productOfProjectiveSpaces -- Cox ring of a product of projective spaces and it Koszul dual exterior algebra

Synopsis

Usage:

(S,E)=productOfProjectiveSpaces N

(S,E)=productOfProjectiveSpaces n
Inputs:
- N, a list, the list \{n_1,...,n_t\} \, of the dimensions of the factors
- n, an integer, Gives n copies of P^1
Optional inputs:
- CoefficientField => a ring, default value ZZ/32003, ground field of S,E
- Variables => a list, default value {x, e}, list of 2 symbols
- CohomologyVariables => a list, default value {h, k}, list of 2 symbols
Outputs:
- S, a polynomial ring, homogeneous coordinate ring of P^{n_1}x ... x P^{n_t}
- E, a polynomial ring, the corresponding exterior algebra

Description

The degrees of the variables for the i-th projective space are indexed x_(i,0),..,x_(i,n_i-1), and have degree (0..0,1,0,..0) with a 1 in the i-th place. The script also caches some values in S.TateData and E.TateData, so that S and E can subsequently find each other and also their cohomology ring.

i1 : (S,E)=productOfProjectiveSpaces{1,2}

o1 = (S, E)

o1 : Sequence

i2 : vars S

o2 = | x_(0,0) x_(0,1) x_(1,0) x_(1,1) x_(1,2) |

             1      5
o2 : Matrix S  <-- S

i3 : vars E

o3 = | e_(0,0) e_(0,1) e_(1,0) e_(1,1) e_(1,2) |

             1      5
o3 : Matrix E  <-- E

i4 : (S,E) = productOfProjectiveSpaces({1,1},
         Variables =>{getSymbol "u",getSymbol"v"},
         CohomologyVariables =>{getSymbol "p",getSymbol "q"},
         CoefficientField => QQ)

o4 = (S, E)

o4 : Sequence

i5 : (coefficientRing S) === (coefficientRing E)

o5 = true

i6 : trim (ideal vars S)^2

             2                                   2                       
o6 = ideal (u   , u   u   , u   u   , u   u   , u   , u   u   , u   u   ,
             1,1   1,0 1,1   0,1 1,1   0,0 1,1   1,0   0,1 1,0   0,0 1,0 
     ------------------------------------------------------------------------
      2               2
     u   , u   u   , u   )
      0,1   0,0 0,1   0,0

o6 : Ideal of S

i7 : trim (ideal vars E)^2

o7 = ideal (v   v   , v   v   , v   v   , v   v   , v   v   , v   v   )
             1,0 1,1   0,1 1,1   0,0 1,1   0,1 1,0   0,0 1,0   0,0 0,1

o7 : Ideal of E

i8 : peek S.TateData

o8 = MutableHashTable{BeilinsonBundles => MutableHashTable{}}
                      CohomRing => ZZ[p..q]
                      Rings => (S, E)

Ways to use productOfProjectiveSpaces:

productOfProjectiveSpaces(List)
productOfProjectiveSpaces(ZZ)

For the programmer

The object productOfProjectiveSpaces is a method function with options.