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, a projective variety $Y \subseteq \mathbb{P}^{s_1}\times\mathbb{P}^{s_2}\times\cdots\times\mathbb{P}^{s_m}$ (if omitted, then the product $Y_1\times\cdots \times Y_m$ is taken)
Outputs:
- a multi-rational map, 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 : ZZ/65521[x_0..x_4];

i2 : f = rationalMap {x_3^2-x_2*x_4,x_2*x_3-x_1*x_4,x_1*x_3-x_0*x_4,x_2^2-x_0*x_4,x_1*x_2-x_0*x_3,x_1^2-x_0*x_2};

o2 : RationalMap (quadratic rational map from PP^4 to PP^5)

i3 : g = rationalMap(f,Dominant=>true);

o3 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)

i4 : Y = (projectiveVariety target g) ** (projectiveVariety target g);

o4 : ProjectiveVariety, 8-dimensional subvariety of PP^5 x PP^5

i5 : multirationalMap {f,g};

o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^5)

i6 : multirationalMap({f,g},Y);

o6 : MultirationalMap (rational map from PP^4 to 8-dimensional subvariety of PP^5 x PP^5)

i7 : assert(factor oo === {f,g} and target oo === Y)

i8 : multirationalMap {f,f,g};

o8 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^5 x PP^5)

i9 : h = last graph f;

o9 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)

i10 : multirationalMap {h};

o10 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)

i11 : multirationalMap {h,h};

o11 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5 x PP^5)

i12 : multirationalMap({h,h,h},Y ** projectiveVariety(target h));

o12 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to 13-dimensional subvariety of PP^5 x PP^5 x PP^5)

i13 : describe oo!

o13 = multi-rational map consisting of 3 rational maps
      source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of degrees ({0, 2},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1})
      target variety: 13-dimensional subvariety of PP^5 x PP^5 x PP^5 cut out by 2 hypersurfaces of degrees ({0, 2, 0},{2, 0, 0})
      base locus: empty subscheme of PP^4 x PP^5
      image: 4-dimensional subvariety of PP^5 x PP^5 x PP^5 cut out by 51 hypersurfaces of degrees ({0, 1, 1},{1, 0, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{0, 0, 2},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{1, 1, 0},{1, 1, 0},{1, 1, 0},{0, 2, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{2, 0, 0})
      dominance: false
      --
      rational map (1/3) defined by multiforms of degree {0, 1}
      target variety: PP^5
      image: smooth quadric hypersurface in PP^5
      dominance: false
      birationality: false
      degree of map: 1
      projective degrees: {51, 28, 14, 6, 2}
      number of minimal representatives: 2, with degrees ({0, 1},{2, 0})
      dimension base locus: -1
      --
      rational map (2/3) defined by multiforms of degree {0, 1}
      target variety: PP^5
      image: smooth quadric hypersurface in PP^5
      dominance: false
      birationality: false
      degree of map: 1
      projective degrees: {51, 28, 14, 6, 2}
      number of minimal representatives: 2, with degrees ({0, 1},{2, 0})
      dimension base locus: -1
      --
      rational map (3/3) defined by multiforms of degree {0, 1}
      target variety: PP^5
      image: smooth quadric hypersurface in PP^5
      dominance: false
      birationality: false
      degree of map: 1
      projective degrees: {51, 28, 14, 6, 2}
      number of minimal representatives: 2, with degrees ({0, 1},{2, 0})
      dimension base locus: -1
      --
      coefficient ring: ZZ/65521

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).

Ways to use `multirationalMap` :

"multirationalMap(List)"
"multirationalMap(List,MultiprojectiveVariety)"
multirationalMap(MultiprojectiveVariety) -- identity map
multirationalMap(MultiprojectiveVariety,MultiprojectiveVariety) -- get the natural inclusion
multirationalMap(MultirationalMap,MultiprojectiveVariety) -- change the target of a rational map

For the programmer

The object multirationalMap is a method function.

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

Synopsis

Description

Caveat

See also

Ways to use multirationalMap :

For the programmer

Ways to use `multirationalMap` :