Secant varieties and birational geometry

Abstract

We show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations. These were first constructed as flips in the case of curves by M. Thaddeus via Geometric Invariant Theory, and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper. We expose the finer structure of a second flip; again for varieties of arbitrary dimension. We also prove a result on the cubic generation of the secant variety and give some conjectures on the behavior of equations defining the higher secant varieties.

Duality, Bridgeland wall-crossing and flips of secant varieties

“…In [30] and [31], Vermeire describes a sequence of flips for the secant varieties of an embedding X ֒→ P N of an algebraic surface. This sequence of flips is constructed in similar fashion to the flips obtained by Thaddeus [28] when studying variation of GIT for moduli spaces of stable pairs on curves.…”

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“…Furthermore, the author proves set theoretic statements for arbitrary smooth varieties in [32]. These statements also contain information about the syzygies among the generators that makes it possible to study the mapj 2 in much the same way asj was studied above.…”

“…The continuation of this process following [28] is taken up in [32]. We need to construct a birational morphismj 2 X M 2 3 P s 2 which contracts the image of 3-secant 2-planes to points, and is an embedding off their union.…”

“…As discussed in [6], a combination of the two techniques should reveal the strongest results. The construction of further £ips is taken up in [32], and the question of vanishing theorems in [33].…”

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“…For varieties of higher dimension, similar problems occur in that the ¢bers in the blow up are copies of the original variety blown up at a point (though the technique should at least reveal information in these cases). A somewhat greater obstacle is the lack of a structure theorem as strong as Theorem 3.1, though parts of this have been worked out in [19] and [20].…”