In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction. In all the other Cases before mentioned, the supreme Court shall have appellate Jurisdiction, both as to Law and Fact, with such Exceptions, and under such Regulations as the Congress shall make.
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Yes, the Supreme Court has original jurisdiction. However, that does not mean the court must exercise its original jurisdiction (and, in practice, the court rarely exercises original jurisdiction). And that’s not even relevant here, since neither party has asked the court to take the case that way. The case has already been heard at the district level. The plaintiffs have asked the Supreme Court to take the case, on appeal, before it is heard at the circuit level. Obama, as a party to the lawsuit, is asking them not to do so. Original jurisdiction has nothing to do with this.