IN THE AGE OF THE GARGANTUAN BLOCKBUSTER, it wasn't immediately clear that the story of a suicidal mathematician in wartime England would make for a successful movie. In fact, it wasn't clear that it would make for a movie at all.

In 2010, Graham Moore was a precocious 28-year-old author who had just written a novel about Sir Arthur Conan Doyle. At a cocktail party in Los Angeles, a producer named Nora Grossman mentioned to him that she and her producing partner were interested in making a film based on a biography of Alan Turing--the English scientist who is credited with developing the first computer but was punished for his homosexuality. Moore was immediately intrigued; he'd been interested in Turing's story since he was a teenager. "I have to be the one to write this!" he told Grossman.

She and her partner, Ido Ostrowsky, agreed, and Moore set to work. After he finished the screenplay, he called his agent. "'Hey, I have this script about a gay English mathematician who killed himself,' " Moore deadpanned to me, recalling that-because of the subject matter--he didn't expect it to be an instant success. But his agent loved the script, recognizing that Moore had managed to turn what could have been a morbid biopic into a riveting thriller. A few months later, Warner Brothers bought a one-year option to make the film.

But selling a screenplay is not the same as making a movie, as Moore would soon learn. Warner Brothers, like many of the major studios today, is largely in the business of making big movies, and the script, despite being very good, did not fit the mold of the tentpole franchises that might do well in, say, China. Moore wondered whether it would ever get made. "It would have been their lowest-budget movie in 30 years," he told me recently.

Nine months into Warner Brothers' year-long option, Moore got a call from Greg Silverman, then an executive vice president at the company, who gave him his script back on good terms and told him to "go make this as the small indie film that you always should have." Technically, Warner Brothers could have sat on the script for another three months, so getting it back when Moore did was a boon. Yet he knew the project faced an uncertain future. Many scripts bounce from studio to studio, cast and crew come and go according to availability, and even a great story can languish for years--or never get told.

But Moore had an important advantage. In 2011, shortly after Warner Brothers optioned his screenplay, it landed in the No. 1 spot on something called the Black List: an anonymous survey in which industry professionals name the scripts they liked the most that year. The Black List was started in 2005 by a 27-year-old film executive from west Georgia named Franklin Leonard, and has become an influential index of the most original and well-written--if not the most bankable--screenplays in Hollywood. Its power to...