# MultiProjCoordRing -- A quick way to build the coordinate ring of a product of projective spaces

## Synopsis

• Usage:
MultiProjCoordRing Dims
MultiProjCoordRing (CoeffRing,Dims)
MultiProjCoordRing (var,Dims)
MultiProjCoordRing (CoeffRing,var,Dims)
• Inputs:
• Dims, a list, representing the dimensions of the projective spaces, i.e. {n_1,...,n_m} corresponds to \PP^{n_1} x.... x \PP^{n_m}
• CoeffRing, a ring, the coefficient ring of the graded polynomial ring to be built by the method, by default this is \ZZ/32749
• var, , to be used for the intermediates of the graded polynomial ring to be built by the method
• Outputs:
• a ring, the graded coordinate ring of the \PP^{n_1} x.... x \PP^{n_m} where {n_1,...,n_m} is the input list of dimensions

## Description

Computes the graded coordinate ring of the \PP^{n_1} x.... x \PP^{n_m} where {n_1,...,n_m} is the input list of dimensions. This method is used to quickly build the coordinate ring of a product of projective spaces for use in computations.

 i1 : S=MultiProjCoordRing(QQ,symbol z,{1,3,3}) o1 = S o1 : PolynomialRing i2 : degrees S o2 = {{1, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, ------------------------------------------------------------------------ 0, 1}, {0, 0, 1}, {0, 0, 1}, {0, 0, 1}} o2 : List i3 : R=MultiProjCoordRing {2,3} o3 = R o3 : PolynomialRing i4 : coefficientRing R ZZ o4 = ----- 32749 o4 : QuotientRing i5 : describe R ZZ o5 = -----[x ..x , Degrees => {3:{1}, 4:{0}}, Heft => {2:1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 2] 32749 0 6 {0} {1} {GRevLex => {7:1} } {Position => Up } i6 : A=ChowRing R o6 = A o6 : QuotientRing i7 : describe A ZZ[h ..h ] 1 2 o7 = ---------- 3 4 (h , h ) 1 2 i8 : Segre(A,ideal random({1,1},R)) 2 3 2 2 3 2 2 3 2 2 o8 = 10h h - 6h h - 4h h + 3h h + 3h h + h - h - 2h h - h + h + h 1 2 1 2 1 2 1 2 1 2 2 1 1 2 2 1 2 o8 : A

## Ways to use MultiProjCoordRing :

• "MultiProjCoordRing(List)"
• "MultiProjCoordRing(Ring,List)"
• "MultiProjCoordRing(Ring,Symbol,List)"
• "MultiProjCoordRing(Symbol,List)"

## For the programmer

The object MultiProjCoordRing is .