Fano manifolds which are not slope stable along curves

TL;DR

This paper characterizes specific Fano manifolds that are not slope stable along curves, identifying them with well-known geometric configurations such as projective space, product spaces, or blow-ups.

Contribution

It provides a complete classification of Fano manifolds that fail slope stability along smooth curves, linking stability properties to explicit geometric structures.

Findings

Fano manifold not slope stable along a curve iff it is one of the classified types.

Identifies geometric conditions for slope stability failure.

Connects stability properties with classical algebraic geometry constructions.

Abstract

We show that a Fano manifold (X,-K_X) is not slope stable with respect to a smooth curve Z if and only if (X,Z) is isomorphic to one of (projective space, line), (product of projective line and projective space, fiber of second projection) or (blow up of projective space along linear subspace of codimension two, nontrivial fiber of blow up).