Restricting semistable bundles on the projective plane to conics

Abstract

Abstract. We study the restrictions of rank 2 semistable vector bundles E on P 2 to conics. A Grauert-Mülich type theorem on the generic splitting is proven. The jumping conics are shown to have the scheme structure of a hypersurface J 2 ⊂ P 5 of degree c 2 (E) when c 1 (E) = 0 and of degree c 2 (E) − 1 when c 1 (E) = −1. Some examples of jumping conics and jumping lines are studied in detail.

Jumping conics on a smooth quadric in $\PP_3$

“…From the theorem 1 in [15], we know that for a semistable vector bundle of rank 2 on P 2 and a general smooth conic f : P 1 ֒→ P 2 , we have…”

“…(13) P(U * ) and let π(ε) be the image of 1 ∈ H 0 (O P * 3 ). Then we can define a homomorphism (15) π : Ext…”

“…Especially, over the projective plane, the moduli space of stable sheaves of rank 2 was studied by W.Barth [1] and K.Hulek [9], using the jumping lines and jumping lines of the second kind. In [15], this idea was generalized to the jumping conics on the projective plane. In this article, we use the concept of jumping conics on the smooth quadric surface.…”

Jumping conics on a smooth quadric in $${\mathbb {P}_3}$$

“…Especially, over the projective plane, the moduli space of stable sheaves of rank 2 was studied by Barth [1] and Hulek [10], using the jumping lines and jumping lines of the second kind. In Vitter [18], this idea was generalized to the jumping conics on the projective plane. In this article, we use the concept of jumping conics on the smooth quadric surface, which was introduced, in the case of trivial first Chern class, by Soberon-Chavez in [17].…”