Beat buttermilk, eggs and dill in a medium bowl. Combine flour, cornmeal, sugar, baking powder, baking soda and salt in a large bowl. Cut butter into dry ingredients until crumbly. Add egg mixture and cheese. Stir to mix well. Knead dough until it holds together. Pat dough to 1 inch thickness. Cut with biscuit cutter. Place on lightly greased baking sheet. Bake at 375 degrees until done, about 25 minutes. Yield: 12 biscuits.