surface(List) -- get a rational surface

• 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.

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