# MultirationalMap -- the class of all multi-rational maps

## Description

A multi-rational map is a rational map between multi-projective varieties, $$\Phi:X\subseteq \mathbb{P}^{r_1}\times\mathbb{P}^{r_2}\times\cdots\times\mathbb{P}^{r_n}\dashrightarrow Y \subseteq \mathbb{P}^{s_1}\times\mathbb{P}^{s_2}\times\cdots\times\mathbb{P}^{s_m} .$$Thus, it can be represented by an ordered list of rational maps$$\Phi_i = (\Phi:X\dashrightarrow Y)\circ(pr_i:Y\to Y_i\subseteq\mathbb{P}^{s_i}) ,$$for $i=1,\ldots,m$. The maps $\Phi_i:X\dashrightarrow Y_i\subseteq\mathbb{P}^{s_i}$, since the target $Y_i$ is a standard projective variety, are implemented with the class RationalMap (more properly, when $n>1$ the class of such maps is called MultihomogeneousRationalMap). Recall that the main constructor for the class RationalMap (as well as for the class MultihomogeneousRationalMap) is the method rationalMap.

The constructor for the class of multi-rational maps is multirationalMap, which can often be abbreviated to rationalMap (see also shortcuts). It takes as input the list of maps $\{\Phi_1:X\dashrightarrow Y_1,\ldots,\Phi_m:X\dashrightarrow Y_m\}$, together with the variety $Y$, and returns the map $\Phi:X\dashrightarrow Y$.

## Functions and methods returning a multi-rational map :

• inverse2 -- inverse of a birational map using a faster algorithm for a special class of maps
• multirationalMap -- the multi-rational map defined by a list of rational maps

## For the programmer

The object MultirationalMap is a type, with ancestor classes MutableHashTable < HashTable < Thing.