In a Strange City by Laura Lippman
It was a case Tess turned down - but now she's caught up in a web of obsession and murder . . .
Every year on 19 January for the past fifty years an unknown visitor has left three red roses and a half-empty bottle of cognac on the grave of Edgar Allan Poe. When a prospective client asks PI Tess Monaghan to investigate, she refuses - after all, no crime is being committed.
But she does go to the 19 January vigil as an observer. In the freezing darkness she watches as two cloaked figures approach the grave, appear to embrace and then part. Then there's a gunshot and one is killed. Tess quickly learns that the dead man is not the regular visitor. So who is he? And why was he there? When it turns out that Tess's would-be client had given her a fake name, she knows she must try to find him.
Then an old friend from her past surfaces, claiming that the shooting was a homophobic hate crime, and things get even more complicated.