# K3(EmbeddedK3surface) -- make a lattice-polarized K3 surface from an embedded K3 surface

## Synopsis

• Function: K3
• Usage:
K3 S
• Inputs:
• S, , a special K3 surface that contains a curve $C$
• Optional inputs:
• CoefficientRing => ..., default value ZZ/65521, make a lattice-polarized K3 surface
• Verbose => ..., default value true, make a lattice-polarized K3 surface
• Outputs:
• , the K3 surface $S$ with rank 2 lattice spanned by $H,C$, where $H$ is the hyperplane section class

## Description

 i1 : S = K3(3,5,-2); o1 : Lattice-polarized K3 surface i2 : S(1,1) o2 = K3 surface of genus 7 and degree 12 in PP^7 o2 : Embedded K3 surface i3 : T = K3 S(1,1) o3 = K3 surface with rank 2 lattice defined by the intersection matrix: | 12 3 | | 3 -2 | o3 : Lattice-polarized K3 surface i4 : T(1,0) o4 = K3 surface of genus 7 and degree 12 in PP^7 o4 : Embedded K3 surface