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# graph -- closure of the graph of a rational map

- Usage:
    `graph phi`
- Inputs:
  - phi, a rational map
- Optional inputs:
  - BlowUpStrategy => ..., default value "Eliminate",
- Outputs:
  - a rational map, the first projection
  - a rational map, the second projection

## Description

    i1 : (ZZ/190181)[x_0..x_4]; phi = rationalMap(minors(2,matrix{{x_0..x_3},{x_1..x_4}}),Dominant=>true)

    o2 = -- rational map --
                        ZZ
         source: Proj(------[x , x , x , x , x ])
                      190181  0   1   2   3   4
                                      ZZ
         target: subvariety of Proj(------[y , y , y , y , y , y ]) defined by
                                    190181  0   1   2   3   4   5
                 {
                  y y  - y y  + y y
                   2 3    1 4    0 5
                 }
         defining forms: {
                             2
                          - x  + x x ,
                             1    0 2

                          - x x  + x x ,
                             1 2    0 3

                             2
                          - x  + x x ,
                             2    1 3

                          - x x  + x x ,
                             1 3    0 4

                          - x x  + x x ,
                             2 3    1 4

                             2
                          - x  + x x
                             3    2 4
                         }

    o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)

    i3 : time (p1,p2) = graph phi;
     -- used 0.0186824s (cpu); 0.0182878s (thread); 0s (gc)

    i4 : p1

    o4 = -- rational map --
                                      ZZ                                 ZZ
         source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y , y , y , y , y ]) defined by
                                    190181  0   1   2   3   4          190181  0   1   2   3   4   5
                 {
                  y y  - y y  + y y ,
                   2 3    1 4    0 5

                  x y  - x y  + x y ,
                   4 2    3 4    2 5

                  x y  - x y  + x y ,
                   3 2    2 4    1 5

                  x y  - x y  + x y ,
                   4 1    3 3    1 5

                  x y  - x y  + x y ,
                   3 1    2 3    0 5

                  x y  - x y  - x y  + x y ,
                   2 1    1 2    1 3    0 4

                  x y  - x y  + x y ,
                   4 0    2 3    1 4

                  x y  - x y  + x y ,
                   3 0    1 3    0 4

                  x y  - x y  + x y
                   2 0    1 1    0 2
                 }
                        ZZ
         target: Proj(------[x , x , x , x , x ])
                      190181  0   1   2   3   4
         defining forms: {
                          x ,
                           0

                          x ,
                           1

                          x ,
                           2

                          x ,
                           3

                          x
                           4
                         }

    o4 : MultihomogeneousRationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4)

    i5 : p2

    o5 = -- rational map --
                                      ZZ                                 ZZ
         source: subvariety of Proj(------[x , x , x , x , x ]) x Proj(------[y , y , y , y , y , y ]) defined by
                                    190181  0   1   2   3   4          190181  0   1   2   3   4   5
                 {
                  y y  - y y  + y y ,
                   2 3    1 4    0 5

                  x y  - x y  + x y ,
                   4 2    3 4    2 5

                  x y  - x y  + x y ,
                   3 2    2 4    1 5

                  x y  - x y  + x y ,
                   4 1    3 3    1 5

                  x y  - x y  + x y ,
                   3 1    2 3    0 5

                  x y  - x y  - x y  + x y ,
                   2 1    1 2    1 3    0 4

                  x y  - x y  + x y ,
                   4 0    2 3    1 4

                  x y  - x y  + x y ,
                   3 0    1 3    0 4

                  x y  - x y  + x y
                   2 0    1 1    0 2
                 }
                                      ZZ
         target: subvariety of Proj(------[y , y , y , y , y , y ]) defined by
                                    190181  0   1   2   3   4   5
                 {
                  y y  - y y  + y y
                   2 3    1 4    0 5
                 }
         defining forms: {
                          y ,
                           0

                          y ,
                           1

                          y ,
                           2

                          y ,
                           3

                          y ,
                           4

                          y
                           5
                         }

    o5 : MultihomogeneousRationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5)

    i6 : assert(p1 * phi == p2 and p2 * phi^-1 == p1)

    i7 : describe p2

    o7 = rational map defined by multiforms of degree {0, 1}
         source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of degrees ({0, 2},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1})
         target variety: smooth quadric hypersurface in PP^5
         dominance: true
         coefficient ring: ZZ/190181

    i8 : projectiveDegrees p2

    o8 = {51, 28, 14, 6, 2}

    o8 : List

When the source of the rational map is a multi-projective variety, the method returns all the projections.

    i9 : time g = graph p2;
     -- used 0.0309725s (cpu); 0.0305745s (thread); 0s (gc)

    i10 : g_0;

    o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^4)

    i11 : g_1;

    o11 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to PP^5)

    i12 : g_2;

    o12 : MultihomogeneousRationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5)

    i13 : describe g_0

    o13 = rational map defined by multiforms of degree {1, 0, 0}
          source variety: 4-dimensional subvariety of PP^4 x PP^5 x PP^5 cut out by 34 hypersurfaces of degrees ({0, 1, 1},{0, 0, 2},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{0, 2, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0})
          target variety: PP^4
          coefficient ring: ZZ/190181

## See also

- graph(RingMap) -- closure of the graph of a rational map
- graphIdeal -- the ideal of the graph of the regular map corresponding to a ring map

## Ways to use graph:

- `graph(RationalMap)`
- graph(RingMap) -- closure of the graph of a rational map

## For the programmer

The object graph is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Cremona/documentation.m2:253:0.