surface(List) -- get a rational surface

Synopsis

• Function: surface

• Usage:

  surface {a,i,j,k,...}

  surface({a,i,j,k,...),K)

  surface({a,i,j,k,...},K,NumNodes=>n,ambient=>m)

• Inputs:
  • a list, a list $\{a,i,j,k,\ldots\}$ of nonnegative integers
• Outputs:
  • an embedded projective variety, the image of the rational map defined by the linear system of curves of degree $a$ in $\mathbb{P}_{K}^2$ having $i$ random base points of multiplicity 1, $j$ random base points of multiplicity 2, $k$ random base points of multiplicity 3, and so on until the last integer in the given list.

Description

In the example below, we take the image of the rational map defined by the linear system of septic plane curves with 3 random simple base points and 9 random double points.

i1 : S = surface {7,3,9};

o1 : ProjectiveVariety, surface in PP^5

i2 : coefficientRing S

       ZZ
o2 = -----
     65521

o2 : QuotientRing

i3 : T = surface({7,3,9},ZZ/33331);

o3 : ProjectiveVariety, surface in PP^5

i4 : X = specialCubicFourfold T;

o4 : ProjectiveVariety, cubic fourfold containing a surface of degree 10 and sectional genus 6

i5 : coefficientRing X

       ZZ
o5 = -----
     33331

o5 : QuotientRing

i6 : describe X

o6 = Special cubic fourfold of discriminant 26
     containing a (smooth) surface of degree 10 and sectional genus 6
     cut out by 10 hypersurfaces of degree 3

See also

• rationalMap(PolynomialRing,List) -- rational map defined by the linear system of hypersurfaces passing through random points with multiplicity
• specialGushelMukaiFourfold(Array,Array) -- construct GM fourfolds by gluing cubic or quartic scrolls to surfaces in PP^6