Is zakat obligatory on desert land purchased for the purpose of cultivation before it yields produce, and how is the zakat on the resulting fruits calculated when there is a dedicated percentage for the operating company?

There is no zakat on land not prepared for trade; rather, zakat is due on the zakatable produce from it. Zakat is obligatory on every staple food, whether grain or fruit, such as wheat, rice, dates, and grapes, according to the Shafi'i school, and this is the fatwa given, because staple foods are necessities. If the produce is zakatable and your share of it reaches the nisab (five wasq), then you must pay its zakat: half of one-tenth if irrigated by machine, and one-tenth if irrigated without a machine. Each partner pays zakat on their share if it reaches the nisab.