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# 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:
• , 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