From an article posted on Ballot Access News, on July 13, 2012:
Virginia petitions to place independent presidential candidates, and the presidential nominees of unqualified parties, are due August 24. The Constitution Party petition will be submitted on Monday, July 16, and will contain approximately 14,000 signatures. The requirement is 10,000, with at least 400 in each U.S. House district.
When a group turns in a Virginia petition with more than the minimum number of signatures, election officials then start to check the signatures. Groups are permitted to then turn in additional signatures. Therefore, it is advantageous for a group to submit at least 10,000 signatures early in the process, because then they can learn how many more they need, if any, and avoid collecting signatures unnecessarily.
See the original article here.