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

- 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Jets.m2:1666:0.