associatedK3surface(SpecialCubicFourfold) -- associated K3 surface to a rational cubic fourfold

Synopsis

Function: associatedK3surface
Usage:

U' = associatedK3surface X
Inputs:
- X, a special cubic fourfold, containing a surface $S\subset\mathbb{P}^5$ that admits a congruence of $(3e-1)$-secant curves of degree $e$
Optional inputs:
- Singular => ..., default value null, whether to transfer computation to Singular
- Strategy => ..., default value null, associated K3 surface to a rational fourfold
- Verbose => ..., default value false, request verbose feedback
Outputs:
- U', the (minimal) K3 surface associated to $X$
Consequences:
- building U' will return the following four objects:
  - the dominant rational map $\mu:\mathbb{P}^5 \dashrightarrow W$ defined by the linear system of hypersurfaces of degree $3e-1$ having points of multiplicity $e$ along $S$;
  - the surface $U\subset W$ determining the inverse map of the restriction of $\mu$ to $X$;
  - the list of the exceptional curves on the surface $U$;
  - a rational map of degree 1 from the surface $U$ to the minimal K3 surface $U'$.

Description

For more details and notation, see the papers Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds and Explicit computations with cubic fourfolds, Gushel-Mukai fourfolds, and their associated K3 surfaces.

i1 : X = specialCubicFourfold "quartic scroll";

o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0

i2 : describe X

o2 = Special cubic fourfold of discriminant 14
     containing a (smooth) surface of degree 4 and sectional genus 0
     cut out by 6 hypersurfaces of degree 2

i3 : time U' = associatedK3surface X;
     -- used 4.08909 seconds

o3 : ProjectiveVariety, K3 surface associated to X

i4 : (mu,U,C,f) = building U';

i5 : ? mu

o5 = multi-rational map consisting of one single rational map
     source variety: PP^5
     target variety: hypersurface in PP^5 defined by a form of degree 2
     dominance: true

i6 : ? U

o6 = surface in PP^5 cut out by 7 hypersurfaces of degrees 2^1 3^6

i7 : last C

o7 = curve in PP^5 cut out by 4 hypersurfaces of degrees 1^3 2^1 

o7 : ProjectiveVariety, curve in PP^5 (subvariety of codimension 1 in U)

i8 : assert(image f == U')

Ways to use this method: