This is K3Surfaces.info, produced by Macaulay2, version 1.20

K3Surfaces -- A package for constructing explicit examples of K3 surfaces
*************************************************************************

References
==========

  * M. H. and G. S., Explicit constructions of K3 surfaces and unirational
    Noether-Lefschetz divisors, available at arXiv:2110.15819 (see
    https://arxiv.org/abs/2110.15819 ) (2021).

Authors
=======

  * Michael Hoff <hahn@math.uni-sb.de>
  * Giovanni Staglianò <giovanni.stagliano@unict.it>

Version
=======

This documentation describes version 1.0 of K3Surfaces.

Source code
===========

The source code from which this documentation is derived is in the file
K3Surfaces.m2 (see common/share/Macaulay2/K3Surfaces.m2 ).

Exports
=======

  * Types
      * EmbeddedK3surface -- the class of all
        embedded K3 surfaces
      * LatticePolarizedK3surface -- the
        class of all lattice-polarized K3 surfaces
  * Functions and commands
      * K3 -- make a lattice-polarized K3 surface
      * mukaiModel -- Mukai models
      * pentagonalK3 -- pentagonal K3 surface
      * project -- project a K3 surface
      * tetragonalK3 -- tetragonal K3 surface
      * trigonalK3 -- trigonal K3 surface
  * Methods
      * coefficientRing(LatticePolarizedK3surface)
        -- coefficient ring of
        a K3 surface
      * "degree(EmbeddedK3surface,ZZ,ZZ)" -- see
        degree(LatticePolarizedK3surface,ZZ,ZZ),
        -- degree of a K3
        surface
      * degree(LatticePolarizedK3surface,ZZ,ZZ)
        -- degree of a K3
        surface
      * "genus(EmbeddedK3surface,ZZ,ZZ)" -- see
        genus(LatticePolarizedK3surface,ZZ,ZZ),
        -- genus of a K3
        surface
      * genus(LatticePolarizedK3surface,ZZ,ZZ)
        -- genus of a K3
        surface
      * "K3(ZZ,ZZ,ZZ)" -- see K3, -- make a lattice-polarized K3
        surface
      * K3(EmbeddedK3surface) -- make a
        lattice-polarized K3 surface from an embedded K3 surface
      * K3(String) -- show available functions to
        construct K3 surfaces of given genus
      * K3(ZZ) -- make a general K3 surface
      * "EmbeddedK3surface Sequence" -- see LatticePolarizedK3surface Sequence,
        -- image of the embedding
        of a K3 surface
      * LatticePolarizedK3surface Sequence
        -- image of the embedding of a K3 surface
      * "map(EmbeddedK3surface,ZZ,ZZ)" -- see
        map(LatticePolarizedK3surface,ZZ,ZZ),
        -- embedding of a K3
        surface
      * map(LatticePolarizedK3surface,ZZ,ZZ)
        -- embedding of a K3
        surface
      * "mukaiModel(ZZ)" -- see mukaiModel, -- Mukai models
      * "pentagonalK3(ZZ)" -- see pentagonalK3, --
        pentagonal K3 surface
      * "project(VisibleList,EmbeddedK3surface)" -- see project,
        -- project a K3 surface
      * "project(VisibleList,LatticePolarizedK3surface,ZZ,ZZ)" -- see
        project, -- project a K3 surface
      * "tetragonalK3(ZZ)" -- see tetragonalK3, --
        tetragonal K3 surface
      * "trigonalK3(ZZ)" -- see trigonalK3, -- trigonal K3
        surface

For the programmer
==================

The object K3Surfaces, is a package (Macaulay2Doc).

coefficientRing(LatticePolarizedK3surface) -- coefficient ring of a K3 surface
******************************************************************************

Synopsis
========

  * Function: coefficientRing (Macaulay2Doc),
  * Usage:
        coefficientRing S
  * Inputs:
      * S, a K3 surface,
  * Outputs:
      * a ring (Macaulay2Doc), the coefficient ring of $S$

Description
===========

+-----------------------------------------------------------------------------+
|i1 : K = ZZ/3331                                                             |
|                                                                             |
|o1 = K                                                                       |
|                                                                             |
|o1 : QuotientRing                                                            |
+-----------------------------------------------------------------------------+
|i2 : S = K3(5,2,-2,CoefficientRing=>K)                                       |
|                                                                             |
|o2 = K3 surface with rank 2 lattice defined by the intersection matrix: | 8 2|
|                                                                        | 2 -|
|     -- (1,0): K3 surface of genus 5 and degree 8 containing rational curve o|
|     -- (2,0): K3 surface of genus 17 and degree 32 containing rational curve|
|     -- (2,1): K3 surface of genus 20 and degree 38 containing rational curve|
|     -- (3,0): K3 surface of genus 37 and degree 72 containing rational curve|
|     -- (3,1): K3 surface of genus 42 and degree 82 containing rational curve|
|                                                                             |
|                                                                             |
|o2 : Lattice-polarized K3 surface                                            |
|-----------------------------------------------------------------------------|
|  |                                                                          |
|2 |                                                                          |
|f degree 2                                                                   |
| of degree 4                                                                 |
| of degree 2 (cubic fourfold)                                                |
| of degree 6                                                                 |
| of degree 4 (GM fourfold)                                                   |
+-----------------------------------------------------------------------------+
|i3 : coefficientRing S                                                       |
|                                                                             |
|o3 = K                                                                       |
|                                                                             |
|o3 : QuotientRing                                                            |
+-----------------------------------------------------------------------------+

Ways to use this method:
========================

  * coefficientRing(LatticePolarizedK3surface)
    -- coefficient ring of a K3
    surface

degree(LatticePolarizedK3surface,ZZ,ZZ) -- degree of a K3 surface
*****************************************************************

Synopsis
========

  * Function: degree (Macaulay2Doc),
  * Usage:
        degree(S,a,b)
  * Inputs:
      * S, a K3 surface,
      * a, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Outputs:
      * an integer (Macaulay2Doc), the degree of $S$ embedded by the
        complete linear system $|a H + b C|$, where $H,C$ is the basis of the
        lattice associated to $S$

Description
===========

+-----------------------------------------------------------------------------+
|i1 : S = K3(5,2,-2)                                                          |
|                                                                             |
|o1 = K3 surface with rank                                                    |
|                                                                             |
|     -- (1,0): K3 surface                                                    |
|     -- (2,0): K3 surface                                                    |
|     -- (2,1): K3 surface                                                    |
|     -- (3,0): K3 surface                                                    |
|     -- (3,1): K3 surface                                                    |
|                                                                             |
|                                                                             |
|o1 : Lattice-polarized K3                                                    |
|-----------------------------------------------------------------------------|
|2 lattice defined by the intersection matrix: | 8 2  |                       |
|                                              | 2 -2 |                       |
|of genus 5 and degree 8 containing rational curve of degree 2                |
|of genus 17 and degree 32 containing rational curve of degree 4              |
|of genus 20 and degree 38 containing rational curve of degree 2              |
|of genus 37 and degree 72 containing rational curve of degree 6              |
|of genus 42 and degree 82 containing rational curve of degree 4              |
|                                                                             |
|                                                                             |
|surface                                                                      |
|-----------------------------------------------------------------------------|
|(cubic fourfold)                                                             |
|                                                                             |
|(GM fourfold)                                                                |
+-----------------------------------------------------------------------------+
|i2 : degree(S,2,1)                                                           |
|                                                                             |
|o2 = 38                                                                      |
+-----------------------------------------------------------------------------+

See also
========

  * genus(LatticePolarizedK3surface,ZZ,ZZ)
    -- genus of a K3 surface

Ways to use this method:
========================

  * "degree(EmbeddedK3surface,ZZ,ZZ)"
  * degree(LatticePolarizedK3surface,ZZ,ZZ)
    -- degree of a K3 surface

EmbeddedK3surface -- the class of all embedded K3 surfaces
**********************************************************

See also
========

  * LatticePolarizedK3surface -- the class of
    all lattice-polarized K3 surfaces
  * LatticePolarizedK3surface Sequence
    -- image of the embedding of a K3 surface

Methods that use an embedded K3 surface :
=========================================

  * "degree(EmbeddedK3surface,ZZ,ZZ)" -- see
    degree(LatticePolarizedK3surface,ZZ,ZZ),
    -- degree of a K3 surface
  * "genus(EmbeddedK3surface,ZZ,ZZ)" -- see
    genus(LatticePolarizedK3surface,ZZ,ZZ),
    -- genus of a K3 surface
  * K3(EmbeddedK3surface) -- make a
    lattice-polarized K3 surface from an embedded K3 surface
  * "EmbeddedK3surface Sequence" -- see LatticePolarizedK3surface Sequence,
    -- image of the embedding of
    a K3 surface
  * "map(EmbeddedK3surface,ZZ,ZZ)" -- see
    map(LatticePolarizedK3surface,ZZ,ZZ),
    -- embedding of a K3 surface
  * "project(VisibleList,EmbeddedK3surface)" -- see project, --
    project a K3 surface

For the programmer
==================

The object EmbeddedK3surface, is a type (Macaulay2Doc),
with ancestor classes EmbeddedProjectiveVariety (MultiprojectiveVarieties),
<
MultiprojectiveVariety (MultiprojectiveVarieties) <
MutableHashTable (Macaulay2Doc) < HashTable (Macaulay2Doc)
< Thing (Macaulay2Doc).

genus(LatticePolarizedK3surface,ZZ,ZZ) -- genus of a K3 surface
***************************************************************

Synopsis
========

  * Function: genus (Macaulay2Doc),
  * Usage:
        genus(S,a,b)
  * Inputs:
      * S, a K3 surface,
      * a, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Outputs:
      * an integer (Macaulay2Doc), the genus of $S$ embedded by the
        complete linear system $|a H + b C|$, where $H,C$ is the basis of the
        lattice associated to $S$

Description
===========

+-----------------------------------------------------------------------------+
|i1 : S = K3(5,2,-2)                                                          |
|                                                                             |
|o1 = K3 surface with rank                                                    |
|                                                                             |
|     -- (1,0): K3 surface                                                    |
|     -- (2,0): K3 surface                                                    |
|     -- (2,1): K3 surface                                                    |
|     -- (3,0): K3 surface                                                    |
|     -- (3,1): K3 surface                                                    |
|                                                                             |
|                                                                             |
|o1 : Lattice-polarized K3                                                    |
|-----------------------------------------------------------------------------|
|2 lattice defined by the intersection matrix: | 8 2  |                       |
|                                              | 2 -2 |                       |
|of genus 5 and degree 8 containing rational curve of degree 2                |
|of genus 17 and degree 32 containing rational curve of degree 4              |
|of genus 20 and degree 38 containing rational curve of degree 2              |
|of genus 37 and degree 72 containing rational curve of degree 6              |
|of genus 42 and degree 82 containing rational curve of degree 4              |
|                                                                             |
|                                                                             |
|surface                                                                      |
|-----------------------------------------------------------------------------|
|(cubic fourfold)                                                             |
|                                                                             |
|(GM fourfold)                                                                |
+-----------------------------------------------------------------------------+
|i2 : genus(S,2,1)                                                            |
|                                                                             |
|o2 = 20                                                                      |
+-----------------------------------------------------------------------------+

See also
========

  * degree(LatticePolarizedK3surface,ZZ,ZZ)
    -- degree of a K3 surface

Ways to use this method:
========================

  * "genus(EmbeddedK3surface,ZZ,ZZ)"
  * genus(LatticePolarizedK3surface,ZZ,ZZ)
    -- genus of a K3 surface

K3 -- make a lattice-polarized K3 surface
*****************************************

Synopsis
========

  * Usage:
        K3(d,g,n)
        K3(d,g,n,CoefficientRing=>K)
  * Inputs:
      * d, an integer (Macaulay2Doc),
      * g, an integer (Macaulay2Doc),
      * n, an integer (Macaulay2Doc),
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing => ..., default value ZZ/65521
      * Verbose => ..., default value true
  * Outputs:
      * a K3 surface, defined over $K$ with
        rank 2 lattice defined by the intersection matrix $\begin{pmatrix} 2g-2
        & d \\ d & n \end{pmatrix}$

Description
===========

+-----------------------------------------------------------------------------+
|i1 : K3(6,1,-2)                                                              |
|                                                                             |
|o1 = K3 surface with rank                                                    |
|                                                                             |
|     -- (1,0): K3 surface                                                    |
|     -- (2,0): K3 surface                                                    |
|                                                                             |
|                                                                             |
|o1 : Lattice-polarized K3                                                    |
|-----------------------------------------------------------------------------|
|2 lattice defined by the intersection matrix: | 10 1  |                      |
|                                              | 1  -2 |                      |
|of genus 6 and degree 10 containing rational curve of degree 1 (GM fourfold) |
|of genus 21 and degree 40 containing rational curve of degree 2              |
|                                                                             |
|                                                                             |
|surface                                                                      |
+-----------------------------------------------------------------------------+

See also
========

  * K3(ZZ) -- make a general K3 surface
  * K3(String) -- show available functions to construct
    K3 surfaces of given genus
  * LatticePolarizedK3surface Sequence
    -- image of the embedding of a K3 surface

Ways to use K3 :
================

  * "K3(ZZ,ZZ,ZZ)"
  * K3(EmbeddedK3surface) -- make a
    lattice-polarized K3 surface from an embedded K3 surface
  * K3(String) -- show available functions to construct
    K3 surfaces of given genus
  * K3(ZZ) -- make a general K3 surface

For the programmer
==================

The object K3, is a method function with options (Macaulay2Doc).

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

Synopsis
========

  * Function: K3,
  * Usage:
        K3 S
  * Inputs:
      * S, an embedded K3 surface, a special K3
        surface that contains a curve $C$
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing, => ..., default value ZZ/65521, make a
        lattice-polarized K3 surface
      * Verbose, => ..., default value true, make a lattice-polarized
        K3 surface
  * Outputs:
      * a K3 surface, 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  |
|                                                                             |
|                                                                             |
|o3 : Lattice-polarized K3 surface                                            |
|-----------------------------------------------------------------------------|
|3  |                                                                         |
|-2 |                                                                         |
+-----------------------------------------------------------------------------+
|i4 : T(1,0)                                                                  |
|                                                                             |
|o4 = K3 surface of genus 7 and degree 12 in PP^7                             |
|                                                                             |
|o4 : Embedded K3 surface                                                     |
+-----------------------------------------------------------------------------+

Ways to use this method:
========================

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

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

Synopsis
========

  * Function: K3,
  * Usage:
        K3 "G"
        K3("G",[Unique=>true])
  * Inputs:
      * G, a string (Macaulay2Doc), the string of an integer
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing, => ..., default value ZZ/65521, make a
        lattice-polarized K3 surface
      * Verbose, => ..., default value true, make a lattice-polarized
        K3 surface
  * Outputs:
      * a list (Macaulay2Doc), a list of terns (d,g,n) such that
        (see (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|
|(K3(11,2,-2))(1,0) -- K3 surface of genus 11 and degree 20 containing rationa|
|(K3(3,6,-2))(1,2) -- K3 surface of genus 11 and degree 20 containing rational|
|(K3(11,3,0))(1,0) -- K3 surface of genus 11 and degree 20 containing elliptic|
|(K3(8,3,0))(1,1) -- K3 surface of genus 11 and degree 20 containing elliptic |
|(K3(5,3,0))(1,2) -- K3 surface of genus 11 and degree 20 containing elliptic |
|(K3(11,4,0))(1,0) -- K3 surface of genus 11 and degree 20 containing elliptic|
|(K3(7,4,0))(1,1) -- K3 surface of genus 11 and degree 20 containing elliptic |
|(K3(3,4,0))(1,2) -- K3 surface of genus 11 and degree 20 containing elliptic |
|(K3(11,5,0))(1,0) -- K3 surface of genus 11 and degree 20 containing elliptic|
|(K3(5,7,-2))(1,1) -- K3 surface of genus 11 and degree 20 containing rational|
|(K3(6,5,0))(1,1) -- K3 surface of genus 11 and degree 20 containing elliptic |
|(K3(5,6,0))(1,1) -- K3 surface of genus 11 and degree 20 containing elliptic |
|(K3(4,7,0))(1,1) -- K3 surface of genus 11 and degree 20 containing elliptic |
|(K3(3,8,0))(1,1) -- K3 surface of genus 11 and degree 20 containing elliptic |
|                                                                             |
|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                                                                    |
|-----------------------------------------------------------------------------|
| curve of degree 1                                                           |
|l curve of degree 2                                                          |
| curve of degree 2                                                           |
| curve of degree 3                                                           |
|curve of degree 3                                                            |
|curve of degree 3                                                            |
| curve of degree 4                                                           |
|curve of degree 4                                                            |
|curve of degree 4                                                            |
| curve of degree 5                                                           |
| curve of degree 5                                                           |
|curve of degree 5                                                            |
|curve of degree 6                                                            |
|curve of degree 7                                                            |
|curve of degree 8                                                            |
+-----------------------------------------------------------------------------+
|i2 : S = K3(5,5,-2)                                                          |
|                                                                             |
|o2 = K3 surface with rank                                                    |
|                                                                             |
|     -- (1,0): K3 surface                                                    |
|     -- (1,1): K3 surface                                                    |
|     -- (1,2): K3 surface                                                    |
|     -- (2,0): K3 surface                                                    |
|     -- (2,1): K3 surface                                                    |
|     -- (2,2): K3 surface                                                    |
|     -- (3,0): K3 surface                                                    |
|     -- (2,3): K3 surface                                                    |
|     -- (2,4): K3 surface                                                    |
|                                                                             |
|                                                                             |
|o2 : Lattice-polarized K3                                                    |
|-----------------------------------------------------------------------------|
|2 lattice defined by the intersection matrix: | 8 5  |                       |
|                                              | 5 -2 |                       |
|of genus 5 and degree 8 containing rational curve of degree 5                |
|of genus 9 and degree 16 containing rational curve of degree 3               |
|of genus 11 and degree 20 containing rational curve of degree 1 (            |
|of genus 17 and degree 32 containing rational curve of degree 10             |
|of genus 26 and degree 50 containing rational curve of degree 8 (            |
|of genus 33 and degree 64 containing rational curve of degree 6              |
|of genus 37 and degree 72 containing rational curve of degree 15             |
|of genus 38 and degree 74 containing rational curve of degree 4 (            |
|of genus 41 and degree 80 containing rational curve of degree 2              |
|                                                                             |
|                                                                             |
|surface                                                                      |
|-----------------------------------------------------------------------------|
|GM fourfold)                                                                 |
|                                                                             |
|GM fourfold)                                                                 |
|                                                                             |
|                                                                             |
|cubic fourfold) (GM fourfold)                                                |
+-----------------------------------------------------------------------------+
|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  |
|                                                                             |
|                                                                             |
|o4 : Lattice-polarized K3 surface                                            |
|-----------------------------------------------------------------------------|
|1  |                                                                         |
|-2 |                                                                         |
+-----------------------------------------------------------------------------+

See also
========

  * K3(ZZ,ZZ,ZZ) -- make a lattice-polarized K3 surface
  * LatticePolarizedK3surface Sequence
    -- image of the embedding of a K3 surface

Ways to use this method:
========================

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

K3(ZZ) -- make a general K3 surface
***********************************

Synopsis
========

  * Function: K3,
  * Usage:
        K3 g
        K3(g,CoefficientRing=>K)
  * Inputs:
      * g, an integer (Macaulay2Doc),
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing, => ..., default value ZZ/65521, make a
        lattice-polarized K3 surface
      * Verbose, => ..., default value true, make a lattice-polarized
        K3 surface
  * Outputs:
      * an embedded K3 surface, a general K3 surface
        defined over $K$ of genus $g$ in $\mathbb{P}^g$

Description
===========

+------------------------------------------------+
|i1 : K3 9                                       |
|                                                |
|o1 = K3 surface of genus 9 and degree 16 in PP^9|
|                                                |
|o1 : Embedded K3 surface                        |
+------------------------------------------------+

See also
========

  * K3(ZZ,ZZ,ZZ) -- make a lattice-polarized K3 surface

Ways to use this method:
========================

  * K3(ZZ) -- make a general K3 surface

LatticePolarizedK3surface -- the class of all lattice-polarized K3 surfaces
***************************************************************************

See also
========

  * K3 -- make a lattice-polarized K3 surface
  * EmbeddedK3surface -- the class of all embedded K3
    surfaces

Functions and methods returning a K3 surface :
==============================================

  * pentagonalK3 -- pentagonal K3 surface
  * tetragonalK3 -- tetragonal K3 surface
  * trigonalK3 -- trigonal K3 surface

Methods that use a K3 surface :
===============================

  * coefficientRing(LatticePolarizedK3surface)
    -- coefficient ring of a K3
    surface
  * degree(LatticePolarizedK3surface,ZZ,ZZ)
    -- degree of a K3 surface
  * genus(LatticePolarizedK3surface,ZZ,ZZ)
    -- genus of a K3 surface
  * LatticePolarizedK3surface Sequence
    -- image of the embedding of a K3 surface
  * map(LatticePolarizedK3surface,ZZ,ZZ)
    -- embedding of a K3 surface
  * "project(VisibleList,LatticePolarizedK3surface,ZZ,ZZ)" -- see
    project, -- project a K3 surface

For the programmer
==================

The object LatticePolarizedK3surface, is a
type (Macaulay2Doc), with ancestor classes HashTable (Macaulay2Doc),
< Thing (Macaulay2Doc).

LatticePolarizedK3surface Sequence -- image of the embedding of a K3 surface
****************************************************************************

Synopsis
========

  * Operator: SPACE (Macaulay2Doc),
  * Usage:
        S(a,b)
  * Inputs:
      * S, a K3 surface, a lattice-polarized
        K3 surface with rank 2 lattice spanned by $H,C$
      * a sequence (Macaulay2Doc), a sequence of two integers
        (a,b)
  * Outputs:
      * an embedded K3 surface, the image of the
        embedding of $S$ by the complete linear system $|a H + b C|$

Description
===========

+-----------------------------------------------------------------------------+
|i1 : S = K3(5,2,-2)                                                          |
|                                                                             |
|o1 = K3 surface with rank                                                    |
|                                                                             |
|     -- (1,0): K3 surface                                                    |
|     -- (2,0): K3 surface                                                    |
|     -- (2,1): K3 surface                                                    |
|     -- (3,0): K3 surface                                                    |
|     -- (3,1): K3 surface                                                    |
|                                                                             |
|                                                                             |
|o1 : Lattice-polarized K3                                                    |
|-----------------------------------------------------------------------------|
|2 lattice defined by the intersection matrix: | 8 2  |                       |
|                                              | 2 -2 |                       |
|of genus 5 and degree 8 containing rational curve of degree 2                |
|of genus 17 and degree 32 containing rational curve of degree 4              |
|of genus 20 and degree 38 containing rational curve of degree 2              |
|of genus 37 and degree 72 containing rational curve of degree 6              |
|of genus 42 and degree 82 containing rational curve of degree 4              |
|                                                                             |
|                                                                             |
|surface                                                                      |
|-----------------------------------------------------------------------------|
|(cubic fourfold)                                                             |
|                                                                             |
|(GM fourfold)                                                                |
+-----------------------------------------------------------------------------+
|i2 : S(1,0)                                                                  |
|                                                                             |
|o2 = K3 surface of genus 5 and degree 8 in PP^5                              |
|                                                                             |
|o2 : Embedded K3 surface                                                     |
+-----------------------------------------------------------------------------+
|i3 : S(2,1)                                                                  |
|                                                                             |
|o3 = K3 surface of genus 20 and degree 38 in PP^20                           |
|                                                                             |
|o3 : Embedded K3 surface                                                     |
+-----------------------------------------------------------------------------+

See also
========

  * map(LatticePolarizedK3surface,ZZ,ZZ)
    -- embedding of a K3 surface

Ways to use this method:
========================

  * "EmbeddedK3surface Sequence"
  * LatticePolarizedK3surface Sequence
    -- image of the embedding of a K3 surface

map(LatticePolarizedK3surface,ZZ,ZZ) -- embedding of a K3 surface
*****************************************************************

Synopsis
========

  * Function: map (Macaulay2Doc),
  * Usage:
        map(S,a,b)
  * Inputs:
      * S, a K3 surface,
      * a, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Optional inputs (Macaulay2Doc),:
      * Degree (Macaulay2Doc), => ..., default value null, null
      * DegreeLift (Macaulay2Doc), => ...,
        default value null, null
      * DegreeMap (Macaulay2Doc), => ...,
        default value null, null
  * Outputs:
      * the birational morphism (Cremona), defined by the
        complete linear system $|a H + b C|$, where $H,C$ is the basis of the
        lattice associated to $S$

Description
===========

+-----------------------------------------------------------------------------+
|i1 : S = K3(3,1,-2)                                                          |
|                                                                             |
|o1 = K3 surface with rank                                                    |
|                                                                             |
|     -- (1,0): K3 surface                                                    |
|     -- (2,0): K3 surface                                                    |
|     -- (3,0): K3 surface                                                    |
|     -- (3,1): K3 surface                                                    |
|     -- (4,0): K3 surface                                                    |
|     -- (4,1): K3 surface                                                    |
|                                                                             |
|                                                                             |
|o1 : Lattice-polarized K3                                                    |
|-----------------------------------------------------------------------------|
|2 lattice defined by the intersection matrix: | 4 1  |                       |
|                                              | 1 -2 |                       |
|of genus 3 and degree 4 containing rational curve of degree 1                |
|of genus 9 and degree 16 containing rational curve of degree 2               |
|of genus 19 and degree 36 containing rational curve of degree 3              |
|of genus 21 and degree 40 containing rational curve of degree 1              |
|of genus 33 and degree 64 containing rational curve of degree 4              |
|of genus 36 and degree 70 containing rational curve of degree 2              |
|                                                                             |
|                                                                             |
|surface                                                                      |
+-----------------------------------------------------------------------------+
|i2 : f = map(S,2,1);                                                         |
|                                                                             |
|o2 : MultirationalMap (rational map from surface in PP^3 to PP^10)           |
+-----------------------------------------------------------------------------+
|i3 : isMorphism f                                                            |
|                                                                             |
|o3 = true                                                                    |
+-----------------------------------------------------------------------------+
|i4 : degree f                                                                |
|                                                                             |
|o4 = 1                                                                       |
+-----------------------------------------------------------------------------+
|i5 : assert(image f == S(2,1))                                               |
+-----------------------------------------------------------------------------+

See also
========

  * LatticePolarizedK3surface Sequence
    -- image of the embedding of a K3 surface

Ways to use this method:
========================

  * "map(EmbeddedK3surface,ZZ,ZZ)"
  * map(LatticePolarizedK3surface,ZZ,ZZ)
    -- embedding of a K3 surface

mukaiModel -- Mukai models
**************************

Synopsis
========

  * Usage:
        mukaiModel g
  * Inputs:
      * g, an integer (Macaulay2Doc), the genus
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing => ..., default value ZZ/65521
  * Outputs:
      * an embedded projective variety (MultiprojectiveVarieties),
        the Mukai model
        of genus $g$ and degree $2g-2$

Description
===========

+-----------------------------------------------------------------------------+
|i1 : X = mukaiModel 9;                                                       |
|                                                                             |
|o1 : ProjectiveVariety, 6-dimensional subvariety of PP^13                    |
+-----------------------------------------------------------------------------+
|i2 : (degree X, sectionalGenus X)                                            |
|                                                                             |
|o2 = (16, 9)                                                                 |
|                                                                             |
|o2 : Sequence                                                                |
+-----------------------------------------------------------------------------+
|i3 : parametrize X                                                           |
|                                                                             |
|o3 = multi-rational map consisting of one single rational map                |
|     source variety: PP^6                                                    |
|     target variety: 6-dimensional subvariety of PP^13 cut out by 21         |
|     dominance: true                                                         |
|                                                                             |
|o3 : MultirationalMap (dominant rational map from PP^6 to X)                 |
|-----------------------------------------------------------------------------|
|hypersurfaces of degree 2                                                    |
+-----------------------------------------------------------------------------+

Ways to use mukaiModel :
========================

  * "mukaiModel(ZZ)"

For the programmer
==================

The object mukaiModel, is a method function with options (Macaulay2Doc).

pentagonalK3 -- pentagonal K3 surface
*************************************

Synopsis
========

  * Usage:
        pentagonalK3 g
  * Inputs:
      * g, an integer (Macaulay2Doc), the genus
  * Optional inputs (Macaulay2Doc),:
      * 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:      |
|                                                                             |
|     -- (1,0): K3 surface of genus 11 and degree 20 containing elliptic      |
|     -- (1,1): K3 surface of genus 16 and degree 30 containing elliptic      |
|     -- (1,2): K3 surface of genus 21 and degree 40 containing elliptic      |
|     -- (1,3): K3 surface of genus 26 and degree 50 containing elliptic      |
|     -- (1,4): K3 surface of genus 31 and degree 60 containing elliptic      |
|     -- (1,5): K3 surface of genus 36 and degree 70 containing elliptic      |
|     -- (1,6): K3 surface of genus 41 and degree 80 containing elliptic      |
|     -- (2,0): K3 surface of genus 41 and degree 80 containing elliptic      |
|                                                                             |
|                                                                             |
|o1 : Lattice-polarized K3 surface                                            |
|-----------------------------------------------------------------------------|
|| 20 5 |                                                                     |
|| 5  0 |                                                                     |
|curve of degree 5 (GM fourfold)                                              |
|curve of degree 5                                                            |
|curve of degree 5                                                            |
|curve of degree 5 (GM fourfold)                                              |
|curve of degree 5                                                            |
|curve of degree 5                                                            |
|curve of degree 5                                                            |
|curve of degree 10                                                           |
+-----------------------------------------------------------------------------+
|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 (Macaulay2Doc).

project -- project a K3 surface
*******************************

Synopsis
========

  * Usage:
        project({i,j,k,...},S,a,b)
        project({i,j,k,...},S(a,b))
  * Inputs:
      * a visible list (Macaulay2Doc), a list
        $\{i,j,k,\ldots\}$ of nonnegative integers
      * S, a K3 surface, a lattice-polarized
        K3 surface with rank 2 lattice spanned by $H,C$
      * a, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Outputs:
      * an embedded projective variety (MultiprojectiveVarieties),
        the projection of
        $S$ embedded by the complete linear system $|a H + b C|$ from $i$
        random points of multiplicity 1, $j$ random points of multiplicity 2,
        $k$ random points of multiplicity 3, and so on until the last integer
        in the given list.

Description
===========

+-----------------------------------------------------------------------------+
|i1 : S = K3(8,2,-2)                                                          |
|                                                                             |
|o1 = K3 surface with rank                                                    |
|                                                                             |
|     -- (1,0): K3 surface                                                    |
|     -- (2,0): K3 surface                                                    |
|     -- (2,1): K3 surface                                                    |
|                                                                             |
|                                                                             |
|o1 : Lattice-polarized K3                                                    |
|-----------------------------------------------------------------------------|
|2 lattice defined by the intersection matrix: | 14 2  |                      |
|                                              | 2  -2 |                      |
|of genus 8 and degree 14 containing rational curve of degree 2 (cubic        |
|of genus 29 and degree 56 containing rational curve of degree 4              |
|of genus 32 and degree 62 containing rational curve of degree 2 (cubic       |
|                                                                             |
|                                                                             |
|surface                                                                      |
|-----------------------------------------------------------------------------|
|fourfold)                                                                    |
|                                                                             |
| fourfold)                                                                   |
+-----------------------------------------------------------------------------+
|i2 : project({5,3,1},S,2,1); -- (5th + 3rd + simple)-projection of           |
|-- *** simulation ***                                                        |
|-- surface of degree 62 and sectional genus 32 in PP^32 (quadrics:           |
|-- surface of degree 37 and sectional genus 22 in PP^17 (quadrics:           |
|-- surface of degree 28 and sectional genus 19 in PP^11 (quadrics:           |
|-- surface of degree 27 and sectional genus 19 in PP^10 (quadrics:           |
|                                                                             |
|-- (degree and genus are as expected)                                        |
|                                                                             |
|o2 : ProjectiveVariety, surface in PP^10                                     |
|-----------------------------------------------------------------------------|
|S(2,1)                                                                       |
|                                                                             |
|435, cubics: 6264)                                                           |
|100, cubics: 979)                                                            |
|28, cubics: 248)                                                             |
|19, cubics: 176)                                                             |
+-----------------------------------------------------------------------------+

See also
========

  * LatticePolarizedK3surface Sequence
    -- image of the embedding of a K3 surface

Ways to use project :
=====================

  * "project(VisibleList,EmbeddedK3surface)"
  * "project(VisibleList,LatticePolarizedK3surface,ZZ,ZZ)"

For the programmer
==================

The object project, is a method function (Macaulay2Doc).

tetragonalK3 -- tetragonal K3 surface
*************************************

Synopsis
========

  * Usage:
        tetragonalK3 g
  * Inputs:
      * g, an integer (Macaulay2Doc), the genus
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing => ..., default value ZZ/65521
  * Outputs:
      * a K3 surface, a random tetragonal K3
        surface of genus $g$

Description
===========

+-----------------------------------------------------------------------------+
|i1 : S = tetragonalK3 11                                                     |
|                                                                             |
|o1 = K3 surface with rank 2 lattice defined by the intersection matrix:      |
|                                                                             |
|     -- (1,0): K3 surface of genus 11 and degree 20 containing elliptic      |
|     -- (1,1): K3 surface of genus 15 and degree 28 containing elliptic      |
|     -- (1,2): K3 surface of genus 19 and degree 36 containing elliptic      |
|     -- (1,3): K3 surface of genus 23 and degree 44 containing elliptic      |
|     -- (1,4): K3 surface of genus 27 and degree 52 containing elliptic      |
|     -- (1,5): K3 surface of genus 31 and degree 60 containing elliptic      |
|     -- (1,6): K3 surface of genus 35 and degree 68 containing elliptic      |
|     -- (1,7): K3 surface of genus 39 and degree 76 containing elliptic      |
|     -- (2,0): K3 surface of genus 41 and degree 80 containing elliptic      |
|     -- (1,8): K3 surface of genus 43 and degree 84 containing elliptic      |
|                                                                             |
|                                                                             |
|o1 : Lattice-polarized K3 surface                                            |
|-----------------------------------------------------------------------------|
|| 20 4 |                                                                     |
|| 4  0 |                                                                     |
|curve of degree 4 (GM fourfold)                                              |
|curve of degree 4                                                            |
|curve of degree 4                                                            |
|curve of degree 4                                                            |
|curve of degree 4 (GM fourfold)                                              |
|curve of degree 4                                                            |
|curve of degree 4 (GM fourfold)                                              |
|curve of degree 4                                                            |
|curve of degree 8                                                            |
|curve of degree 4                                                            |
+-----------------------------------------------------------------------------+
|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
  * pentagonalK3 -- pentagonal K3 surface

Ways to use tetragonalK3 :
==========================

  * "tetragonalK3(ZZ)"

For the programmer
==================

The object tetragonalK3, is a method function with options (Macaulay2Doc).

trigonalK3 -- trigonal K3 surface
*********************************

Synopsis
========

  * Usage:
        trigonalK3 g
  * Inputs:
      * g, an integer (Macaulay2Doc), the genus
  * Optional inputs (Macaulay2Doc),:
      * CoefficientRing => ..., default value ZZ/65521
  * Outputs:
      * a K3 surface, 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:      |
|                                                                             |
|     -- (1,0): K3 surface of genus 11 and degree 20 containing elliptic      |
|     -- (1,1): K3 surface of genus 14 and degree 26 containing elliptic      |
|     -- (1,2): K3 surface of genus 17 and degree 32 containing elliptic      |
|     -- (1,3): K3 surface of genus 20 and degree 38 containing elliptic      |
|     -- (1,4): K3 surface of genus 23 and degree 44 containing elliptic      |
|     -- (1,5): K3 surface of genus 26 and degree 50 containing elliptic      |
|     -- (1,6): K3 surface of genus 29 and degree 56 containing elliptic      |
|     -- (1,7): K3 surface of genus 32 and degree 62 containing elliptic      |
|     -- (1,8): K3 surface of genus 35 and degree 68 containing elliptic      |
|     -- (1,9): K3 surface of genus 38 and degree 74 containing elliptic      |
|     -- (1,10): K3 surface of genus 41 and degree 80 containing elliptic     |
|     -- (2,0): K3 surface of genus 41 and degree 80 containing elliptic      |
|     -- (1,11): K3 surface of genus 44 and degree 86 containing elliptic     |
|                                                                             |
|                                                                             |
|o1 : Lattice-polarized K3 surface                                            |
|-----------------------------------------------------------------------------|
|| 20 3 |                                                                     |
|| 3  0 |                                                                     |
|curve of degree 3 (GM fourfold)                                              |
|curve of degree 3 (cubic fourfold) (GM fourfold)                             |
|curve of degree 3                                                            |
|curve of degree 3 (cubic fourfold)                                           |
|curve of degree 3                                                            |
|curve of degree 3 (GM fourfold)                                              |
|curve of degree 3                                                            |
|curve of degree 3 (cubic fourfold)                                           |
|curve of degree 3 (GM fourfold)                                              |
|curve of degree 3 (cubic fourfold) (GM fourfold)                             |
| curve of degree 3                                                           |
|curve of degree 6                                                            |
| curve of degree 3 (cubic fourfold)                                          |
+-----------------------------------------------------------------------------+
|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
  * tetragonalK3 -- tetragonal K3 surface
  * pentagonalK3 -- pentagonal K3 surface

Ways to use trigonalK3 :
========================

  * "trigonalK3(ZZ)"

For the programmer
==================

The object trigonalK3, is a method function with options (Macaulay2Doc).