On a question of Koll\'ar and Kov\'acs

TL;DR

This paper constructs a flat projective morphism with specific fiber properties, demonstrating non-constancy of the first cohomology of the structure sheaf, using AI tools like ChatGPT 5.5 pro and Rethlas.

Contribution

It provides a counterexample to a question by Kollár and Kovács, utilizing generative AI to achieve a novel geometric construction.

Findings

Constructed a flat projective morphism with Cohen--Macaulay, reduced fibers

Demonstrated non-constancy of the first cohomology of fibers

Used AI tools to assist in complex geometric construction

Abstract

We answer a question of Koll\'ar and Kov\'acs 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.