Does the Supreme Court has jurisdiction over state courts?

The Supreme Court only has two types of jurisdiction. There is original jurisdiction and appellate jurisdiction. SCOTUS review of state court judgments concerns its appellate jurisdiction as originally conferred in the Judiciary Act of 1789 and since amended, now codified at 28 U.S.C. 1257 (2000). Unless a case is of the type specifically identified in Article III, sect. 2, cl. 2, the Court takes the case by way of a petition for a writ of certiorari, as specified in 28 U.S.C. 1254.

Since the term general jurisdiction is usually associated with determinations in civil procedure of whether a state court can require a person to defend against a suit (see in personam jurisdiction pre-International Shoe, and as I recall that problem has been since resolved through the adoption of minimum contacts analysis and the evolving jurisdictional analysis in the 20th Century) or is based on the fact that an event or suit arises within the geographical area for which the court is responsible, I doubt it's general jurisdiction. Suffice it to say that so long as a person has certain minimum levels of contacts with the forum state, a court can acquire jurisdiction and thus require the defendant to defend against the suit else be held liable through default.

Plenary jurisdiction, on the other hand, connotes a form of complete control over a subject. Federal question jurisdiction is usually a form of plenary jurisdiction of the federal courts (thus why we allow cases raising a federal issue by the plaintiff to be filed initially in federal courts and why we allow removal of federal issues from state courts into federal court... but see the Mottley well pleaded complaint rule). Because the court has both a plenary and non-plenary docket, and review of state court judgments falls in both categories, jurisdiction over state court judgments is not an instance of plenary jurisdiction.

While The Court has a plenary docket, but that merely connotes that the Court is taking the case for briefing, oral argument and the full process of hearing and deciding a case/controversy. Other cases accepted on certiorari may be accepted on briefs alone, with argument scheduled only if the case presents a different issue than the Rule of 4 thought initially or some other issue or circumstance arises.

Since SCOTUS review of state court matters is authorized by Congressional statute, and extends only to the federal issue implicated (as per Michigan v. Long and others), SCOTUS jurisdiction in this instance would appear to be an exercise of its appellate jurisdiction.