# mapToProjectiveSpace -- compute the map to projective space associated with the global sections of a Cartier divisor

## Synopsis

• Usage:
mapToProjectiveSpace( D )
• Inputs:
• Optional inputs:
• KnownCartier => , default value true, specify whether the divisor is known to be Cartier
• Variable => , default value YY
• Variable => , default value YY, specify the variable to use in the construction of the target projective space
• Outputs:
• ,

## Description

Given a Cartier divisor $D$ on a projective variety (represented by a divisor on a normal standard graded ring), this function returns the map to projective space induced by the global sections of $O(D)$. If KnownCartier is set to false (default is true), the function will also check to make sure the divisor is Cartier away from the irrelevant ideal.

 i1 : R = QQ[x,y,u,v]/ideal(x*y-u*v); i2 : D = divisor( ideal(x, u) ) o2 = Div(x, u) o2 : WeilDivisor on R i3 : mapToProjectiveSpace(D) o3 = map (R, QQ[YY ..YY ], {v, x}) 1 2 o3 : RingMap R <--- QQ[YY ..YY ] 1 2

The user may also specify the variable name of the new projective space.

 i4 : R = ZZ/7[x,y,z]; i5 : D = divisor(x*y) o5 = Div(y) + Div(x) o5 : WeilDivisor on R i6 : mapToProjectiveSpace(D, Variable=>"Z") ZZ 2 2 2 o6 = map (R, --[Z ..Z ], {x , x*y, x*z, y , y*z, z }) 7 1 6 ZZ o6 : RingMap R <--- --[Z ..Z ] 7 1 6