Geminis function by using various spot removal cards to help establish a strong field presence in the later stages of the game. Dark Valkyria and other core monsters sit at a respectable attack status that stands above many other popular normal summon monsters. Heavy Knight of the Flame in particular gets an attack boost from Vampire Kingdom, which can apply a fair amount of pressure to Vampire opponents. Evocator Chevalier and Gemini Spark can target both backrow and monsters, putting opposing control/stun decks in a situation where they cannot afford to play a long waiting game. Geminis' primary weakness is needing to draw enough disruption to survive against more aggressive decks, and cards like Hey, Trunade! are a real concern.
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Due to Supervise's graveyard revival effect activating on a new chain, you don’t need to have a deceased monster at the exact point of Supervise hitting the grave to get its recursive ability, there just needs to be a monster in the graveyard before the new chain with Supervise starts. This means that if you use Evocator Chevalier’s effect sending Supervise for cost targeting your opponent’s card (with an empty graveyard), you can chain a card like Gemini Spark, Enemy Controller, or anything else that can put the Chevalier in grave to setup the effect of Supervise. When the new chain starts there will be a monster in the graveyard that Supervise can target.
If the activation of Supervise is negated, it will not allow you to use its effect to revive a monster from the graveyard. It will not happen even if you chain Gemini Spark (tributing the equipped monster) to the card that is negating Supervise’s activation. For example, if you equip Supervise onto your monster and the opponent responds by activating Wiz, Sage Fur Hire’s effect to negate Supervise and you chain Gemini Spark (tributing the equipped monster), the Gemini Spark will resolve all fine and dandy but Supervise’s graveyard effect will not trigger.