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We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolovs inequality for semistable sheaves on a smooth projective variety of any dimension $ge 2$ without using any restriction theorems.
An important classification problem in Algebraic Geometry deals with pairs $(E,phi)$, consisting of a torsion free sheaf $E$ and a non-trivial homomorphism $phicolon (E^{otimes a})^{oplus b}lradet(E)^{otimes c}otimes L$ on a polarized complex project
We bound the slope of sweeping curves in the fourgonal locus of the moduli space of genus g algebraic curves. Our results follow from some Bogomolov-type inequalities for weakly positive rank two vector bundles on ruled surfaces.
We introduce the notion of mixed-$omega$-sheaves and use it for the study of a relative version of Fujitas freeness conjecture.
It is conjectured that the canonical models of varieties (not of general type) are bounded when the Iitaka volume is fixed. We confirm this conjecture when the general fibers of the corresponding Iitaka fibration are in a fixed bounded family of pola
We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the speci