As with any other light source, the position of an umbrella in relation to the subject will always have an effect on the shadows produced.
Generally speaking, you want the light source level with or somewhat above the eyes of your subject.
The higher your light source in relation to the subject the longer the shadows under the nose and chin will be.
Learn to study photographs and determine where the lights were positioned by the shadows. You can also judge this by looking at the catch-lights in the eyes.
I'll post some photos here and describe how they were lit, umbrella's were used in all of these photos.
Take notice of the shadows and catch lights. Then do some experimenting on your own and determine where to place your light source to achieve the results you want.
This was shot with two umbrellas, one for the main and one for rim-light.
As you can see by the shadows and catchlights, the main was positioned at camera right and a little above the models head. Notice the shadow under the chin and nose.
On this image, main light was umbrella positioned above model and reflector used for fill. The catch lights show the position of both.
Main light provided by a single umbrella. Rim light provided by another single umbrella;
This image was lit with a single umbrella. It lit both the model and the background.
If you look hard enough you'll notice a tiny pin-point catch light in the eye (right next to the larger catch light) that was produced by using an on camera flash to trip the main light. The oncamera flash was aimed at the umbrella and not at the model and as such didn't influence the light on the subject hardly at all.
It should be pretty easy to tell where the main light was in this photo...you can see the umbrella in her glasses. I had another light behind her and to camera left...this light was positioned even higher then the main light.
In the right hands, umbrellas can be great sources of light. It does take a bit of practice and knowledge to learn how to control the light they produce however.
I hope this helps.