If a Corn plant (of any growth stage) is in the ground on the 28th of Summer, it will continue growing on the 1st of Fall as though nothing had changed. Fertilizer in the ground underneath a Corn plant on Summer 28 will not disappear on Fall 1.
Note that the game does not consider [[Oil]] to be an [[Artisan Goods|Artisan Good]], so it does not benefit from the [[Farming#Farming Skill|Artisan Profession]].