# rationalMap(Ring,Tally) -- rational map defined by an effective divisor

## Synopsis

• Function: rationalMap
• Usage:
rationalMap D
rationalMap(R,D)
• Inputs:
• R, a ring, the coordinate ring of a locally factorial projective variety $X\subseteq\mathbb{P}^n$
• D, , a multiset of homogeneous ideals in $R$ (or in ambient $R$) defining pure codimension 1 subschemes of $X$ with no embedded components; so that D is interpreted as an effective divisor on $X$
• Optional inputs:
• Outputs:
• , the rational map defined by the complete linear system $|D|$

## Description

In the example below, we take a smooth complete intersection $X\subset\mathbb{P}^5$ of three quadrics containing a conic $C\subset\mathbb{P}^5$. Then we calculate the map defined by the linear system $|2H+C|$, where $H$ is the hyperplane section class of $X$.

 i1 : P5 = ZZ/65521[x_0..x_5]; i2 : C = ideal(x_1^2-x_0*x_2,x_3,x_4,x_5) 2 o2 = ideal (x - x x , x , x , x ) 1 0 2 3 4 5 o2 : Ideal of P5 i3 : X = quotient ideal(-x_1^2+x_0*x_2-x_1*x_3+x_3^2-x_0*x_5+x_1*x_5+x_3*x_5,-x_0*x_3-x_1*x_3+x_2*x_4-x_3*x_4-x_4^2-x_1*x_5-x_2*x_5+x_5^2,-x_1^2+x_0*x_2+x_2*x_3+x_1*x_4-x_3*x_4-x_4*x_5); i4 : H = ideal random(1,X) o4 = ideal(- 32646x - 28377x + 26433x - 29566x + 3783x + 26696x ) 0 1 2 3 4 5 o4 : Ideal of X i5 : D = new Tally from {H => 2,C => 1}; i6 : time phi = rationalMap D -- used 0.0498385 seconds o6 = -- rational map -- ZZ source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by 65521 0 1 2 3 4 5 { 2 2 - x + x x - x x + x - x x + x x + x x , 1 0 2 1 3 3 0 5 1 5 3 5 2 2 - x x - x x + x x - x x - x - x x - x x + x , 0 3 1 3 2 4 3 4 4 1 5 2 5 5 2 - x + x x + x x + x x - x x - x x 1 0 2 2 3 1 4 3 4 4 5 } ZZ target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y ]) 65521 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 defining forms: { x x x , 0 4 5 2 x x , 0 5 x x x , 1 2 5 x x x , 1 4 5 2 x x , 1 5 2 x x , 2 5 x x x , 2 3 5 x x x , 2 4 5 2 x x , 2 5 2 x x , 3 5 x x x , 3 4 5 2 x x , 3 5 2 x x , 4 5 2 x x , 4 5 3 x , 5 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 3 x x + x x + x x x + x x - 3x x x - x x x - x x + x x - x x + 2x x - x - x x + 2x x x + 2x x x + x x + 2x x x - x x + 4x x x - 3x x x + x x + x x - 3x x - 2x x - 2x x + x , 0 1 0 2 0 1 2 0 2 0 2 4 1 2 4 2 4 0 4 1 4 2 4 4 0 5 0 2 5 1 2 5 2 5 2 3 5 3 5 1 4 5 3 4 5 4 5 0 5 1 5 2 5 4 5 5 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 3 x x + x x x + x x + x x + 2x x + 32760x - 32760x x x - x x x - 32760x x + 32757x x x - 32760x x + 32760x x + x x - x x - 32760x x x - 32758x x x + 32760x x x + 32760x x + 2x x x - 3x x + 32758x x x + 7x x x + 32760x x x - 4x x x - 32760x x - 32758x x - 5x x - x x + 32759x x - 4x x - 32759x , 0 2 0 1 2 0 2 1 2 2 3 3 0 2 4 1 2 4 2 4 2 3 4 3 4 2 4 3 4 0 5 0 1 5 0 2 5 1 2 5 2 5 2 3 5 3 5 0 4 5 1 4 5 2 4 5 3 4 5 4 5 0 5 1 5 2 5 3 5 4 5 5 2 2 3 2 2 x x x + x x + x x + x + x x x - 2x x + x x , 0 1 2 0 2 1 2 2 1 2 4 2 4 2 4 2 2 2 - 2x x x + x x x - 2x x x + x x x + x x x - x x + x x + x x , 0 2 5 0 4 5 1 4 5 2 4 5 3 4 5 4 5 1 5 4 5 2 2 2 2 2 2 2 2 3 - x x + 2x x x + 2x x x + x x + 2x x x - x x + 4x x x - 3x x x + x x + x x - 3x x - 2x x - 2x x + x , 0 5 0 2 5 1 2 5 2 5 2 3 5 3 5 1 4 5 3 4 5 4 5 0 5 1 5 2 5 4 5 5 2 2 x x x + x x x + x x x + x x + x x x - 2x x x + x x 0 1 5 0 2 5 1 2 5 2 5 1 4 5 2 4 5 4 5 } o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20) i7 : time ? image(phi,"F4") -- used 0.989511 seconds o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153 hypersurfaces of degree 2

See also the package Divisor, which provides general tools for working with divisors.