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

i1 : K3("11",Verbose=>true)
(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