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# TautologicalLineBundle -- a symbol used internally as a key

## Description

If $X$ is an abstract variety and its tautological line bundle has already been built, it is stored in X.TautologicalLineBundle. However, because building this bundle can be computationally intensive, it is often not built unless it is requested by tautologicalLineBundle. The programmer is warned not to make direct reference to this key for this reason.

    i1 : X = abstractProjectiveSpace 3

    o1 = X

    o1 : a flag bundle with subquotient ranks {1, 3}

    i2 : try X.TautologicalLineBundle else print "bundle not found"
    bundle not found

    i3 : L = tautologicalLineBundle X

    o3 = L

    o3 : an abstract sheaf of rank 1 on X

    i4 : L === X.TautologicalLineBundle

    o4 = true

    i5 : rank L

    o5 = 1

    i6 : chern L

    o6 = 1 + H
              2,1

                           QQ[][h, H   ..H   ]
                                    2,1   2,3
    o6 : -------------------------------------------------------
         (- h - H   , - h*H    - H   , - h*H    - H   , -h*H   )
                 2,1       2,1    2,2       2,2    2,3      2,3

Here are the preferred ways to get the tautological line bundle:

    i7 : OO_X(1)

    o7 = L

    o7 : an abstract sheaf of rank 1 on X

    i8 : L === oo

    o8 = true

    i9 : tautologicalLineBundle X

    o9 = L

    o9 : an abstract sheaf of rank 1 on X

    i10 : L === oo

    o10 = true

## See also

- tautologicalLineBundle -- get the tautological line bundle of a flag bundle or abstract variety

## For the programmer

The object TautologicalLineBundle is a symbol.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Schubert2.m2:1857:0.