the Senate, where the math of majority can be hard to follow: The 63-seat chamber actually includes 32 Democrats, but one Democrat — Senator Simcha Felder of Brooklyn — caucuses with the Republicans, giving that party power.

Democrats hope to pick up at least one seat to seize the majority, but even so, they also will probably have to work with the Independent Democratic Conference, a breakaway group of Democratic senators who have often worked closely with Republicans.

But the Democrats are aiming larger, focusing on seemingly safe Republican seats like the Eighth District on the South Shore of Long Island. But because that district is held by Senator Michael Venditto, the son of John Venditto, Democrats feel that the electoral map has expanded.

On Thursday, Democrats began new internal pollingin the Eighth District on behalf of their candidate, John E. Brooks, to determine a path to victory. And on Friday, Mr. Brooks unveiled a four-point ethics plan, saying that “corrupt officials have lined their pockets at the expense of Long Island taxpayers, and people like Senator Mike Venditto have stood by and let it happen.”

As Mr. Gianaris suggested, the issue of corruption had already been in play in several Nassau races, including the Ninth District, which was touched by scandal last year as well: It is the former seat of Dean G. Skelos, a Republican who was the Senate majority leader before being convicted of federal corruption charges in December. The incumbent, Todd Kaminsky, a Democrat who won a special election in April, is a former federal