How do you get a component from the tag using the user has selected from
theUI.SelectionManager.SelectTaggedObject("Select a curve", "", Selection.SelectionScope.AnyInAssembly, false, curveType, out type, out cursor);
The Curve class has a property called "OwningComponent" that should allow you to determine what component the selected curve belongs to.
The .SelectTaggedObject method returns a TaggedObject, not a tag (though you can get the tag from the TaggedObject if needed). You may need to cast the TaggedObject to a Curve object before accessing the .OwningComponent property as suggested by Steve.