Billionaire art collectors rarely air their deals or dirty laundry out in public. But a battle between a pair of boldfaced names over a Giacometti sculpture is now playing out in the open, shining a rare light on a global market featuring unlicensed agents, multi-million-dollar handshake deals and pervasive secrecy.

The Chinese crypto entrepreneur Justin Sun and the American entertainment executive David Geffen are slinging written accusations at each other in federal court in New York, each claiming to be the rightful owner of Alberto Giacometti’s “Le Nez” (“The Nose”). Along a trail stretching from New York to Singapore to Paris and back to the United States, Mr. Sun says the sculpture was secretly sold by a dishonest adviser and sued for its return from Mr. Geffen, who recently struck back, accusing Mr. Sun of devising an elaborate fraud because he regrets the sale.

Adding to the tangle is an agent who might or might not be to blame, and might or might not be in a Chinese detention center.

At a 2021 auction in New York, Mr. Sun, who founded the crypto platform Tron in Singapore, paid $78.4 million for “Le Nez,” a midcentury work of bronze, steel and iron depicting a head, suspended in an open cage and boasting a very long nose.

Mr. Geffen, in turn, bought the sculpture last year while it was on loan to the Giacometti Institute in Paris.

But early this year, Mr. Sun sued, alleging that an adviser who sold it to Mr. Geffen, for $10.5 million and two unnamed paintings, had done so without his permission, forging documents and inventing a nonexistent Chinese lawyer to make the deal. The accused adviser was publicly connected with Mr. Sun and his art purchases for years and spoke about advising him on his bid for the Giacometti and other pieces.