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# tangentSpace -- tangent space to a projective variety at a point

- Usage:
    `tangentSpace(X,p)`
    `tangentSpace(p,X)`
- Inputs:
  - X, an embedded projective variety
  - p, an embedded projective variety, a point on $X$
- Outputs:
  - an embedded projective variety, the embedded tangent space $T_p(X)$ to $X$ at the point $p$

## Description

    i1 : X = PP_(ZZ/333331)^(3,2);

    o1 : ProjectiveVariety, threefold in PP^9

    i2 : p := point X

    o2 = point of coordinates [-63251, 140212, 25299, -105446, -91880, -117577, 129587, 38839, 41433, 1]

    o2 : ProjectiveVariety, a point in PP^9

    i3 : tangentSpace(X,p)

    o3 = threefold in PP^9 cut out by 6 hypersurfaces of degree 1

    o3 : ProjectiveVariety, threefold in PP^9

## See also

- tangentCone(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety)
- singularLocus(MultiprojectiveVariety) -- the singular locus of the variety
- dual(EmbeddedProjectiveVariety) -- the variety projectively dual to an embedded projective variety
- point(MultiprojectiveVariety) -- pick a random rational point on a multi-projective variety

## Ways to use tangentSpace:

- `tangentSpace(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety)`

## For the programmer

The object tangentSpace is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MultiprojectiveVarieties.m2:3801:0.