schubertCycle -- take a random Schubert cycle

Synopsis

Usage:

schubertCycle(a,G)
Inputs:
- a, a visible list, a list of integers $a = (a_0,\ldots,a_k)$ with $n-k\geq a_0 \geq \cdots \geq a_k \geq 0$
- G, a ring, the coordinate ring Grass$(k,n)$ of the Grassmannian of $k$-dimensional subspaces of $\mathbb{P}^n$
Outputs:
- an ideal, the Schubert cycle $\Sigma_a(\mathcal P)\subset\mathbb{G}(k,n)$ associated to a random complete flag $\mathcal P$ of nested projective subspace $\emptyset\subset P_0\subset \cdots \subset P_{n-1} \subset P_{n} = \mathbb{P}^n$ with $dim(P_i)=i$

Description

For the general theory, see e.g. the book 3264 & All That - Intersection Theory in Algebraic Geometry, by D. Eisenbud and J. Harris.

i1 : G = Grass(1,5,ZZ/33331,Variable=>"x");

i2 : S = schubertCycle({2,1},G)

o2 = ideal (x    - 8610x    + 10298x    + 5789x    - 2504x   , x    -
             0,5        1,5         2,5        3,5        4,5   0,4  
     ------------------------------------------------------------------------
     8610x    + 10298x    + 5789x    + 8150x   , x    - 8610x    + 10298x   
          1,4         2,4        3,4        4,5   0,3        1,3         2,3
     ------------------------------------------------------------------------
     + 2504x    + 8150x   , x    - 13774x    + 5598x    + 4612x    + 8887x   
            3,4        3,5   1,2         1,3        2,3        1,4        2,4
     ------------------------------------------------------------------------
     - 4557x    - 15329x    + 13604x    - 10197x    - 15599x   , x    -
            3,4         1,5         2,5         3,5         4,5   0,2  
     ------------------------------------------------------------------------
     2442x    - 3635x    + 12099x    - 8402x    - 5183x    + 8070x    +
          1,3        2,3         1,4        2,4        3,4        1,5  
     ------------------------------------------------------------------------
     13456x    - 2316x    + 16540x   , x    + 6295x    - 14426x    + 205x   
           2,5        3,5         4,5   0,1        1,3         2,3       1,4
     ------------------------------------------------------------------------
     - 8600x    + 2062x    + 5724x    + 3799x    - 16056x    - 16413x   )
            2,4        3,4        1,5        2,5         3,5         4,5

o2 : Ideal of G

i3 : cycleClass S

o3 = s
      2,1

o3 : ZZ[s   , s   ]
         3,0   2,1

By calling the function as below, it returns as second output an automorphism of the Grassmannian which sends the random Schubert cycle to a standard Schubert cycle.

i4 : (S,f) = schubertCycle({2,1},G,"standard");

i5 : f;

o5 : RationalMap (linear rational map from 8-dimensional subvariety of PP^14 to 8-dimensional subvariety of PP^14)

i6 : S

o6 = ideal (x    + 1141x    + 10832x    - 3672x    + 1772x   , x    +
             0,5        1,5         2,5        3,5        4,5   0,4  
     ------------------------------------------------------------------------
     1141x    + 10832x    - 3672x    + 12906x   , x    + 1141x    + 10832x   
          1,4         2,4        3,4         4,5   0,3        1,3         2,3
     ------------------------------------------------------------------------
     - 1772x    + 12906x   , x    + 10020x    - 5740x    + 3630x    +
            3,4         3,5   1,2         1,3        2,3        1,4  
     ------------------------------------------------------------------------
     1469x    - 8597x    + 9316x    + 16226x    + 6818x    - 14991x   , x   
          2,4        3,4        1,5         2,5        3,5         4,5   0,2
     ------------------------------------------------------------------------
     - 287x    - 13195x    - 8786x    - 11351x    + 9863x    + 3033x    -
           1,3         2,3        1,4         2,4        3,4        1,5  
     ------------------------------------------------------------------------
     2255x    - 13215x    + 5928x   , x    + 14576x    - 13365x    -
          2,5         3,5        4,5   0,1         1,3         2,3  
     ------------------------------------------------------------------------
     12192x    + 13321x    + 4110x    - 2450x    + 5669x    - 8920x    +
           1,4         2,4        3,4        1,5        2,5        3,5  
     ------------------------------------------------------------------------
     6120x   )
          4,5

o6 : Ideal of G

i7 : f S

o7 = ideal (x   , x   , x   , x   , x   , x   )
             4,5   3,5   2,5   1,5   0,5   3,4

o7 : Ideal of G

Ways to use `schubertCycle` :

"schubertCycle(VisibleList,Ring)"
"schubertCycle(VisibleList,Ring,String)"
schubertCycle(VisibleList,EmbeddedProjectiveVariety) -- take a random Schubert cycle
"schubertCycle(VisibleList,EmbeddedProjectiveVariety,String)" -- see schubertCycle(VisibleList,EmbeddedProjectiveVariety) -- take a random Schubert cycle

For the programmer

The object schubertCycle is a method function.

schubertCycle -- take a random Schubert cycle

Synopsis

Description

See also

Ways to use schubertCycle :

For the programmer

Ways to use `schubertCycle` :