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universalExtension -- Universal extension of vector bundles on P^1

Synopsis

Description

Every vector bundle E on $\PP^1$ splits as a sum of line bundles OO(a_i). If La is a list of integers, we write E(La) for the direct sum of the line bundle OO(La_i). Given two such bundles specified by the lists La and Lb this script constructs a module representing the universal extension of E(Lb) by E(La). It is defined on the product variety Ext^1(E(La), E(Lb)) x $\PP^1$, and represented here by a graded module over the coordinate ring S = A[y_0,y_1] of this variety; here A is the coordinate ring of Ext^1(E(La), E(Lb)), which is a polynomial ring.

i1 : M = universalExtension({-2}, {2})

o1 = cokernel {2, 0} | x_0 x_1 x_2 |
              {1, 1} | y_0 0   0   |
              {1, 1} | y_1 y_0 0   |
              {1, 1} | 0   y_1 y_0 |
              {1, 1} | 0   0   y_1 |

      ZZ                                       ZZ                 5
o1 : ---[x ..x ][y ..y ]-module, quotient of (---[x ..x ][y ..y ])
     101  0   2   0   1                       101  0   2   0   1
 
i2 : M = universalExtension({-2,-3}, {2,3})

o2 = cokernel {2, 0} | x_0y_1 x_1y_1 x_2y_1 x_3y_1 x_4y_1 x_5y_1 x_6y_1 x_7y_1 x_8y_1 |
              {3, 0} | x_9    x_10   x_11   x_12   x_13   x_14   x_15   x_16   x_17   |
              {2, 1} | y_0    0      0      0      0      0      0      0      0      |
              {2, 1} | y_1    y_0    0      0      0      0      0      0      0      |
              {2, 1} | 0      y_1    y_0    0      0      0      0      0      0      |
              {2, 1} | 0      0      y_1    y_0    0      0      0      0      0      |
              {2, 1} | 0      0      0      y_1    0      0      0      0      0      |
              {2, 1} | 0      0      0      0      y_0    0      0      0      0      |
              {2, 1} | 0      0      0      0      y_1    y_0    0      0      0      |
              {2, 1} | 0      0      0      0      0      y_1    y_0    0      0      |
              {2, 1} | 0      0      0      0      0      0      y_1    y_0    0      |
              {2, 1} | 0      0      0      0      0      0      0      y_1    y_0    |
              {2, 1} | 0      0      0      0      0      0      0      0      y_1    |

      ZZ                                        ZZ                  13
o2 : ---[x ..x  ][y ..y ]-module, quotient of (---[x ..x  ][y ..y ])
     101  0   17   0   1                       101  0   17   0   1

It is interesting to consider the loci in Ext where the extension has a particular splitting type. See the documentation for directImageComplex for a conjecture about the equations of these varieties.

See also

Ways to use universalExtension :

For the programmer

The object universalExtension is a method function.