24 research outputs found
Maximal lengths of exceptional collections of line bundles
In this paper we construct infinitely many examples of toric Fano varieties
with Picard number three, which do not admit full exceptional collections of
line bundles. In particular, this disproves King's conjecture for toric Fano
varieties.
More generally, we prove that for any constant there exist
infinitely many toric Fano varieties with Picard number three, such that
the maximal length of exceptional collection of line bundles on is strictly
less than c\rk K_0(Y). To obtain varieties without exceptional collections of
line bundles, it suffices to put
On the other hand, we prove that for any toric nef-Fano DM stack with
Picard number three, there exists a strong exceptional collection of line
bundles on of length at least \frac34 \rk K_0(Y). The constant
is thus maximal with this property.Comment: 27 pages, no figures; misprints and typos corrected, an arithmetic
mistake in the proof of Theorem 6.2 corrected, consequently Theorem 6.3
slightly modified, new Lemma 4.4 added, description of the constructed
varieties extended, references adde
On trivial line bundles on toric DM stacks of dim
We study line bundles on smooth toric DM stacks
of arbitrary dimension. A sufficient condition
is given for when infinitely many line bundles on
have trivial cohomology. In dimension three, the
sufficient condition is also a necessary condition in the case that
has no more than one pair of collinear rays.Comment: 16 pages, 6 figures. Reference added. arXiv admin note: text overlap
with arXiv:1812.0175
A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles
We show that there exists a cubic threefold defined by an invertible
polynomial that, when quotiented by the maximal diagonal symmetry group, has a
derived category which does not have a full exceptional collection consisting
of line bundles. This provides a counterexample to a conjecture of Lekili and
Ueda.Comment: 8 pages, minor revision, to appear in Forum of Math, Sigm
Exceptional collections on toric Fano threefolds and birational geometry
Bernardi and Tirabassi show the existence of full strong exceptional
collections consisting of line bundles on smooth toric Fano -folds under
assuming Bondal's conjecture, which states that the Frobenius push-forward of
the structure sheaf \mc O_X generates the derived category for
smooth projective toric varieties .
In this article, we show Bondal's conjecture for smooth toric Fano -folds
and also improve their result, using birational geometry.Comment: 6 figure