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# EmbeddedProjectiveVariety -- the class of all embedded projective varieties

## Description

The embedded projective varieties are exactly the multi-projective varieties embedded in a single projective space; so that $X$ is an embedded projective variety if and only if # shape X == 1.

    i1 : X = PP_QQ^(2,2);

    o1 : ProjectiveVariety, surface in PP^5

    i2 : class X

    o2 = EmbeddedProjectiveVariety

    o2 : Type

    i3 : Y = X ** X;

    o3 : ProjectiveVariety, X x X

    i4 : class Y

    o4 = MultiprojectiveVariety

    o4 : Type

## See also

- MultiprojectiveVariety -- the class of all multi-projective varieties
- WeightedProjectiveVariety -- the class of all weighted projective varieties
- ambient(MultiprojectiveVariety) -- the ambient multi-projective space of the variety
- shape(MultiprojectiveVariety) -- shape of the ambient multi-projective space of a multi-projective variety

## Types of embedded projective variety:

- GrassmannianVariety -- the class of all Grassmannians of linear subspaces of projective spaces

## Functions and methods returning an embedded projective variety:

- coneOfLines -- cone of lines on a subvariety passing through a point

## Methods that use an embedded projective variety:

- chowForm(EmbeddedProjectiveVariety) -- chow forms of a projective variety
- `coneOfLines(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety)` -- see coneOfLines -- cone of lines on a subvariety passing through a point
- conormalVariety(EmbeddedProjectiveVariety) -- the conormal variety of a projective variety
- `cycleClass(EmbeddedProjectiveVariety)` -- see cycleClass -- determine the expression of the class of a cycle as a linear combination of Schubert classes
- dual(EmbeddedProjectiveVariety) -- the variety projectively dual to an embedded projective variety
- EmbeddedProjectiveVariety ! -- print a more detailed description of an embedded projective variety
- `sumUp(EmbeddedProjectiveVariety)` -- see EmbeddedProjectiveVariety ! -- print a more detailed description of an embedded projective variety
- EmbeddedProjectiveVariety ++ EmbeddedProjectiveVariety -- join of projective varieties
- `EmbeddedProjectiveVariety <=== EmbeddedProjectiveVariety` -- see EmbeddedProjectiveVariety ===> EmbeddedProjectiveVariety -- try to find an isomorphism between two projective varieties
- EmbeddedProjectiveVariety ===> EmbeddedProjectiveVariety -- try to find an isomorphism between two projective varieties
- `Fano(EmbeddedProjectiveVariety)` -- see Fano(ZZ,EmbeddedProjectiveVariety) -- Fano scheme of a projective variety
- Fano(ZZ,EmbeddedProjectiveVariety) -- Fano scheme of a projective variety
- `GG(EmbeddedProjectiveVariety)` -- see GG -- the Grassmannian of k-dimensional linear subspaces of an n-dimensional projective space
- `GG(ZZ,EmbeddedProjectiveVariety)` -- see GG -- the Grassmannian of k-dimensional linear subspaces of an n-dimensional projective space
- hilbertPolynomial(EmbeddedProjectiveVariety) -- the Hilbert polynomial of the variety
- `linearlyNormalEmbedding(EmbeddedProjectiveVariety)` -- see linearlyNormalEmbedding -- get the linearly normal embedding
- `linearSpan(EmbeddedProjectiveVariety)` -- see linearSpan -- the linear span of an embedded projective variety
- `RationalMap ^** EmbeddedProjectiveVariety` -- see MultirationalMap ^** MultiprojectiveVariety -- inverse image via a multi-rational map
- `rationalMap(EmbeddedProjectiveVariety,Tally,ZZ)` -- see rationalMap(MultiprojectiveVariety,Tally) -- rational map defined by an effective divisor
- `sectionalGenus(EmbeddedProjectiveVariety)` -- see sectionalGenus -- the sectional genus of an embedded projective variety
- tangentialChowForm(EmbeddedProjectiveVariety,ZZ) -- higher Chow forms of a projective variety
- `tangentSpace(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety)` -- see tangentSpace -- tangent space to a projective variety at a point
- variety(EmbeddedProjectiveVariety) -- convert an embedded projective variety into a built-in projective variety

## For the programmer

The object EmbeddedProjectiveVariety is a type, with ancestor classes MultiprojectiveVariety < MutableHashTable < HashTable < Thing.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MultiprojectiveVarieties.m2:2461:0.