Macaulay2 » Documentation
Packages » MultiprojectiveVarieties :: tangentCone(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety)
next | previous | forward | backward | up | index | toc

# tangentCone(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety) -- tangent cone to a projective variety at a point

- Function: tangentCone
- Usage:
    `tangentCone(X,p)`
    `tangentCone(p,X)`
- Inputs:
  - X, an embedded projective variety
  - p, an embedded projective variety, a point on $X$
- Optional inputs:
  - Strategy (missing documentation) => ..., default value Local,
- Outputs:
  - an embedded projective variety, the embedded tangent cone $C_p(X)$ to $X$ at the point $p$

## Description

    i1 : Y = random(3,0_(PP_(ZZ/333331)^6)), q = point Y, j = parametrize tangentSpace(Y,q);

    i2 : (X, p) = (j^* Y, j^* q);

    i3 : C = tangentCone(X,p);

    o3 : ProjectiveVariety, hypersurface in PP^5

    i4 : describe C

    o4 = ambient:.............. PP^5
         dim:.................. 4
         codim:................ 1
         degree:............... 2
         generators:........... 2^1
         purity:............... true
         dim sing. l.:......... 0
         gens sing. l.:........ 1^5

    i5 : assert(isSubset(C,tangentSpace(X,p)) and coneOfLines(C,p) == C)

## See also

- tangentSpace(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety) -- tangent space to a projective variety at a point
- tangentCone(Ideal)
- coneOfLines -- cone of lines on a subvariety passing through a point

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MultiprojectiveVarieties.m2:3817:0.