Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDerived categories of Quot schemes of locally free quotients, I
91
0
0.0
(
0
)
Authors
Qingyuan Jiang
Ask ChatGPT about the research
No Arabic abstract
This paper studies the derived category of the Quot scheme of rank $d$ locally free quotients of a sheaf $mathscr{G}$ of homological dimension $le 1$ over a scheme $X$. In particular, we propose a conjecture about the structure of its derived category and verify the conjecture in various cases. This framework allows us to relax certain regularity conditions on various known formulae -- such as the ones for blowups (along Koszul-regular centers), Cayleys trick, standard flips, projectivizations, and Grassmannain-flips -- and supplement these formulae with the results on mutations and relative Serre functors. This framework also leads us to many new phenomena such as virtual flips, and structural results for the derived categories of (i) $mathrm{Quot}_2$ schemes, (ii) flips from partial desingularizations of $mathrm{rank}le 2$ degeneracy loci, and (iii) blowups along determinantal subschemes of codimension $le 4$.
rate research
Read More
We prove a formula for Chow groups of $Quot$-schemes which resolve degeneracy loci of a map between vector bundles, under expected dimension conditions. This result provides a unified way to understand known formulae for various geometric situations such as blowups, Cayleys trick, projectivizations, Grassmannian bundles, flops from Springer type resolutions, as well as provide new phenomena such as formulae for Grassmannain type flips/flops and virtual flips. We also give applications to blowups of determinantal ideals, moduli spaces of linear series on curves, and Hilbert schemes of points on surfaces.
We provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the base and the remaining non-trivial component is equivalent to the derived category of a flat and finite of degree 5 scheme over the base. We introduce two methods for the construction of the decomposition. One is the moduli space approach following the work of Kuznetsov on the sextic del Pezzo fibrations and the components are given by the derived categories of fine relative moduli spaces. The other approach is that one can realize the fibration as a linear section of a Grassmannian bundle and apply Homological Projective Duality.
A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by describing the complement. As an application, we can lift the quadratic Fano correspondence (due to Galkin-Shinder) in the Grothendieck ring of varieties between a smooth cubic hypersurface, its Fano variety of lines, and its Hilbert square, to a semiorthogonal decomposition. We also show that the Hilbert square of a cubic hypersurface of dimension at least 3 is again a Fano variety, so in particular the Fano variety of lines on a cubic hypersurface is a Fano visitor. The most interesting case is that of a cubic fourfold, where this exhibits the first higher-dimensional hyperkahler variety as a Fano visitor.
We prove two existing conjectures which describe the geometrical McKay correspondence for a finite abelian G in SL3(C) such that C^3/G has a single isolated singularity. We do it by studying the relation between the derived category mechanics of computing a certain Fourier-Mukai transform and a piece of toric combinatorics known as `Reids recipe, effectively providing a categorification of the latter.
Our main goal is to give a sense of recent developments in the (stable) rationality problem from the point of view of unramified cohomology and 0-cycles as well as derived categories and semiorthogonal decompositions, and how these perspectives intertwine and reflect each other. In particular, in the case of algebraic surfaces, we explain the relationship between Blochs conjecture, Chow-theoretic decompositions of the diagonal, categorical representability, and the existence of phantom subcategories of the derived category.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
