Abstract

We study a certain class of degenerations of Gushel–Mukai fourfolds as conic bundles, which we call tame degenerations and which are natural if one wants to prove that very general Gushel–Mukai fourfolds are irrational using the degeneration method due to Voisin, Colliot-Thélène–Pirutka, Totaro et al. However, we prove that no such tame degenerations exist.

Highlights

  • We work over the complex numbers C unless otherwise stated

  • The problem we consider in this article is if this theorem is applicable to the conic bundles arising from GM fourfolds, and our answer will be negative if one restricts attention to a rather natural class of “tame degenerations”

  • The roadmap to this paper is as follows: in Sect. 2 we prove that general Gushel–Mukai fourfolds are birational to certain types of conic bundles over P3, see

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Summary

Introduction

We work over the complex numbers C unless otherwise stated. Definition 1.1 We fix a five-dimensional vector space W. A Gushel–Mukai fourfold (GM fourfold for short) X is a smooth dimensionally transverse intersection. X = Q ∩ Gr(2, W ) ∩ H of the Grassmannian Gr(2, W ) ⊂ P( 2W ), a hyperplane H and a quadric Q in P( 2W )

B Christian Böhning
Admissible degenerations
Null-correlation bundles
Tame degenerations
A Sextic curves of arithmetic genus 5 in P3
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