Looks like they have jurisdiction "upon the streets, sidewalks, and highways, immediately adjacent thereto"
From ยง 23-234. Powers and duties; jurisdiction.
http://leg1.state.va.us/cgi-bin/legp504.exe?000+cod+23-234
(i) upon any property owned or controlled by the relevant public or private institution of higher education, or, upon request, any property owned or controlled by another public or private institution of higher education and upon the streets, sidewalks, and highways, immediately adjacent thereto,