Democrats have a big problem in Illinois just two days after the primaries.  Scott Lee Cohen, who won the nomination for Lieutenant Governor on Tuesday, has an arrest for domestic violence on his record from just five years ago.  Police arrested Cohen for holding a knife to his ex-girlfriend’s throat:

Scott Lee Cohen — a pawnbroker who shocked state Democratic leaders Tuesday night by winning the party’s nomination for lieutenant governor — was arrested about four-and-a-half years ago and accused of holding a knife to a former live-in girlfriend’s neck, newly obtained court records show.

The misdemeanor charge against Cohen was dropped weeks later when the woman — who had just been found guilty of prostitution — failed to show up to testify, according to those records. …

Cohen’s Oct. 14, 2005, arrest came five months after his wife filed for divorce and convinced a judge to give her a temporary order of protection, records show. A status hearing in the divorce case took place Wednesday, hours after Cohen’s election-night triumph.

Cohen — who records show also had federal tax troubles that he says he has settled — denied in a written statement that he ever hurt the ex-girlfriend or his family. Cohen disclosed his domestic violence arrest when he announced his candidacy, but the details about the knife and prostitution case didn’t surface in the campaign, as Cohen was considered a longshot.

Democrats were going to have a tough year anyway, even in Illinois, thanks to the national unpopularity of the Democratic agenda and the debacle of the Rod Blagojevich corruption scandal.  His successor, Pat Quinn, is already unpopular in Illinois, and this won’t improve matters.  The Chicago Sun-Times reports that one of Cohen’s rivals for the nomination warned both Quinn and his opponent Dan Hynes about Cohen’s past, and neither of them took any action to push Cohen back to the sidelines.

Now Democrats have a problem in both races that could undermine the entire ticket in November no matter who gets the Democratic nomination for governor.  That kind of trouble could make it difficult for Alex Giannoulias to beat Rep. Mark Kirk, who has a six-point lead already in the latest Rasmussen poll:

Republican Mark Kirk holds a modest 46% to 40% lead over Democrat Alexi Giannoulias in the race for the Illinois Senate following Tuesday’s party primaries.

The first post-primary Rasmussen Reports Election 2010 telephone survey of the Kirk-Giannoulias race finds just four percent (4%) of likely voters in the state prefer some other candidate, while another 10% are undecided.

Among voters not affiliated with either of the major parties, the Republican holds a sizable 59% to 22% lead.

That’s a nine-point swing from December, when voters gave Giannoulias a narrow three-point edge over Kirk.  Giannoulias also has lower favorability numbers: 49% to Kirk’s 56%, and Kirk has a few more undecideds, 15% to 12%.  Barack Obama’s popularity has dropped as well, with his home state only giving him a 54% approval rating.  That means that Obama may not be much help for Democrats in Illinois this year — and he may not want to get too connected to Cohen in any circumstance.