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

Synopsis

• Function: rationalMap

• Usage:

  rationalMap(X,D)

• Inputs:
  • X, a multi-projective variety
  • D, a tally, 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:
  • Dominant => ..., default value null,
• Outputs:
  • a multi-rational map, 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.

See also

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