# toMap -- rational map defined by a linear system

## Description

When the input represents a list of homogeneous elements $F_0,\ldots,F_m\in R=K[t_0,\ldots,t_n]/I$ of the same degree, then the method returns the ring map $\phi:K[x_0,\ldots,x_m] \to R$ that sends $x_i$ into $F_i$.

 i1 : QQ[t_0,t_1]; i2 : linSys=gens (ideal(t_0,t_1))^5 o2 = | t_0^5 t_0^4t_1 t_0^3t_1^2 t_0^2t_1^3 t_0t_1^4 t_1^5 | 1 6 o2 : Matrix (QQ[t ..t ]) <--- (QQ[t ..t ]) 0 1 0 1 i3 : phi=toMap linSys 5 4 3 2 2 3 4 5 o3 = map (QQ[t ..t ], QQ[x ..x ], {t , t t , t t , t t , t t , t }) 0 1 0 5 0 0 1 0 1 0 1 0 1 1 o3 : RingMap QQ[t ..t ] <--- QQ[x ..x ] 0 1 0 5

If a positive integer $d$ is passed to the option Dominant, then the method returns the induced map on $K[x_0,\ldots,x_m]/J_d$, where $J_d$ is the ideal generated by all homogeneous elements of degree $d$ of the kernel of $\phi$ (in this case kernel(RingMap,ZZ) is called).

 i4 : phi'=toMap(linSys,Dominant=>2) QQ[x ..x ] 0 5 5 4 3 2 2 3 4 5 o4 = map (QQ[t ..t ], --------------------------------------------------------------------------------------------------------------------------, {t , t t , t t , t t , t t , t }) 0 1 2 2 2 2 0 0 1 0 1 0 1 0 1 1 (x - x x , x x - x x , x x - x x , x x - x x , x - x x , x x - x x , x x - x x , x - x x , x x - x x , x - x x ) 4 3 5 3 4 2 5 2 4 1 5 1 4 0 5 3 1 5 2 3 0 5 1 3 0 4 2 0 4 1 2 0 3 1 0 2 QQ[x ..x ] 0 5 o4 : RingMap QQ[t ..t ] <--- -------------------------------------------------------------------------------------------------------------------------- 0 1 2 2 2 2 (x - x x , x x - x x , x x - x x , x x - x x , x - x x , x x - x x , x x - x x , x - x x , x x - x x , x - x x ) 4 3 5 3 4 2 5 2 4 1 5 1 4 0 5 3 1 5 2 3 0 5 1 3 0 4 2 0 4 1 2 0 3 1 0 2

If the input is a pair consisting of a homogeneous ideal $I$ and an integer $v$, then the output will be the map defined by the linear system of hypersurfaces of degree $v$ which contain the projective subscheme defined by $I$.

 i5 : I=kernel phi 2 2 o5 = ideal (x - x x , x x - x x , x x - x x , x x - x x , x - x x , x x 4 3 5 3 4 2 5 2 4 1 5 1 4 0 5 3 1 5 2 3 ------------------------------------------------------------------------ 2 2 - x x , x x - x x , x - x x , x x - x x , x - x x ) 0 5 1 3 0 4 2 0 4 1 2 0 3 1 0 2 o5 : Ideal of QQ[x ..x ] 0 5 i6 : toMap(I,2) 2 2 2 2 o6 = map (QQ[x ..x ], QQ[y ..y ], {x - x x , x x - x x , x x - x x , x x - x x , x - x x , x x - x x , x x - x x , x - x x , x x - x x , x - x x }) 0 5 0 9 4 3 5 3 4 2 5 2 4 1 5 1 4 0 5 3 1 5 2 3 0 5 1 3 0 4 2 0 4 1 2 0 3 1 0 2 o6 : RingMap QQ[x ..x ] <--- QQ[y ..y ] 0 5 0 9

This is identical to toMap(I,v,1), while the output of toMap(I,v,e) will be the map defined by the linear system of hypersurfaces of degree $v$ having points of multiplicity $e$ along the projective subscheme defined by $I$.

 i7 : toMap(I,2,1) 2 2 2 2 o7 = map (QQ[x ..x ], QQ[y ..y ], {x - x x , x x - x x , x x - x x , x x - x x , x - x x , x x - x x , x x - x x , x - x x , x x - x x , x - x x }) 0 5 0 9 4 3 5 3 4 2 5 2 4 1 5 1 4 0 5 3 1 5 2 3 0 5 1 3 0 4 2 0 4 1 2 0 3 1 0 2 o7 : RingMap QQ[x ..x ] <--- QQ[y ..y ] 0 5 0 9 i8 : toMap(I,2,2) o8 = map (QQ[x ..x ], QQ[], {}) 0 5 o8 : RingMap QQ[x ..x ] <--- QQ[] 0 5 i9 : toMap(I,3,2) 3 2 2 2 2 2 2 2 2 3 2 2 o9 = map (QQ[x ..x ], QQ[y ..y ], {x - 2x x x + x x + x x - x x x , x x - x x - x x x + x x + x x x - x x x , x x - x x - x x x + x x x + x x - x x x , x - 2x x x + x x + x x - x x x }) 0 5 0 3 3 2 3 4 1 4 2 5 1 3 5 2 3 2 4 1 3 4 0 4 1 2 5 0 3 5 2 3 1 3 1 2 4 0 3 4 1 5 0 2 5 2 1 2 3 0 3 1 4 0 2 4 o9 : RingMap QQ[x ..x ] <--- QQ[y ..y ] 0 5 0 3

## Ways to use toMap :

• "toMap(Ideal)"
• "toMap(Ideal,List)"
• "toMap(Ideal,ZZ)"
• "toMap(Ideal,ZZ,ZZ)"
• "toMap(List)"
• "toMap(Matrix)"
• "toMap(RingMap)"

## For the programmer

The object toMap is .