# K3(String) -- show available functions to construct K3 surfaces of given genus

## Synopsis

• Function: K3
• Usage:
K3 "G"
K3("G",[Unique=>true])
• Inputs:
• G, , the string of an integer
• Optional inputs:
• CoefficientRing => ..., default value ZZ/65521, make a lattice-polarized K3 surface
• Verbose => ..., default value true, make a lattice-polarized K3 surface
• Outputs:
• a list, a list of terns (d,g,n) such that ((K3(d,g,n))(a,b) is a K3 surface of genus G, for some integers a,b

## Description

 i1 : K3 "11" (K3(5,5,-2))(1,2) -- K3 surface of genus 11 and degree 20 containing rational curve of degree 1 (K3(11,2,-2))(1,0) -- K3 surface of genus 11 and degree 20 containing rational curve of degree 2 (K3(3,6,-2))(1,2) -- K3 surface of genus 11 and degree 20 containing rational curve of degree 2 (K3(11,3,0))(1,0) -- K3 surface of genus 11 and degree 20 containing elliptic curve of degree 3 (K3(8,3,0))(1,1) -- K3 surface of genus 11 and degree 20 containing elliptic curve of degree 3 (K3(5,3,0))(1,2) -- K3 surface of genus 11 and degree 20 containing elliptic curve of degree 3 (K3(11,4,0))(1,0) -- K3 surface of genus 11 and degree 20 containing elliptic curve of degree 4 (K3(7,4,0))(1,1) -- K3 surface of genus 11 and degree 20 containing elliptic curve of degree 4 (K3(3,4,0))(1,2) -- K3 surface of genus 11 and degree 20 containing elliptic curve of degree 4 (K3(11,5,0))(1,0) -- K3 surface of genus 11 and degree 20 containing elliptic curve of degree 5 (K3(5,7,-2))(1,1) -- K3 surface of genus 11 and degree 20 containing rational curve of degree 5 (K3(6,5,0))(1,1) -- K3 surface of genus 11 and degree 20 containing elliptic curve of degree 5 (K3(5,6,0))(1,1) -- K3 surface of genus 11 and degree 20 containing elliptic curve of degree 6 (K3(4,7,0))(1,1) -- K3 surface of genus 11 and degree 20 containing elliptic curve of degree 7 (K3(3,8,0))(1,1) -- K3 surface of genus 11 and degree 20 containing elliptic curve of degree 8 o1 = {(5, 5, -2), (11, 2, -2), (3, 6, -2), (11, 3, 0), (8, 3, 0), (5, 3, 0), ------------------------------------------------------------------------ (11, 4, 0), (7, 4, 0), (3, 4, 0), (11, 5, 0), (5, 7, -2), (6, 5, 0), (5, ------------------------------------------------------------------------ 6, 0), (4, 7, 0), (3, 8, 0)} o1 : List i2 : S = K3(5,5,-2) o2 = K3 surface with rank 2 lattice defined by the intersection matrix: | 8 5 | | 5 -2 | -- (1,0): K3 surface of genus 5 and degree 8 containing rational curve of degree 5 -- (1,1): K3 surface of genus 9 and degree 16 containing rational curve of degree 3 -- (1,2): K3 surface of genus 11 and degree 20 containing rational curve of degree 1 (GM fourfold) -- (2,0): K3 surface of genus 17 and degree 32 containing rational curve of degree 10 -- (2,1): K3 surface of genus 26 and degree 50 containing rational curve of degree 8 (GM fourfold) -- (2,2): K3 surface of genus 33 and degree 64 containing rational curve of degree 6 -- (3,0): K3 surface of genus 37 and degree 72 containing rational curve of degree 15 -- (2,3): K3 surface of genus 38 and degree 74 containing rational curve of degree 4 (cubic fourfold) (GM fourfold) -- (2,4): K3 surface of genus 41 and degree 80 containing rational curve of degree 2 o2 : Lattice-polarized K3 surface i3 : S(1,2) o3 = K3 surface of genus 11 and degree 20 in PP^11 o3 : Embedded K3 surface i4 : K3 S(1,2) o4 = K3 surface with rank 2 lattice defined by the intersection matrix: | 20 1 | | 1 -2 | o4 : Lattice-polarized K3 surface