4. Lifting

This package provides an implementation of the global Newton lifting algorithm. Its verbose flag is HenselVerbose.

4.1 Splitting before lifting

intrinsic: LiftSplitLF (lf::Rec) -> SeqEnum

• lf, is an isolated lifting fiber.

It returns a sequence of isolated lifting fibers corresponding to different subvarieties behaving differently wrt the the lifting process. This splitting must be achieved before performing any lifting.

4.2 Lift curves

Common requirements for all the lifting functions:

• lf must be shape lemma.
• The variety must be isolated wrt to the lf`LiftingSystem.
• lf must be irreducible with respect to the lifting process.

intrinsic: LiftCurveLF (~lf::Rec, destpoint::SeqEnum, precision::RngIntElt)

• lf, an isolated lifting fiber.
• destpoint, destination point.
• precision, integer.

This procedure lifts the curve from the lifting point to the destination point destpoint and up to precision.
Error condition: cf. common requirements above.

intrinsic: LiftCurveLF (~llf::[], destpoint::SeqEnum, precision::RngIntElt)

It iterates LiftCurveLF over each element of llf.

intrinsic: LiftLastFreeVariableLF (~lf::Rec)

• lf, an isolated lifting fiber.

This procedure lifts the last free variable of lf. lf`ResolutionField becomes an univariate rational function field over the resolution field.
Error condition: cf. common requirements above.

intrinsic: LiftLastFreeVariableLF (~llf::[])

It iterates LiftLastFreeVariableLF over each element of llf.

4.3 Change the magic point

intrinsic: ChangeMagicPointLF (~lf::Rec, magicpoint)

• lf, an isolated lifting fiber.
• magicpoint, the destination magic point.

Computes the lifting fiber for the magic point magicpoint and returns it in lf.
Error conditions:

• #magicpoint must be equal to the dimension of lf
• cf. common requirements above.

intrinsic: ChangeMagicPointLF (~llf::[], magicpoint)

It iterates ChangeMagicPointLF over each element of llf.

4.4 Check lifting

intrinsic: HasLiftingPointLF (lf::Rec) -> BoolElt

• lf, an isolated fiber.

It tells whether the magic point is a lifting point. If the variety is multiple wrt its lifting system then its generic trace must be known.
Error conditions:

• cf. common requirements above.

intrinsic: HasLiftingPointLF (llf::[]) -> BoolElt

It iterates HasLiftingPointLF over each element of llf.