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.
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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.
The object universalExtension is a method function.