my is an adjective, and adjectives modify nouns, not verbs. So,
Ex: That would be my killing
Ex: That would be my kill
Gerunds--because they have verbal qualities--can take an object as well as adverbs:
Originally Posted by unregistered
Ex: That would be my killing her slowly. The gerund killing takes the pronoun her as its object, and the adverb slowly modifies the gerund killing.
<gerund + object>
Non-gerundival nouns, like kill, can also take objects, but they are prepositional in form:
Does that help?
Ex: It's the kill of the day
<non-gerundival noun + object>