map(CongruenceOfCurves) -- compute the parameter space of a congruence

Function: map
Usage:

map f
Inputs:
- f, a congruence of curves, a congruence of curves to a surface inside a variety $Y$
Optional inputs:
- Degree => ..., default value null,
- DegreeLift => ..., default value null,
- DegreeMap => ..., default value null,
Outputs:
- a multi-rational map, a dominant map from $Y$ to the parameter space of $f$ whose general fibers are the curves of the congruence

Description

i1 : S = PP_(ZZ/65521)[2,2];

o1 : ProjectiveVariety, surface in PP^5

i2 : Y = ambient S;

o2 : ProjectiveVariety, PP^5

i3 : X = specialCubicFourfold S;

o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0

i4 : f = detectCongruence(X,1);

o4 : Congruence of 2-secant lines to S in Y

i5 : F = map f;

o5 : MultirationalMap (dominant rational map from Y to hypersurface in PP^5)

i6 : Q = target F

o6 = Q

o6 : ProjectiveVariety, hypersurface in PP^5

i7 : f;

o7 : Congruence of 2-secant lines to S in Y; parameter space: Q

i8 : p = point Y;

o8 : ProjectiveVariety, a point in PP^5

i9 : assert(f p == F^* F p)

The source of this document is in SpecialFanoFourfolds.m2:3336:0.