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# projectiveImage -- Projective image of the map defined by a divisor or matrix

## Synopsis

• Usage:
I = projectiveImage Dplus
I = projectiveImage (Dplus, Dminus)
I = projectiveImage (DplusList, DminusList, C)
I = projectiveImage (DplusList, C)
• Inputs:
• Dplus, an ideal, in the homogeneous coordinate ring A of a plane curve C
• Dminus, an ideal, in A
• DplusList, a list,
• DminusList, a list, lists representing the coordinates of points in P^2.
• C, a ring, the homogeneous coordinate ring of a plane curve
• Optional inputs:
• Conductor => ..., default value null
• Outputs:
• I, an ideal, the ideal of the image curve

## Description

The output ideal is the ideal of polynomial relations among the generators of the linear series |Dplus-Dminus|.

If C is a general curve of genus 6, then C can be represented as a plane sextic with 4 nodes. Its canonical embedding is then the projective image of C by the space of cubic forms vanishing at the 4 nodes. This lies on the surface that is the image of P2 under the linear series consisting of the cubics vanishing at the 4 nodes, a del Pezzo surface of degree 5.

 i1 : P5 = ZZ/101[x_0..x_5] o1 = P5 o1 : PolynomialRing i2 : P2 = ZZ/101[a,b,c] o2 = P2 o2 : PolynomialRing i3 : fourpoints = { {0,0,1}, {1,0,0}, {0,1,0}, {1,1,1}} o3 = {{0, 0, 1}, {1, 0, 0}, {0, 1, 0}, {1, 1, 1}} o3 : List i4 : fourPointsIdeals = apply (fourpoints, L -> fromCoordinates(L,P2)) o4 = {ideal (a, b), ideal (b, c), ideal (a, c), ideal (- a + b, - a + c)} o4 : List i5 : nodes = intersect apply(fourPointsIdeals, p -> p) o5 = ideal (a*c - b*c, a*b - b*c) o5 : Ideal of P2 i6 : sings' = intersect apply(fourPointsIdeals, p -> p^2) 2 2 2 2 2 2 2 2 2 2 2 2 o6 = ideal (a c - 2a*b*c + b c , a b*c - a*b c - a*b*c + b c , a b - ------------------------------------------------------------------------ 2 2 2 2a*b c + b c ) o6 : Ideal of P2 i7 : C0 = P2/(ideal random(6, sings')) o7 = C0 o7 : QuotientRing i8 : sings = sub (sings', C0) 2 2 2 2 2 2 2 2 2 2 2 2 o8 = ideal (a c - 2a*b*c + b c , a b*c - a*b c - a*b*c + b c , a b - ------------------------------------------------------------------------ 2 2 2 2a*b c + b c ) o8 : Ideal of C0 i9 : conductor C0 == sub(nodes, C0) o9 = true i10 : B' = gens image basis (3,nodes) o10 = | a2c-abc abc-b2c ac2-bc2 a2b-abc ab2-b2c abc-bc2 | 1 6 o10 : Matrix P2 <-- P2 i11 : B = sub(B',C0); 1 6 o11 : Matrix C0 <-- C0 i12 : canonicalSeries(C0) == B o12 = true

Now the image of C under B lies on the image of P^2 under B'. Since "projective image defines a ring", we need to make sure the two ideals are in the same ring to compare them:

 i13 : X = projectiveImage B' o13 = X o13 : QuotientRing i14 : C = projectiveImage B o14 = C o14 : QuotientRing i15 : betti res ideal C 0 1 2 3 4 o15 = total: 1 6 10 6 1 0: 1 . . . . 1: . 6 5 . . 2: . . 5 6 . 3: . . . . 1 o15 : BettiTally i16 : betti res ideal X 0 1 2 3 o16 = total: 1 5 5 1 0: 1 . . . 1: . 5 5 . 2: . . . 1 o16 : BettiTally i17 : isSubset(sub(ideal X, ring ideal C), ideal C) o17 = true