Let L = {a_0,..a_{(m-1)}}, and let P = P^{(#L-1+ sum L)}. Just as the ordinary scroll S(L) is the union of planes joining rational normal curves C_i of degree a_i according to some chosen isomorphism among them (a (1,1,..,1) correspondence), the generalized Scroll is the union of planes joining the points that correspond under an arbitrary correspondence, specified by I.
Thus if I is the ideal of the small diagonal of (P^1)^m, then generalized Scroll(I,L) is equal to S(L). If #L = 2, and I is the square of the ideal of the diagonal, we get a K3 carpet:
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Here is how to make the generalized scroll corresponding to a general elliptic curve in (P^1)^3. First, the general elliptic curve, as a plane cubic through three given points:
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Here the irrelevant ideal is the intersection of the 4 ideals of coordinates (P^2 and the three copies of P^1). Next, define the pairs of sections on the curve giving the three projections:
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And create the equations of the incidence variety
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This must be saturated with respect to the irrelevant ideal, and then the y variables are eliminated, to get the curve in (P^1)^3.
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Finally, we compute the ideal of the generalized Scroll:
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The script currently uses an elimination method, but could be speeded up by replacing that with the easy direct description of the equations that come from the correspondence I.
The object correspondenceScroll is a method function.