As I understand it, Fumbari is a word used to describe something that dramatically gets wider as it approaches its base. For example it can be used to describe the so called 'horse stance' that some martial artists use where their legs are wide apart but the top of their body is not. When you are talking about the japanese sword, Fumbari refers to the shape of the blade as it approaches the machi, or the notches where the habaki is slid up to.
A diagram showing the base of the sword and the swelling that is referred to as 'fumbari'.
Tachi are one type of japanese sword that always have fumbari. Being a longer and overall more slender sword, the fumbari no doubt added strength as well as affected the balance of what would otherwise be quite an unwieldy sword especially if it was used from horseback. Katana on the other hand, being much shorter don't actually have or need fumbari*. It isnt unheard of however for an instance of a katana to have it but this seems to be the exception rather than the rule and occurring only with modern reproductions. In years long past however, there may be another reason that katana do not exhibit this sudden tapering.
* This isn't actually 100% true. It would be more accurate to say that katana do not have enough fumbari for it to be immediately noticeable. so little in fact as to look as if none exists at all.
One reason that katana may not show fumbari might be that the blade started as tachi and has undergone shortening or suriage. This is usually done at the base of the blade so the entire tip will not have to be reworked. When this was done, if there was any fumbari present this would of been the first amount of metal that would have been removed. This type of shortening was done simply to reduce the overall length of the blade and such a blade could be worn afterwards as a katana would be.
Fumbari is often paired with koshi-sori or curvature that starts closer to the base of the blade and is something that was almost always seen with tachi. Shortening the tachi would most certainly remove this curvature along with the fumbari. It's easy to see how the shape of a katana strongly resembles the shape of tachi that had undergone suriage, in fact a lot of japanese sword enthusiasts use the term 'fumbari' to explain whether or not the sword has been shortened or not rather than to describe the actual shape of the sword.
As much as fumbari had a certain desired aesthetic, the real purpose was to keep the weight of the sword closer to the base and to add strength and stiffness to the blade. Tachi took a real beating in combat especially when you consider that they were used against armoured combatants and often from horseback. This additional strength was an absolute necessity in order for them to survive for more than five minutes.
From a modern target cutting perspective, fumbari is simply not necessary. The length of the blades that we tend to use, the targets we cut and the modern materials used in the construction of our swords mean that even if fumbari were present on historic examples of katana, we simply don't need it in this day and age. I find that if the weight of a sword doesn't feel right, this can be changed by altering the tsuka and/or the koshirae, and by simply shifting the weight further from or closer to the leading hand, we can tailor the sword to our personal needs.