mechanical sitation is no different than if you stored it in a pouch or on a string around your neck.