The head of the Geluk school is not (as is commonly supposed) the Dalai Lama, but the Khri Rinpoche (or throne-holder), an office passed on by educational attainment, not by incarnation.
"Geluk." The Concise Oxford Dictionary of World Religions. . Encyclopedia.com. (November 7, 2018). https://www.encyclopedia.com/religion/dictionaries-thesauruses-pictures-and-press-releases/geluk
"Geluk." The Concise Oxford Dictionary of World Religions. . Retrieved November 07, 2018 from Encyclopedia.com: https://www.encyclopedia.com/religion/dictionaries-thesauruses-pictures-and-press-releases/geluk