On a question of Kollár and Kovács

We answer a question of Kollár and Kovács by constructing a flat projective morphism to a smooth curve whose fibers are Cohen–Macaulay and reduced, whose generic fiber is smooth, and for which the first cohomology of the structure sheaf of the fibers is not constant. The main result of this paper is obtained by generative AI, particularly Chatgpt 5.5 pro and the Rethlas system.