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

Synopsis

• 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 => ..., 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