DNA Test on Mary to Show Descent from Jesus?

Rating: 5/10 (20 May 2006)

Perhaps I missed something in the movie (or at least in the book, which I haven't read), but it seemed like it was said that the "Holy Grail" -- Mary Magdalene's sarcophagus -- was important because it would provide proof via DNA testing that there was a living descendant of Jesus.
But (assuming Mary's sarcophagus contains Mary), wouldn't that just show that there is a living descendant of Mary? How would that show Jesus was the father? Not much is known about Mary; I doubt many (in the church or not) would be surprised one or the other whether she had children via another man.
It seems more than a small plot problem -- it makes the whole movie not make any sense.
Also, it sounds like the Priory had been trying to protect Mary's sarcophagus for two thousand years to preserve evidence of a bloodline. Did the medieval people think, "Hey, I bet one day there will be DNA testing..."?
Also, how would you know the body in the sarcophagus is really Mary Magdalene?