I think Lawdog is correct in saying that the sharpness of the blade contributes more to it's cutting ability than it's curve.
As was mentioned before, the advantage in having a curved blade while cutting is in drawing the cut across the body, and increasing the edge that is used. For a simple 'try at home' experiment, we all know that using the 'sawing' action helps us cut things in the kitchen. A curved blade adds a little of this 'saw' to a cut with a sword.
One thing Lawdog neglected to mention, though, was the balance of a sword. That is one of the things that greatly differentiates swords. A good european longsword balances about 3-4 inches out from the cross guard, which makes it suitable for both offence and defense. With the circumstances of battle that the longsword was being used for, this was critical. This balance point lets the blade move fast enough to defend yourself, and with enough power to injure your opponent.
A scimitar, on the other hand, is balanced about 1-2 feet out from the cross guard. This balance point gives the sword tremendous power in its cut, but once it gets moving, it is very hard to change the direction of its swing, making it ill suited to defense. The scimitar is very akin to the modern machete. Good at cutting, but not good in a sword-only fight.
The katana is an entity a cultural world apart from the longswords and scimitars that saw each other often during the crusades. The katana was made as a vessel for a single, sharp edge that could be used on whatever it needed to be used for. With its balance point and curve, it cuts well, but it is straight enough and balanced close enough to be defended with. It might not be the best at each of these things, but it strikes a healthy compromise, and with the relatively low amount of steel in japan, one that makes a great deal of sense.
If you want to take a closer look at what cutting with a european longsword will actually do, AEMMA conducted a test (with discussion) a few years ago, and posted their videos online.
As a comparison, I have seen videos of the same tests being done with a katana, and the results are remarkably similar.
So, in conclusion, it is the sharpness of the edge, and to a lesser extent, the balance of the sword and curve, that affect the cut. The skill of striking with the node only maximizes the cut.
If you'd like to see for yourself, I second going to the FACT open house. Lots of fun, lots of cool toys. You can learn more about the open house here: