Related papers: A Phantom on a Rational Surface
We construct a universal phantom subcategory on the blow-up of the complex projective plane in 11 general points. This phantom subcategory is the orthogonal complement of a non-full exceptional collection of line bundles of maximal length.…
A conjecture of Bondal-Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We…
The purpose of the article is to explain a new method to establish the existence of an exceptional collection of length three for a fake projective plane M with non-trivial automorphism group, related to a conjecture of…
We investigate Hermitian metrics on the anti-canonical bundle of a rational surface obtained by blowing up the projective plane at nine points. For that purpose, we pose a modified variant of an argument made by Ueda on the complex analytic…
A fullness conjecture of Kuznetsov says that if a smooth projective variety $X$ admits a full exceptional collection of line bundles of length $l$, then any exceptional collection of line bundles of length $l$ is full. In this paper, we…
We show that there exist exceptional collections of length 3 consisting of line bundles on the three fake projective planes that have a 2-adic uniformisation with torsion free covering group. We also compute the Hochschild cohomology of the…
We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface $S$. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional…
We use constructions of surfaces as abelian covers to write down exceptional collections of line bundles of maximal length for every surface $X$ in certain families of surfaces of general type with $p_g=0$ and $K_X^2=3,4,5,6,8$. We also…
We prove that the bounded derived category of the surface S constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli space of…
We show that blowups of the projective plane at points lying on a smooth cubic curve do not contain phantoms, provided the points are chosen in very general position on this curve.
For a vector bundle $\mathcal E \to \mathbb P^\ell$ we investigate exceptional sequences of line bundles on the total space of the projectivisation $X = \mathbb P(\mathcal E)$. In particular, we consider the case of the cotangent bundle of…
We construct exceptional collections of maximal length on four families of surfaces of general type with $p_g=q=0$ which are isogenous to a product of curves. From these constructions we obtain new examples of quasiphantom categories as…
A conjecture of Batyrev and Manin predicts the asymptotic behaviour of rational points of bounded height on smooth projective varieties over number fields. We prove some new cases of this conjecture for conic bundle surfaces equipped with…
We provide an explicit description of exceptional collection of maximal length in the derived category $D^b(Y)$ for a particular class of elliptic surfaces $Y$. The existence of non\,-\,trivial semiorthogonal complement (a "\,phantom\,") of…
Following Krah's method, we construct new examples of phantom categories as semiorthogonal components of the derived categories of two types of rational surfaces: the blowup of the plane at 11 points in general position, and the blowup of…
We prove that any numerically exceptional collection of maximal length, consisting of line bundles, on a smooth del Pezzo surface is a standard augmentation in the sense of L.Hille and M.Perling. We deduce that any such collection is…
Prompted by results of Guardo, Van Tuyl and the second author for lines in projective 3 space, we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r dimensional…
Line bundles of rational degree are defined using Perfectoid spaces, and their co-homology computed via standard \v{C}ech complex along with Kunneth formula. A new concept of `braided dimension' is introduced, which helps convert the curse…
We study projective manifolds with nonamenable and non-residually finite fundamental groups. We generalize the uniformization theorem of our earlier note. We generalize a classical theorem of Maltsev about finitely generated subgroups of…
By utilizing elementary techniques from toric geometry, we prove sharp cohomological vanishing results for line bundles defined on the blow-up of projective space $\mathbb{P}^n$ at no more than $n+1$ points.