# project -- project a K3 surface

## Synopsis

• Usage:
project({i,j,k,...},S,a,b)
project({i,j,k,...},S(a,b))
• Inputs:
• , a list $\{i,j,k,\ldots\}$ of nonnegative integers
• S, , a lattice-polarized K3 surface with rank 2 lattice spanned by $H,C$
• a, an integer
• b, an integer
• Outputs:
• , the projection of $S$ embedded by the complete linear system $|a H + b C|$ from $i$ random points of multiplicity 1, $j$ random points of multiplicity 2, $k$ random points of multiplicity 3, and so on until the last integer in the given list.

## Description

 i1 : S = K3(8,2,-2) o1 = K3 surface with rank 2 lattice defined by the intersection matrix: | 14 2 | | 2 -2 | -- (1,0): K3 surface of genus 8 and degree 14 containing rational curve of degree 2 (cubic fourfold) -- (2,0): K3 surface of genus 29 and degree 56 containing rational curve of degree 4 -- (2,1): K3 surface of genus 32 and degree 62 containing rational curve of degree 2 (cubic fourfold) o1 : Lattice-polarized K3 surface i2 : project({5,3,1},S,2,1); -- (5th + 3rd + simple)-projection of S(2,1) -- *** simulation *** -- surface of degree 62 and sectional genus 32 in PP^32 (quadrics: 435, cubics: 6264) -- surface of degree 37 and sectional genus 22 in PP^17 (quadrics: 100, cubics: 979) -- surface of degree 28 and sectional genus 19 in PP^11 (quadrics: 28, cubics: 248) -- surface of degree 27 and sectional genus 19 in PP^10 (quadrics: 19, cubics: 176) -- (degree and genus are as expected) o2 : ProjectiveVariety, surface in PP^10