# multirationalMap -- the multi-rational map defined by a list of rational maps

## Synopsis

• Usage:
multirationalMap Phi
multirationalMap(Phi,Y)
• Inputs:
• Phi, a list of rational maps, $\{\Phi_1:X\dashrightarrow Y_1\subseteq\mathbb{P}^{s_1},\ldots,\Phi_m:X\dashrightarrow Y_m\subseteq\mathbb{P}^{s_m}\}$, all having the same source $X\subseteq \mathbb{P}^{r_1}\times\mathbb{P}^{r_2}\times\cdots\times\mathbb{P}^{r_n}$
• Y, $Y \subseteq \mathbb{P}^{s_1}\times\mathbb{P}^{s_2}\times\cdots\times\mathbb{P}^{s_m}$ (if omitted, then the product $Y_1\times\cdots \times Y_m$ is taken)
• Outputs:
• , the unique rational map $\Phi:X\subseteq \mathbb{P}^{r_1}\times\mathbb{P}^{r_2}\times\cdots\times\mathbb{P}^{r_n}\dashrightarrow Y \subseteq \mathbb{P}^{s_1}\times\mathbb{P}^{s_2}\times\cdots\times\mathbb{P}^{s_m}$ such that $pr_i\circ\Phi = \Phi_i$, where $pr_i:Y\subseteq \mathbb{P}^{s_1}\times\mathbb{P}^{s_2}\times\cdots\times\mathbb{P}^{s_m} \to Y_i\subseteq \mathbb{P}^{s_i}$ denotes the i-th projection

## Description

 i1 : R = ring PP_(ZZ/65521)^{2,1}; i2 : f = rationalMap for i to 3 list random({1,1},R); o2 : MultihomogeneousRationalMap (rational map from PP^2 x PP^1 to PP^3) i3 : g = rationalMap(for i to 4 list random({0,1},R),Dominant=>true); o3 : MultihomogeneousRationalMap (dominant rational map from PP^2 x PP^1 to curve in PP^4) i4 : h = rationalMap for i to 2 list random({1,0},R); o4 : MultihomogeneousRationalMap (rational map from PP^2 x PP^1 to PP^2) i5 : Phi = multirationalMap {f,g,h} o5 = Phi o5 : MultirationalMap (rational map from PP^2 x PP^1 to 6-dimensional subvariety of PP^3 x PP^4 x PP^2) i6 : describe Phi -- long description o6 = multi-rational map consisting of 3 rational maps source variety: PP^2 x PP^1 target variety: 6-dimensional subvariety of PP^3 x PP^4 x PP^2 cut out by 3 hypersurfaces of multi-degree (0,1,0) base locus: empty subscheme of PP^2 x PP^1 dominance: false image: threefold in PP^3 x PP^4 x PP^2 cut out by 14 hypersurfaces of multi-degrees (0,1,0)^3 (1,0,2)^4 (1,1,1)^6 (1,3,0)^1 multidegree: {3, 6, 12, 24} degree: 1 degree sequence (map 1/3): [(1,1)] degree sequence (map 2/3): [(0,1)] degree sequence (map 3/3): [(1,0)] coefficient ring: ZZ/65521 i7 : ? Phi -- short description o7 = multi-rational map consisting of 3 rational maps source variety: PP^2 x PP^1 target variety: 6-dimensional subvariety of PP^3 x PP^4 x PP^2 cut out by 3 hypersurfaces of multi-degree (0,1,0) base locus: empty subscheme of PP^2 x PP^1 dominance: false image: threefold in PP^3 x PP^4 x PP^2 cut out by 14 hypersurfaces of multi-degrees (0,1,0)^3 (1,0,2)^4 (1,1,1)^6 (1,3,0)^1 multidegree: {3, 6, 12, 24} degree: 1 i8 : X = projectiveVariety R; o8 : ProjectiveVariety, PP^2 x PP^1 i9 : Phi; o9 : MultirationalMap (morphism from X to 6-dimensional subvariety of PP^3 x PP^4 x PP^2) i10 : Y = target Phi; o10 : ProjectiveVariety, 6-dimensional subvariety of PP^3 x PP^4 x PP^2 i11 : Phi; o11 : MultirationalMap (morphism from X to Y) i12 : Z = (image multirationalMap {f,g}) ** target h; o12 : ProjectiveVariety, 5-dimensional subvariety of PP^3 x PP^4 x PP^2 i13 : Psi = multirationalMap({f,g,h},Z) o13 = Psi o13 : MultirationalMap (rational map from X to Z) i14 : assert(image Psi == image Phi)

## Caveat

Be careful when you pass the target Y as input, because it must be compatible with the maps but for efficiency reasons a full check is not done automatically. See check(MultirationalMap).