Former Virginia Gov. Tim Kaine took Twitter by storm today when numerous people tweeted that he told his University of Richmond class that he was going to run for Senate. The Democratic National Committee poured cold water on the story but said Kaine was “increasingly likely” to run. When Sen. Jim Webb announced he was retiring, that left Kaine as the strongest candidate if Democrats are going to retain the seat. Former Sen. George Allen, the man Webb beat in 2006, is the Republican frontrunner.

The bench after Kaine isn’t very deep, so the DNC’s enthusiasm for a Kaine candidacy is obvious. So are the reasons for Kaine’s hestiation: Webb barely beat Allen in a great Democratic year even after Macaca-gate, and Virginia has trended back to the Republicans in 2009 and 2010. Kaine, like Webb and Mark Warner, was elected as a moderate. But he has since been DNC chairman and thus identified with the entirety of the Obama agenda. (Webb’s pro-Obama voting record was likely a factor in his retirement.) So even with their strongest candidate, the Virginia Senate race should be an uphill climb for the Democrats unless political conditions change again or there are Republican errors.