# linearlyNormalEmbedding -- get the linearly normal embedding

## Synopsis

• Usage:
linearlyNormalEmbedding X
• Inputs:
• Outputs:
• an isomorphism from X to a linearly normal variety, whose inverse is a linear projection

## Description

 i1 : K = ZZ/333331; i2 : X = PP_K^(1,7); -- rational normal curve of degree 7 o2 : ProjectiveVariety, curve in PP^7 i3 : time f = linearlyNormalEmbedding X; -- used 0.0179022 seconds o3 : MultirationalMap (automorphism of X) i4 : Y = (rationalMap {for i to 3 list random(1,ring ambient X)}) X; -- an isomorphic projection of X in PP^3 o4 : ProjectiveVariety, curve in PP^3 i5 : time g = linearlyNormalEmbedding Y; -- used 1.06849 seconds o5 : MultirationalMap (birational map from Y to curve in PP^7) i6 : assert(isIsomorphism g) i7 : describe g o7 = multi-rational map consisting of one single rational map source variety: curve in PP^3 cut out by 6 hypersurfaces of degree 4 target variety: curve in PP^7 cut out by 21 hypersurfaces of degree 2 base locus: empty subscheme of PP^3 dominance: true multidegree: {7, 7} degree: 1 degree sequence (map 1/1): [3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4] coefficient ring: K

## Caveat

This is an experimental function.

## Ways to use linearlyNormalEmbedding :

• "linearlyNormalEmbedding(EmbeddedProjectiveVariety)"

## For the programmer

The object linearlyNormalEmbedding is .