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Then Z ′ is a line in P 4 . Choose a line l( = Z ′ ) in Q with l ∩ Z ′ = ∅. Let C be a component of ϕ −1 (l) intersecting Z. Then (−K X ) · C ≤ 2 deg l = 2 and C ∩ Z = ∅ -
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- Publisher
- Wiley
- Copyright
- © London Mathematical Society
- ISSN
- 0024-6093
- eISSN
- 1469-2120
- DOI
- 10.1112/blms/bdr020
- Publisher site
- See Article on Publisher Site
Abstract
Ross and Thomas introduced the concept of slope stability to study K‐stability, which has conjectural relation with the existence of constant scalar curvature Kähler metric. This paper presents a study of slope stability of Fano manifolds of dimension n⩾3 with respect to smooth curves. The question turns out to be easy for curves of genus at least 1 and the interest lies in the case of smooth rational curves. Our main result classifies completely the cases when a polarized Fano manifold (X,−KX) is not slope stable with respect to a smooth curve. Our result also states that a Fano three‐fold X with Picard number 1 is slope stable with respect to every smooth curve unless X is the projective space.
Journal
Bulletin of the London Mathematical Society – Wiley
Published: Oct 1, 2011