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 P¹ factors
- y, a symbol
- 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, ZZⁿ - graded

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