This is a good place to ask. Your sword says
藤原統行 Fujiwara Muneyuki
Possibly a smith from Bungo Province, 1600s. As you may know, there are many, many (many!) fake signatures in the sword world. Hard to authenticate just by looking at photos, and usually the sword would need to get looked at in hand by one of the authentication authorities. NBTHK in Japan is the gold standard.