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# mapOntoImage -- the induced map from a variety to the closure of its image under a rational map

## Synopsis

• Usage:
h = mapOntoImage(f)
psi = mapOntoImage(phi)
• Inputs:
• f, , the ring map corresponding to a rational map $\phi$ of projective varieties
• phi, an instance of the type RationalMapping, a rational map $\phi$ of projective varieties
• Optional inputs:
• QuickRank => , default value true, whether to compute rank via the package FastMinors
• Outputs:
• h, , the map of rings corresponding to $X \to \overline{\phi(X)}$.
• psi, an instance of the type RationalMapping, the rational map

## Description

Given $f : X \to Y$ mapOntoImage returns $X \to \overline{\phi(X)}$. Alternately, given $f: S \to R$, mapOntoImage just returns $S/(kernel f) \to R$. mapOntoImage first computes whether the kernel is $0$ without calling ker, which can have speed advantages.

 i1 : R = QQ[x,y]; i2 : S = QQ[a,b,c]; i3 : f = map(R, S, {x^2, x*y, y^2}); o3 : RingMap R <--- S i4 : mapOntoImage(f) S 2 2 o4 = map (R, --------, {x , x*y, y }) 2 b - a*c S o4 : RingMap R <--- -------- 2 b - a*c i5 : phi = rationalMapping f 2 2 o5 = Proj R - - - > Proj S {x , x*y, y } o5 : RationalMapping i6 : mapOntoImage(phi) / S \ 2 2 o6 = Proj R - - - > Proj|--------| {x , x*y, y } | 2 | \b - a*c/ o6 : RationalMapping

## Ways to use mapOntoImage :

• "mapOntoImage(RationalMapping)"
• "mapOntoImage(RingMap)"

## For the programmer

The object mapOntoImage is .