One numerical obstruction for rational maps between hypersurfaces

TL;DR

This paper establishes a numerical inequality for rational maps between generic hypersurfaces, providing a new obstruction criterion related to their invariants.

Contribution

It introduces a Noether-Fano type inequality for rational maps between hypersurfaces, under specific assumptions, linking their numerical invariants.

Findings

Proves a numerical inequality $m_Y \,\ge\, m_X$ for rational maps between hypersurfaces.

Provides an effectively computable criterion for obstructions to such maps.

Extends classical ideas to higher-dimensional hypersurfaces.

Abstract

Given a rational dominant map $\phi: Y \dashrightarrow X$ between two generic hypersurfaces $Y,X \subset \mathbb{P}^n$ of dimension $\ge 3$, we prove (under an addition assumption on $\phi$) a "Noether-Fano type" inequality $m_Y \ge m_X$ for certain (effectively computed) numerical invariants of $Y$ and $X$.