Small bandwidth ℂ*-actions and birational geometry

Abstract

In this paper we study smooth projective varieties and polarized pairs with an action of a one dimensional complex torus. As a main tool, we define birational geometric counterparts of these actions, that, under certain assumptions, encode the information necessary to reconstruct them. In particular, we consider some cases of actions of low complexity—measured in terms of two invariants of the action, called bandwidth and bordism rank—and discuss how they are determined by well known birational transformations, namely Atiyah flips and Cremona transformations.

Morelli-Włodarczyk cobordism and examples of rooftop flips

“…Step 1: We want to prove that Y ≃ X/C * . Thanks to [13,Remark 4.2], the contraction f : P(E) → X is C * -equivariant, in particular the geometric quotients of (P(E), O P(E) (1)) and (X, L) with respect to the C * -action are isomorphic. Since the former is a P 1 -bundle on Y , and therefore its geometric quotient is isomorphic to Y , we conclude.…”

“…In this section we recall the characterization of smooth drums, that is smooth polarized pairs admitting a C * -action of bandwidth 1. We refer to [13,Section 4].…”

“…The relation between birational geometry and C * -actions has been deeply studied over the years (see for instance [17], [18] and in recent years [13]). One famous example for instance is the notion of Morelli-W lodarczyk cobordism (cfr.…”

“…Smooth projective varieties with two different projective bundle structures have been already studied in the literature (see for instance [12]): in particular, in [13,Lemma 4.4] the authors have constructed a correspondence between them and smooth projective varieties X of Picard number 1 admitting a C * -action having only two fixed point components. Motivated by the example of Atiyah flop and its connection with a variety admitting two projective bundle structures, we introduce the notion of rooftop flip.…”

What is... a Białynicki-Birula Decomposition?

“…An explicit remark in [7] says that, given a ℂ * -action on a nonsingular projective variety 𝑋, we can associate a birational map among projective varieties.…”