Back in January, when a little-known Republican candidate named Scott Brown broke a long drought for his party, taking the Senate seat held for 47 years by the late Edward M. Kennedy, hope flowered among Republicans in this part of the state.

The analysis showed some 60 per cent of voters in the 10th Congressional District, which covers the Cape and Islands, went to Brown. Absent the influence of a strong incumbent, the 10th was winnable for Republicans.

And in the upcoming November midterm elections for the House of Representatives, the seat will be without a strong incumbent; after seven terms, Rep. William Delahunt, a Democrat, decided earlier this year not to run again.

The G.O.P. is targeting the seat, the only open seat in Massachusetts this term, as its perhaps best chance of a win in the state.

All of which helps explain why no fewer than four Republicans are in the contest for next Tuesday’s primary for the district, and why the contest for the 10th district will be the most closely watched of the 13 contests Island voters will determine in Tuesday’s state primary.

Polls across the state are open from 7 a.m. to 8 p.m. on Tuesday. All voters may participate even if they are not registered supporters of either party. But they can fill in only one party’s ballot.

The four Republican hopefuls for the 10th congressional district nomination are: Robert E. Hayden 3rd, a former Plymouth district attorney; Raymond Kasperowitz, an accountant and small businessman; Joseph Daniel Malone, former state treasurer; and Jeffrey Davis Perry, a four-term state representative, lawyer and former police officer.

All are touting conservative credentials, stressing smaller government and campaigning against out-of-touch Washington politicians.

Mr. Perry has a state legislative record of tough positions on illegal immigrants and opposition to health care reform at the state level.

On the Democratic side, there are two candidates: Robert A. O’Leary, the current Cape and Islands state representative, and William R. Keating, Norfolk district attorney.

Mr. Keating is the only candidate in favor of Cape Wind. Mr. O’Leary, while opposed to Cape Wind, championed the state Oceans Act, which provides for commercial-scale wind generation in waters close to the Vineyard.

After the contest for federal representative, the next most closely-watched primary race will likely be for state senator. Here the Democrats have two hopefuls with contrasting backgrounds: Sheila R. Lyons, a Barnstable county commissioners and social worker, and the high-profile head of Cape Air, Daniel A. Wolf. She stresses community involvement; he stresses business acumen.

The Republicans also have two candidates, of similar backgrounds: James H. Crocker and Eric R. Steinhilber both have backgrounds in real estate. Mr. Steinhilber also works as development director for the Alliance to Protect Nantucket Sound.

For the office of treasurer, Democratic primary voters have a choice of two well-established party players: Steven Grossman and Stephen J. Murphy. One of them will face Republican Karyn E. Polito in November.

It’s a bigger field for auditor, with Suzanne M. Bump, Guy William Glodis and Mike Lake competing for the Democratic nomination, and Mary Z. Connaughton and Kamal Jain vying for the Republican nomination.

The most hotly contested Democratic field is for councillor, for which five are running, among them an Islander — Thomas J. Hallahan of Oak Bluffs. There are two Republican hopefuls.

Most of the other contests on Tuesday will be one-horse races. For governor, it is the Democratic incumbent Deval L. Patrick, and the Republican Charles D. Baker. For lieutenant governor the Democratic candidate is Timothy L. Murray; the Republican candidate is Richard R. Tisei. Secretary of State William Francis Galvin seeks the Democratic nomination, while William C. Campbell seeks the Republican one.

State attorney general Martha Coakley, who was blamed for running a poor campaign against Scott Brown in January, is up for re-election. There is no Republican candidate on the state primary ballot.

Likewise for the state representative for the Cape and Islands, where incumbent Democrat Timothy Madden is the lone candidate on the primary ballot. Dukes County Sheriff. Michael A. McCormack, a Democrat, is the lone candidate on the primary ballot, although he faces a challenge in November from retired state police Sgt. Neal Maciel, an independent who is also an Islander.

Polling places in Island towns on Tuesday are as follows: Aquinnah, old town hall; Chilmark, Community Center; Edgartown, town hall; Oak Bluffs, town library; Tisbury, American Legion Hall; West Tisbury, public safety building.