jets(ZZ,AffineVariety) -- the jets of an affine variety

Synopsis

Function: jets
Usage:

jets (n,V)
Inputs:
- n, an integer,
- V, an affine variety,
Optional inputs:
- Projective => ..., default value false, Option for jets
Outputs:
- a variety, an AffineVariety or a ProjectiveVariety

Description

Returns the jets of an AffineVariety as an AffineVariety.

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(y^2-x^2*(x+1))

              3    2    2
o2 = ideal(- x  - x  + y )

o2 : Ideal of R

i3 : A = Spec(R/I)

o3 = A

o3 : AffineVariety

i4 : jets(2,A)

         /                                   QQ[x0, y0][x1, y1][x2, y2]                                   \
o4 = Spec|------------------------------------------------------------------------------------------------|
         |       2                                  2     2        2                        3     2     2 |
         \((- 3x0  - 2x0)x2 + 2y0*y2 + (- 3x0 - 1)x1  + y1 , (- 3x0  - 2x0)x1 + 2y0*y1, - x0  - x0  + y0 )/

o4 : AffineVariety

If jets(...,Projective=>...) is set to true, then jets are computed with the grading introduced in Proposition 6.6 (c) of P. Vojta, Jets via Hasse-Schmidt Derivations, and the function returns a ProjectiveVariety.

i5 : jets(2,A,Projective=>true)

         /                                   QQ[x0, y0][x1, y1][x2, y2]                                   \
o5 = Proj|------------------------------------------------------------------------------------------------|
         |       2                                  2     2        2                        3     2     2 |
         \((- 3x0  - 2x0)x2 + 2y0*y2 + (- 3x0 - 1)x1  + y1 , (- 3x0  - 2x0)x1 + 2y0*y1, - x0  - x0  + y0 )/

o5 : ProjectiveVariety

Caveat

With Projective=>true the jet variables of order zero have degree 0, therefore no heft vector exist for the ambient ring of the jets. As a result, certain computations will not be supported, and others may run more slowly. See heft vectors for more information.

Note: jets of projective varieties are currently not implemented.

Ways to use this method: