Macaulay2 » Documentation
Packages » Varieties :: tangentSheaf
next | previous | forward | backward | up | index | toc

tangentSheaf -- tangent sheaf of a projective variety

Description

This method computes the tangent sheaf of the projective variety $X$. It is computed by taking the dual of the cotangentSheaf.

Tangent sheaf of the projective plane:

i1 : P2 = Proj QQ[a,b,c]

o1 = P2

o1 : ProjectiveVariety
 
i2 : TP = tangentSheaf P2

o2 = image {-2} | a  b 0 |
           {-2} | -c 0 b |
           {-2} | 0  c a |

                                           3
o2 : coherent sheaf on P2, subsheaf of OO   (2)
                                         P2
 
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 : N = Proj QQ[a,b,c]/(b^2*c-a^2*(a+c))

o5 = N

o5 : ProjectiveVariety
 
i6 : TN = tangentSheaf N

o6 = image {0}  | 0  0       |
           {-1} | 3b 3a2+3ac |
           {-1} | 2a 2bc     |

                                         1       2
o6 : coherent sheaf on N, subsheaf of OO   ++ OO  (1)
                                        N       N
 
i7 : HH^0(TN), HH^1(TN)

        1
o7 = (QQ , 0)

o7 : Sequence
 
i8 : C = Proj QQ[a,b,c]/(b^2*c-a^3)

o8 = C

o8 : ProjectiveVariety
 
i9 : TC = tangentSheaf C

o9 = image {1}  | 0   0   |
           {-1} | -2a -2b |
           {-2} | 3bc 3a2 |

                                         1           1          1
o9 : coherent sheaf on C, subsheaf of OO  (-1) ++ OO  (1) ++ OO  (2)
                                        C           C          C
 
i10 : HH^0(TC), HH^1(TC)

         2
o10 = (QQ , 0)

o10 : Sequence

See also

Ways to use tangentSheaf:

  • tangentSheaf(ProjectiveVariety)

For the programmer

The object tangentSheaf is a method function with options.

The source of this document is in Varieties/doc-sheaves.m2:501:0.