A compactification of the space of twisted cubics

Abstract

We give an elementary, explicit smooth compactification of a parameter space for the family of twisted cubics. The construction also applies to the family of subschemes defined by determinantal nets of quadrics, e.g., cubic ruled surfaces in P 4 , Segre varieties in P 5 . It is suitable for applications of Bott's formula to a few enumerative problems.

Enumeration of cones over cubic scrolls

“…Recall that any two quadrics containing a cubic scroll cut a residual codimension-2 subspace. The main idea of the construction of the parameter space in [8] summarized below is to reverse the process: look at pencils of quadrics through a codimension-2 subspace. A Hilbert scheme construction is also available, though less elementary, cf.…”

“…Continuing, we recall from [7] (see also [8]) the construction of a parameter space for the family of cscs in the fibers of a P 4 -fibration, P (E) → X. It is obtained by a sequence of three explicit blowups,…”

“…It seems to become quite involved and ineffective otherwise, e.g., for the case of incidence to the maximal number of lines, say for the family of cubic scrolls in P 4 . The latter in turn can be handled by more classical tools, e.g., [8].…”

“…The condition of incidence of a codimension two cone to a line in P n is rephrased in terms of the condition of incidence of the abstract base of the cone to a line within the grassmannian. Next, borrowing from [7], [8] we summarize the construction of a suitable parameter space for the family of cubic surface scrolls in P 4 . This is aimed at the precise geometric characterization of blowup centers and natural vector bundles needed to feed into Bott's residue formula.…”

Numbers for reducible cubic scrolls

“…It can be rendered flat by a single blowup along a suitable smooth subvariety of X using techniques as in (Vainsencher and Xavier 2002), but this is not needed in the sequel.…”