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

## Synopsis

• Usage:
E = universalExtension(La, Lb)
• Inputs:
• La, a list, of integers
• Lb, a list, of integers
• Outputs:
• E, , representing the extension

## 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.