# rationalMap(Ideal,ZZ,ZZ) -- makes a rational map from an ideal

## Synopsis

• Function: rationalMap
• Usage:
rationalMap(I,d,e)
rationalMap(I,d)
rationalMap I
• Inputs:
• I, an ideal, a homogeneous ideal in the coordinate ring of a projective variety $X\subseteq\mathbb{P}^n$
• d, an integer, a positive integer (or a list of integers in the case $X$ is embedded in a product of projective spaces)
• e, an integer, a positive integer (if omitted, it is taken to be 1)
• Optional inputs:
• Outputs:
• , the rational map $X\dashrightarrow \mathbb{P}^m$ defined by the linear system of hypersurfaces of degree $d$ having points of multiplicity $e$ along the projective subscheme of $X$ defined by $I$.

## Description

In most cases, the command rationalMap(I,d,e) yields the same output as rationalMap(saturate(I^e),d), but the former is implemented using pure linear algebra.

The command rationalMap I is basically equivalent to rationalMap(I,max degrees I).

In the following example, we calculate the rational map defined by the linear system of cubic hypersurfaces in $\mathbb{P}^6$ having double points along a Veronese surface $V\subset\mathbb{P}^5\subset\mathbb{P}^6$.

 i1 : ZZ/33331[x_0..x_6]; V = ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6); ZZ o2 : Ideal of -----[x ..x ] 33331 0 6 i3 : time phi = rationalMap(V,3,2) -- used 0.296142 seconds o3 = -- rational map -- ZZ source: Proj(-----[x , x , x , x , x , x , x ]) 33331 0 1 2 3 4 5 6 ZZ target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y , y , y , y ]) 33331 0 1 2 3 4 5 6 7 8 9 10 11 12 13 defining forms: { 3 x , 6 2 x x , 5 6 2 x x , 4 6 2 x x , 3 6 2 x x , 2 6 2 x x , 1 6 2 x x , 0 6 2 x x - x x x , 4 6 3 5 6 x x x - x x x , 2 4 6 1 5 6 x x x - x x x , 2 3 6 1 4 6 2 x x - x x x , 2 6 0 5 6 x x x - x x x , 1 2 6 0 4 6 2 x x - x x x , 1 6 0 3 6 2 2 2 x x - 2x x x + x x + x x - x x x 2 3 1 2 4 0 4 1 5 0 3 5 } o3 : RationalMap (cubic rational map from PP^6 to PP^13) i4 : describe phi! o4 = rational map defined by forms of degree 3 source variety: PP^6 target variety: PP^13 image: 6-dimensional variety of degree 16 in PP^13 cut out by 21 hypersurfaces of degree 2 dominance: false birationality: false degree of map: 1 projective degrees: {1, 3, 6, 12, 16, 16, 16} number of minimal representatives: 1 dimension base locus: 4 degree base locus: 3 coefficient ring: ZZ/33331