If left alone, the pigeon population limits itself to whatever the nesting sites will support. That is why we are not up to our necks in pigeons. That is why habitat modification works so well. Think about it. The reason most endangered species become that way is because man is destroying their habitat.
Poisoning is also indiscriminate.
Poisoned birds can fly far after ingesting their last supper. Cats, dogs, and other nontarget animals, or even children, could get sick or die from eating the lethal carrion. On a different note, but an important one, many birds and other animals succumb to diseases from tainted water. You will help save a lot of lives. Foremost, I would wish that everyone could share the joy and friendship I get from our nonhuman earthmates.