Hi Beth,

If your ancestor was a Demartini from "Genoa" then most likely he was more specifically from Lorsica, a community in the Valfontanbuona, near Genoa, that supplied a disproportionate share of the "Demartini" in the US. When did your relative come over? If it was sometime around the turn of the 20th century, have you tried the online Ellis Island records? It doesn't cost anything. it's searchable, and gives you access to copies of the relevant passenger lists

Michael