One numerical obstruction for rational maps between hypersurfaces

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