Exceptional object

Abstract: Using cyclotomic specializations of the equivariant $K$-theory with respect to a torus action we derive congruences for discrete invariants of exceptional objects in derived categories of coherent sheaves on a class of varieties that includes Grassmannians and smooth quadrics. For example, we prove that if $X={\Bbb P}^{n_1-1}\times...\times{\Bbb P}^{n_k-1}$, where $n_i$'s are powers of a fixed prime number $p$, then the rank of an exceptional object on $X$ is congruent to $\pm 1$ modulo $p$.

Abstract: We first prove that the subcategory of fixed points of mutation determined by an exceptional object E in a triangulated category coincide with the perpendicular category of E. Based on this characterisation, we prove that the subcategory of fixed points of mutation in the derived category of the coherent sheaves on weighted projective line with genus one is equivalent to the derived category of a hereditary algebra. Meanwhile, we induce two new recollements by left and right mutations from a given recollement.

Abstract: Structure theorems for exceptional objects and exceptional collections of the bounded derived category of coherent sheaves on del Pezzo surfaces are established by Kuleshov and Orlov. In this paper we propose conjectures which generalize these results to weak del Pezzo surfaces. Unlike del Pezzo surfaces, an exceptional object on a weak del Pezzo surface is not necessarily a shift of a sheaf and is not determined by its class in the Grothendieck group. Our conjectures explain how these complications are taken care of by spherical twists, the categorification of $(-2)$-reflections acting on the derived category. This paper is devoted to solving the conjectures for the prototypical weak del Pezzo surface $\Sigma _{ 2 }$, the Hirzebruch surface of degree $2$. Specifically, we prove the following results: Any exceptional object is sent to the shift of the uniquely determined exceptional vector bundle by a product of spherical twists which acts trivially on the Grothendieck group of the derived category. Any exceptional collection on $\Sigma _{ 2 }$ is part of a full exceptional collection. We moreover prove that the braid group on $4$ strands acts transitively on the set of exceptional collections of length $4$ (up to shifts).

Abstract: The concept of exceptional sequences was developed by Gorodentsev and Rudakov in [3] and generalized by Bondal in [1]. It is a crucial tool to classify exceptional modules over hereditary algebras and exceptional sheaves over weighted projective lines. In fact, let A be the category of finitely generated modules over a hereditary algebras or the category of coherent sheaves over a weighted projective line and let E be an exceptional sequence in A. Then the length of E is smaller than or equal to the rank n of the Grotherndieck group of A and E is called a complete exceptional sequence if the length of E is equal to n. It is known that any exceptional sequence can be enlarged to a complete exceptional sequence ([2, Lemma 1], [5, Lemma 3.1.3] ) and the braid group Bn on n strings acts transitively on the set of complete exceptional sequences ([2], [5, Theorem 3.3.1]). Fixing a complete exceptional sequence E , for any exceptional object X ∈ A we can find an element σ ∈ Bn such that X belongs to σE . Hence we are able to classify exceptional objects in A by a fixed complete exceptional sequence E with the action of the braid group Bn. In this paper, we shall consider exceptional sequences in the category of finitely generated graded modules over a graded ring. To compare with above cases, exceptional sequences for graded modules may have infinite length. Moreover the braid group B on infinite strings acts on the set of exceptional sequences of infinite length, but the action may not be transitive. Therefore we have to consider the following condition for the exceptional sequence E to classify exceptional modules;