This article includes SPOILERS for Crisis On Earth X Parts I & II
Lucky for Oliver and co, Arrow brought out a kryptonite arrow in Crisis on Earth X - but how did he get it? Crisis, widely touted as the most ambitious Arrowverse crossover thus far and easily the one closest in tone and scale to the Silver and Bronze Age DC Comics "event" crossovers they're inspired by, certainly made good on the promise of surprises, twists and clever character beats for devoted fans: Evil Oliver and Evil Kara are an item? Alex Danvers hooks up with White Canary? Tommy Merlyn (briefly) back? Felicity rejecting a proposal from Oliver - loudly? The whole thing going down at a superhero wedding - infamously one of the most attack-prone venues in the genre? Somebody was doing their homework.
Most interestingly of all, the logic for the (somewhat controversial) presence of the Earth-X villains themselves - Nazi-aligned incarnations of the familiar main heroes of the other CW series, hailing from an alternate Earth where the Third Reich won World War II and the setting for the Ray animated series - had an unexpected logic behind it; Overgirl (aka "Nazi Supergirl") is dying and in need of a heart transplant, and they're looking to steal a replacement from her non-Nazi counterpart while she's away from her own universe visiting Arrowverse-proper for Barry Allen's wedding.
With two episodes down and the story set to conclude with Tuesday night's episodes of The Flash and Legends of Tomorrow (though the whole enterprise is this time structured much more like a single miniseries than a crossover) there were also plenty of questions for fans to parse over as well, but one likely stood out more than most; during a skirmish with their evil counterparts, Oliver surprisingly managed to get one, er... "over" on Overgirl when he pulled a new trick out of his arsenal - a Kryptonite arrow! Clever gag, both in terms of the in-show reality (he literally has it "In case an evil you ever showed up," Oliver matter-of-factly explains when Kara asks why he'd have one of those) and as a shout-out for fans who'd recognize a callback to Green Arrow's role in The Dark Knight Returns, but it also raises a big question: where did he even GET Kryptonite?
Setting aside the fact that the glowing green - or blue, or red, or silver, or sometimes pink but probably not anymore because that joke now seems in poor taste - space-rock is generally presented as extremely rare and difficult to produce artificially (in Dark Knight Returns, Bruce Wayne describes the version he gifts to Green Arrow as having "cost millions" and taken "years" to synthesize), it should be next to impossible for Oliver Queen to have gotten it under any circumstances given that it has no reason to exist in his universe. Or, at least, it shouldn't be anywhere near Earth where he (or anyone else) can acquire it.
Though it really only ever comes up during these crossover events, Supergirl takes place in a different reality than Arrow and Flash (and isn't generally on the agenda for the time and reality-jumping Legends, either). Originally created as a standalone series with no overt connection to any other DC TV or film continuity for the CBS network, Supergirl's extra-universal adjacency to the rest of the crew was established during a surreptitiously-launched crossover with Flash in its first season that employed DC Comics' well-worn "multiverse" concept to set the story in motion. Despite Supergirl's jump to the CW with the other series, that universe-separation has remained intact - presumably in order to help the other shows avoid having to constantly answer why Supergirl or her equally-powerful cousin don't just solve every problem that comes up.
That means that, at least as far as anyone knows, Kara Zor-El and Kal-El never landed on Earth in this timeline and, indeed, it's possible that Krypton either didn't explode at all or that nothing from it ever reached the planet - and since "Kryptonite" is traditionally explained as being cosmically-irradiated remnants of Krypton scattered as debris by the explosion, this begs the question of how it exists for Oliver to have found it and made a special arrow out of it in the first place.
Granted, this being a superhero series playing around with alternate timelines and dimension-jumping-through-running-very-fast, it's not as though Arrow need adhere to any kind of real-world logic about these things. If an explanation is deemed to be required (which is itself an open question, given how the crossovers tend to be the place where CW DC creators get to openly flaunt the genre's "just because"-sim) one can likely be conjured easily enough. It's conceivable that synthetic Kryptonite isn't as difficult to create here as in other continuities, for example, and it would be just as plausible for the series to suggest that he merely borrowed it from the reality-jumping Legends team at some point offscreen after the previous crossover introduced him to Kara.
Or, perhaps we're getting an early tease at bigger integrations to come for the CW/DC project. That's certainly going to be a theory many fans will gravitate to the longer this goes unanswered, particularly given how much Flash has come to lean on callbacks to Flashpoint, an event comic that depicted (among other things) a supposedly Superman-less DCU Earth where the Kryptonian visitor actually had arrived but had been immediately imprisoned and covered-up by the government. At this point, all that's certain is that the Arrowverse rules can change at any time - and it won't always be to the benefit of the good guys.
Crisis On Earth-X wraps up on Tuesday with The Flash @ 8pm ET and DC’s Legends of Tomorrow @ 9pm ET on The CW.