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

Synopsis

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

See also

Ways to use projectiveImage :

For the programmer

The object projectiveImage is a method function with options.