The police refuse to give details of the painting, as investigations continue. But it is not thought to be a well-known work, so it would have been relatively easy to sell the fragments as studies or smaller works by the artist or one of his followers.
This is no isolated case. Dividing stolen artworks into sections makes them easier to smuggle and the parts may be easier to sell than the whole work. This is particularly the case if the work is well known. Religious pictures packed with figures and different scenes provide especially rich pickings - faces, figures, an animal, even isolated hands, can be separated and passed off as individual works, or as parts of previously undiscovered but damaged paintings.
One source in the Italian police said that 'perhaps two or three out of every 100 works stolen' are being cut up. Another, in a different force, estimates that up to 10 per cent of large stolen paintings are cut up.
Although pictures have been cut up for centuries, as fashions changed or when a work was damaged, only recently has it been done for illegal gain. It is difficult to tell whether a work that is not well documented was cut yesterday or a century ago.
This could explain the fate of stolen items that seem to vanish completely, according to Philip Saunders of Trace magazine, which liaises with the police, the art world and even the criminal fraternity in recovering stolen art. Trace has a database of some 20,000 missing works. The average recovery rate, from up to pounds 3bn-worth stolen each year, is 3.5 per cent.
There is no evidence yet that any major works have been cut up, but thieves might take a knife to them if they could not find a buyer for the whole painting. Few believe in the 'secret collector' theory usually trotted out when a work too famous to sell openly is stolen. Roland Kollawijn of Sotheby's in Rome said: 'There may be one secret collector. But I don't see them as a major cause of this kind of robbery. A millionaire who can afford to pay thieves to steal great works can also afford to buy them on the legitimate market.
'When a Tiepolo was stolen from a church in Venice last summer, everyone suspected an American millionaire. But then it was found a few days later. It had been taken by a drug addict who didn't know what to do with it.'
If a secret collector hasn't got the Vermeer and other Old Masters stolen from a Boston museum three years ago, it is not entirely inconceivable that they have been divided and given forged provenances.
Caravaggio's Nativity with Saints Francis and Lawrence, stolen in 1969 from the Church of San Lorenzo in Palermo, is said by some to be in the hands of the Mafia. If not, perhaps the Saints Francis and Lawrence no longer share the same picture.
Julien Stock of Sotheby's in London estimates that the Caravaggio would be worth between pounds 20m and pounds 40m (depending on its condition). If the thieves tried to sell fragments of it, disguising them with some expert overpainting, perhaps saying they were from another version of the composition, they could be worth anything from pounds 10,000 for a single hand to pounds 3m for the angel hovering above the other figures.
But although Caravaggio painted versions of his pictures, they were not identical. Even if a master forger overpainted the fragments, X-rays and cleaning would give the game away.
Mr Saunders says that it is easy to divide up a painting: 'Just find an old frame and cut the canvas to fit.' To re-lay the canvas, you find some wood from the right period - from an old bookcase or wardrobe, for example. Concocting a story is even easier: 'It has been in a private collection for 60 years . . .' To indicate that it is the only bit of a painting to have survive a fire, he suggests that the edges of the fragment could be slightly charred.
As photographs are rarely clear enough to identify a fragment as coming from the original stolen work, 'it can be sold as a working study for the final work' - 'Everyone will be delighted at the discovery.'