Is there any genetic reason for the difference in size between the upper arm and lower arm (i.e., the forearm)?
In most women, it seems that the upper arm is larger in circumference than the forearm. However, in males it seems to vary - some men (like myself) have an upper arm with a diameter smaller than their forearm, while for example in body builders the opposite may be true, as they often have large biceps.
The first answer explains that proximal limb segments do more "work" * using quotes to be careful w.r.t. technical definitions of work. Genetically, this is correct--it's the ancestral condition for tetrapods. It is also worthwhile to answer why it's beneficial for why proximal segments get bulky to do this work, instead of their distal counterparts -- i.e. why did natural selection favor this arrangement in the first place.
There is a biomechanical reason for distal limb segments to be less massive than proximal limb segments, especially in tetrapods: That moving masses farther away from the center of rotation, when force is applied between the center of rotation and the mass, takes a LOT more force.
Let's consider the arm from your example. We'll treat the forearm as a "load" which is supported by the upper arm and shoulder.
In this case, the whole arm is a Type 3 lever, with the force applied (by upper arm and shoulder) inbetween the load center of mass and the fulcrum (joint). Image from https://alexeinstein.wordpress.com/2014/09/03/lever-of-human-body/:
In the picture, it's showing the hand (could be a weight in the hand, too) as the load, but for an empty hand the center of mass of the lower arm is probably slightly more proximal (but not more proximal than the muscle insertion).
In order to support the load that is the lower arm, the torques about the joint need to be balanced. So the torque produced by the muscles (by force AF in the diagram), needs to equal (or for bicep curls, exceed) the opposing torque produced by the mass of the load.
Torque is the product of the rotational force acting perpendicular to the lever and the distance along the lever to the fulcrum. (Reference is Wikipedia, but there's no argument about the definition of torque.)
Lets make up some numbers, and work through the example. Say that the weight of the lower arm (R in the diagram) is 10. And the distance from the center of mass to the elbow is also 10. Then, the biceps in this image needs to produce an opposing torque of at least 100. But the insertion of the biceps is very near the fulcrum, say at a distance of 1. So it needs to produce 100 times the weight of the load to create a balancing torque. For most joints in the body, the muscle insertion is nearer to the joint than the load--so the muscle will ALWAYS need to produce more than the load. So you can see how having a lighter forearm would be of benefit.
Now let's consider the whole arm, relative to the shoulder. I drew a picture for us. 
Suppose we only need 10 muscle mass to do day to day moving of things. Let's imagine a world where that 10 muscle mass is in the lower arm. If we have an upper arm of length 10, and a lower arm of length 10, and we hold our arm perpendicularly in front of us (so the center of mass of the 10 muscle mass is approximately at the middle of the lower arm(15 length from the shoulder), the weight of the arm is generating a torque of 150. This needs to be matched by the rotator cuff muscles, and it's a lot. The shoulder girdle insertion is quite close to the joint (let's say 2 length from the shoulder), so these muscles must produce 150 Torque / 2 length = 75 force perpendicular to the arm -- which means even more force overall, because the muscle is not acting perpendicular to the arm. That's a lot of force. (PS: red is showing lines of action of forces)
Now, suppose instead we put the 10 required muscle in the upper arm. (Another drawing for you. 
We'll say the center of mass of the whole arm is now probably just before the elbow, at length 9 from the shoulder. The whole arm is producing a torque of 10 * 9 = 90 at the shoulder. The shoulder girdle muscles (with the same insertion distance of 2) must produce 90 Torque / 2 length = 45 force perpendicular -- much less. This holds in general--for a type 3 lever, the muscles will always have to do more work if the load is farther away from the joint.
Thus, placing muscle mass distally means the center of mass of the entire limb s farther from the body -- which means the body must produce a lot more force to support the mass than if the load is closer to the body. This is the same reason why it is easier to carry a heavy backpack in your arms than at the end of your arms. On an evolutionary time scale, it makes sense that the bulk of muscle mass has been placed closer to the center of mass.
Yes, there are genetic reasons for this. But your observation is either incorrect or abnormal.
The shoulder (and hip) muscles do the heavier work, thus have larger (and bulkier) muscles. The more distal the muscles, the more involved with finer motor skills (yes, even in the foot), so we're after finesse here, not strength. You can pick up 100 pounds primarily because of your upper arms. You can curls weights (not 100 pounds for most humans) because of your distal arm musculature. You can grip the weight because of your hand and lower arm muscles.
It's like this for most vertebrates. Have you ever seen a dog, cat, horse or other animal with skinny "thighs" and fat "lower legs"? A horse's thighs (between their back and their knee) are huge, their gaskins (between the knee and the ankle, also called the hock) are prominent, the rest barely visible.
Here is a Grizzly Bear:
Here is a frog:
This is the plan for almost all terrestrial vertebrates, including humans. Certain vertebrates may have different musculature depending on how they live.
If it is true that your forearms are larger than your upper arm, then it simply means you are using your forearms much more than your upper arms, something not common for man before the present age.
Horse: http://thinklikeahorse.org
Grizzly: The Grizzly Bear
Frog: antique drawing, French.
