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

## Synopsis

• Function: rationalMap
• Usage:
rationalMap(X,D)
• Inputs:
• D, , a multiset of 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 = PP_(ZZ/65521)^5; o1 : ProjectiveVariety, PP^5 i2 : C = random({{2},3:{1}},0_P5); o2 : ProjectiveVariety, curve in PP^5 i3 : X = random({3:{2}},C); o3 : ProjectiveVariety, surface in PP^5 i4 : H = random(1,0_X); -- it's interpreted as X * H o4 : ProjectiveVariety, hypersurface in PP^5 i5 : D = tally {H, H, C} o5 = Tally{C => 1} H => 2 o5 : Tally i6 : phi = rationalMap(X,D) o6 = phi o6 : MultirationalMap (rational map from X to PP^20) i7 : assert(phi == rationalMap(X,tally {X*H, X*H, C}))

This function is based internally on the function rationalMap(Ring,Tally), provided by the package Cremona.