Can we all admit to the inability to convince each other of our differing views?
And, for the sake of gathering information pertinent to the subject, realise that it may well be the case that none of us have THE definitive answer to the question?
I ask this on the assumption that your educational institution and Professor would appreciate a balanced survey of the differing views of all who have an interest in the question, and why those views are held, rather than a pedantic rant that assumes its own infallibility and disallows any contrary argument.
I'd prefer your assignment qualified for a distinction, not merely a passing grade.
I agree; however, I see no place for faith based assumptions in such a paper. I look at this as a historian. I want to see evidence for the claims that are being made, and not simply tradition that was passed down later on, by people who already had faith of the claims they were supporting.
 
There are a variety of good reasons to read the genealogy preserved in Luke as being Mary's.
Firstly, it was accepted as such by the early church fathers. see Ireaeus XXII
ANF01. The Apostolic Fathers with Justin Martyr and Irenaeus | Christian Classics Ethereal Library
Irenaeus is in no doubt as to whose genealogy is recorded in Luke and as to why it is there recorded.
As to why this was later obscured is another question, but suffice to say Mary's lineage is there, and it is there for a reason, it means something and shows a link to David.
I'm not very interested in what Early Church Fathers have to say in this case though. They dealt in apologetics. For the most part, they aren't always even basing their information on facts. More so, Christians don't agree with everything they wrote anymore.
The Gospel of Luke is enough for me. Looking at the passage, it is clear who the genealogy is. It specifically states that it is of Joseph. That is why they use Joseph as the starting point, and tell us that he was the son of Heli. There really is no wiggle room here. We are told that Jesus is the son of Joseph (or at least that was thought to be the case), and that Joseph was the son of Heli, and so on and so forth. There is no mention of Mary. Not even an off hand suggestion.
More so, there is no reason for a genealogy to go through the mothers side. It would show nothing, as the significance was through the father's side (even if it was just an adoptive father). And there is no reason to mention Joseph in that account unless it was his genealogy. Logic, and a brief knowledge of Jewish customs shows us beyond a doubt that the genealogy in Luke is that of Joseph. Again, that is why it specifically states that.
 
The same passage from Irenaeus quotes Paul, the very verses that you fragment.
That Irenaeus and I read the verses in the same way shows that the long held understanding of those verses has remained consistent.
I have not added to their content.
Ireneaus was wrong about the case above though. So why should we assume he is right about what Paul is stating? I can point out many scholars, throughout history, who would agree with my reading, but that really shows nothing. In order for you point to be valid, you would have to show that was the general consensus up through the ages, and not just that of one Church Father who is known to have twisted the facts in order to support his cause.
And since Paul never mentions anything of the such, nothing about Mary, or Jesus having a divine birth, I see no reason to assume that is what Paul is talking about. Without the Gospels influencing your thought, it is highly unlikely that you would see that verse in the same light.
 
I do not rely on tradition.
There is no reason to assume that Paul was isolated from conversation with the other Apostles, any argument that assumes such is flawed at its root, imo.
You are relying on tradition. Your use of the Church Fathers is part of that tradition.
We have no evidence that Paul ever met with the writers of the Gospels. The writer of the Gospels were not the Apostles. That idea is part of the Christian tradition.
The only individual who may have known Paul, from the evidence we have, is that author of Luke, and that is even debated. As for Mark, Matthew, and John, there is no suggestion that those authors knew of Paul. If you want to hold such an idea, you need to show a reason to actually believe such. You need to provide some evidence for such a belief. As it stands, there is no evidence, and thus it is not logical to assume that they knew each other. That is making an unfounded assumption, and I see little room for that in a historical matter.
 
Paul says that Jesus was born of a woman and he also says that Jesus was born by the direct agency of God.
You seem to be unaware of the duality of Jesus' nature as elaborated on by Paul.
The resurrection is Paul's proof that Jesus was born with a dual nature as it is also, most clearly so, in John.
 
Paul never states anything about the direct agency of God in the birth of Jesus. I am aware that Paul had an idea of Jesus having a dual nature. One that was physical, and one that was spiritual. However, there is no suggestion in Paul that the two crossed. Paul, for the most part, talks of Jesus after the resurrection. Paul is interested in a post-resurrection Jesus that is spiritual.
When Paul speaks of Jesus as a physical being, during the life of Jesus, nothing miraculous is being described. Very little is actually mentioned. We are told that he was born of a woman, born of the flesh, born under the law. To say anything else from Paul about the birth of Jesus is adding additional material to Paul. Because as it stands, you can't show with Paul the idea Mary was a virgin (that never even comes up), or that Paul thought God impregnated Mary. Neither idea is stated or suggested.