# productOfProjectiveSpaces -- Constructs the multi-graded ring of a product of copies of P^1 (pp is a synonym)

## Synopsis

• Usage:
R = productOfProjectiveSpaces(L,y,kkk)
R = productOfProjectiveSpaces(L,y)
R = productOfProjectiveSpaces L
R = productOfProjectiveSpaces (n, y)
R = productOfProjectiveSpaces n
• Inputs:
• L, a list, of positive integers, the dimensions of the projective spaces
• n, an integer, positive integer, number of P^1 factors
• y, ,
• kkk, a ring, ground field to use in the construction variable name for R (defaults to "x" if none given)
• Optional inputs:
• Characteristic => an integer, default value 32003, the characteristic of the ground field
• Outputs:
• R, a ring, ZZ^n - graded

## Description

The variables are in 1+L_i -tuples, x_(0,i).. x_(L_i,i) with degree {0..0,1,0..0}, the 1 being in the i-th place.

 i1 : R = productOfProjectiveSpaces{1,3} o1 = R o1 : PolynomialRing i2 : v = gens R o2 = {x , x , x , x , x , x } 0,0 1,0 0,1 1,1 2,1 3,1 o2 : List i3 : v/degree o3 = {{1, 0}, {1, 0}, {0, 1}, {0, 1}, {0, 1}, {0, 1}} o3 : List i4 : gens productOfProjectiveSpaces({1,1},symbol y) o4 = {y , y , y , y } 0,0 1,0 0,1 1,1 o4 : List i5 : gens productOfProjectiveSpaces 2 o5 = {x , x , x , x } 0,0 1,0 0,1 1,1 o5 : List

## Ways to use productOfProjectiveSpaces :

• "productOfProjectiveSpaces(List)"
• "productOfProjectiveSpaces(List,Symbol)"
• "productOfProjectiveSpaces(List,Symbol,Ring)"
• "productOfProjectiveSpaces(ZZ)"
• "productOfProjectiveSpaces(ZZ,Symbol)"

## For the programmer

The object productOfProjectiveSpaces is .