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VirtualResolutions :: curveFromP3toP1P2

curveFromP3toP1P2 -- creates the ideal of a curve in P^1xP^2 from the ideal of a curve in P^3

Synopsis

Description

Given an ideal J defining a curve C in 3, curveFromP3toP1P2 produces the ideal of the curve in 1×ℙ2 defined as follows: consider the projections 3→ℙ2 and 3→ℙ1 from the point [0:0:0:1] and the line [0:0:s:t], respectively. The product of these defines a map from 3 to 1×ℙ2. The curve produced by curveFromP3toP1P2 is the image of the input curve under this map.

This computation is done by first constructing the graph in 3×(ℙ1xℙ2) of the product of the two projections 3→ℙ2 and 3→ℙ1 defined above. This graph is then intersected with C×(ℙ1×ℙ2). A curve in 1×ℙ2 is then obtained from this by saturating and then eliminating.

Note the curve in 1×ℙ2 will have degree and genus equal to the degree and genus of C as long as C does not intersect the base locus of the projection. If the option PreserveDegree is set to true, curveFromP3toP1P2 will check whether C intersects the base locus. If it does, the function will return an error. If PreserveDegree is set to false, this check is not performed and the output curve in 1×ℙ2 may have degree and genus different from C.

i1 : R = ZZ/101[z_0,z_1,z_2,z_3];
i2 : J = ideal(z_0*z_2-z_1^2, z_1*z_3-z_2^2, z_0*z_3-z_1*z_2);

o2 : Ideal of R
i3 : curveFromP3toP1P2(J)

             2
o3 = ideal (x    - x   x   , - x   x    + x   x   , - x   x    + x   x   )
             1,1    1,0 1,2     0,1 1,1    0,0 1,2     0,1 1,0    0,0 1,1

               ZZ
o3 : Ideal of ---[x   , x   , x   , x   , x   ]
              101  0,0   0,1   1,0   1,1   1,2

Caveat

This creates a ring F[x0,0,x0,1,x1,0,x1,1,x1,2] in which the resulting ideal is defined.

Ways to use curveFromP3toP1P2 :