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# pentagonalK3 -- pentagonal K3 surface

- Usage:
    `pentagonalK3 g`
- Inputs:
  - g, an integer, the genus
- Optional inputs:
  - CoefficientRing => ..., default value ZZ/65521
- Outputs:
  - a K3 surface, a random pentagonal K3 surface of genus $g$

## Description

    i1 : S = pentagonalK3 11

    o1 = K3 surface with rank 2 lattice defined by the intersection matrix: | 20 5 |
                                                                            | 5  0 |
         -- (1,0): K3 surface of genus 11 and degree 20 containing elliptic curve of degree 5 (GM fourfold)
         -- (1,1): K3 surface of genus 16 and degree 30 containing elliptic curve of degree 5
         -- (1,2): K3 surface of genus 21 and degree 40 containing elliptic curve of degree 5
         -- (1,3): K3 surface of genus 26 and degree 50 containing elliptic curve of degree 5 (GM fourfold)
         -- (1,4): K3 surface of genus 31 and degree 60 containing elliptic curve of degree 5
         -- (1,5): K3 surface of genus 36 and degree 70 containing elliptic curve of degree 5
         -- (1,6): K3 surface of genus 41 and degree 80 containing elliptic curve of degree 5
         -- (2,0): K3 surface of genus 41 and degree 80 containing elliptic curve of degree 10

    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)

## See also

- K3(ZZ,ZZ,ZZ) -- make a lattice-polarized K3 surface
- trigonalK3 -- trigonal K3 surface
- tetragonalK3 -- tetragonal K3 surface

## Ways to use pentagonalK3:

- `pentagonalK3(ZZ)`

## For the programmer

The object pentagonalK3 is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/K3Surfaces.m2:1012:0.