# tangentSheaf(ProjectiveVariety) -- tangent sheaf of a projective variety

## Description

Computes the tangent sheaf of the projective variety X

Tangent sheaf of the projective plane:

 i1 : P = Proj(QQ[a,b,c]) o1 = P o1 : ProjectiveVariety i2 : TP = tangentSheaf(P) o2 = image {-2} | a b 0 | {-2} | -c 0 b | {-2} | 0 c a | 3 o2 : coherent sheaf on P, subsheaf of OO (2) P i3 : HH^0(TP(-1)) 3 o3 = QQ o3 : QQ-module, free i4 : HH^1(TP(-3)) 1 o4 = QQ o4 : QQ-module, free
Tangent sheaf of a plane nodal and cuspidal curve:
 i5 : Node = Proj(QQ[a,b,c]/ideal(b^2*c-a^2*(a+c))) o5 = Node o5 : ProjectiveVariety i6 : Cusp = Proj(QQ[a,b,c]/ideal(b^2*c-a^3)) o6 = Cusp o6 : ProjectiveVariety i7 : TNode = tangentSheaf(Node) o7 = image {0} | 0 0 | {-1} | b a2+ac | {-1} | 2a 2bc | 1 2 o7 : coherent sheaf on Node, subsheaf of OO ++ OO (1) Node Node i8 : HH^0(TNode) 1 o8 = QQ o8 : QQ-module, free i9 : HH^1(TNode) o9 = 0 o9 : QQ-module i10 : TCusp = tangentSheaf(Cusp) o10 = image {1} | 0 0 | {-1} | -2a -2b | {-2} | 3bc 3a2 | 1 1 1 o10 : coherent sheaf on Cusp, subsheaf of OO (-1) ++ OO (1) ++ OO (2) Cusp Cusp Cusp i11 : HH^0(TCusp) 2 o11 = QQ o11 : QQ-module, free i12 : HH^1(TCusp) o12 = 0 o12 : QQ-module