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

## 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.