# trigonalK3 -- trigonal K3 surface

## Synopsis

• Usage:
trigonalK3 g
• Inputs:
• Optional inputs:
• CoefficientRing => ..., default value ZZ/65521
• Outputs:
• , a random trigonal K3 surface of genus $g$

## Description

See also the paper A remark on the generalized franchetta conjecture for K3 surfaces, by Beauville.

 i1 : S = trigonalK3 11 o1 = K3 surface with rank 2 lattice defined by the intersection matrix: | 20 3 | | 3 0 | -- (1,0): K3 surface of genus 11 and degree 20 containing elliptic curve of degree 3 (GM fourfold) -- (1,1): K3 surface of genus 14 and degree 26 containing elliptic curve of degree 3 (cubic fourfold) (GM fourfold) -- (1,2): K3 surface of genus 17 and degree 32 containing elliptic curve of degree 3 -- (1,3): K3 surface of genus 20 and degree 38 containing elliptic curve of degree 3 (cubic fourfold) -- (1,4): K3 surface of genus 23 and degree 44 containing elliptic curve of degree 3 -- (1,5): K3 surface of genus 26 and degree 50 containing elliptic curve of degree 3 (GM fourfold) -- (1,6): K3 surface of genus 29 and degree 56 containing elliptic curve of degree 3 -- (1,7): K3 surface of genus 32 and degree 62 containing elliptic curve of degree 3 (cubic fourfold) -- (1,8): K3 surface of genus 35 and degree 68 containing elliptic curve of degree 3 (GM fourfold) -- (1,9): K3 surface of genus 38 and degree 74 containing elliptic curve of degree 3 (cubic fourfold) (GM fourfold) -- (1,10): K3 surface of genus 41 and degree 80 containing elliptic curve of degree 3 -- (2,0): K3 surface of genus 41 and degree 80 containing elliptic curve of degree 6 -- (1,11): K3 surface of genus 44 and degree 86 containing elliptic curve of degree 3 (cubic fourfold) o1 : Lattice-polarized K3 surface i2 : S' = S(1,0); o2 : Embedded K3 surface i3 : map(S',0,1) o3 = multi-rational map consisting of one single rational map source variety: K3 surface of genus 11 and degree 20 in PP^11 target variety: PP^1 o3 : MultirationalMap (rational map from S' to PP^1)