On the finiteness theorem for rational maps on a variety of general type

TL;DR

This paper discusses the finiteness of dominant rational maps of finite degree from a fixed variety to varieties of general type, emphasizing recent advances and seeking effective estimates for their number.

Contribution

It refines classical methods to provide more effective estimates for the finite set of such rational maps.

Findings

Confirmed the finiteness of the set of rational maps of finite degree.

Provided updated bounds and estimates for the number of these maps.

Abstract

The dominant rational maps of finite degree from a fixed variety to varieties of general type, up to birational isomorphisms, form a finite set. This has been known as the Iitaka-Severi conjecture, and is nowdays an established result, in virtue of some recent advances in the theory of pluricanonical maps. We study the question of finding some effective estimate for the finite number of maps, and to this aim we provide some update and refinement of the classical treatment of the subject.