All Questions
Tagged with birational-geometry derived-categories
17 questions
22
votes
2
answers
2k
views
Applications of derived categories to "Traditional Algebraic Geometry"
I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical ...
8
votes
1
answer
1k
views
Progress on Bondal–Orlov derived equivalence conjecture
In their 1995 paper, Bondal and Orlov posed the following conjecture:
If two smooth $n$-dimensional varieties $X$ and $Y$ are related by a flop, then their bounded derived categories of coherent ...
- 305
8
votes
1
answer
870
views
Why is proving fully-faithfulness of an integral functor locally analytically sufficient?
More than once I've come across a statement in a paper about derived categories in which it says something to the effect of "in order to prove that $\Phi:D^b(X)\rightarrow D^b(Y)$ is fully-faithful we ...
- 2,108
6
votes
1
answer
386
views
Derived categories of smooth proper varieties?
We know several amazing techniques about the derived category $Perf (X)$ of a smooth projective variety such as the whole theory of Fourier-Mukai transforms. On the other hand, from a dg-categorical ...
- 256
5
votes
2
answers
980
views
Does birational imply D-equivalent?
It is well-known that there are Calabi-Yau's who are not birational but are derived equivalent. However I am interested in seeing D-equivalence as a weakening of birationality.
Q. If $X$ and $Y$ ...
4
votes
1
answer
295
views
When is the birational Torelli problem for CY threefolds true?
I am aware from Borisov, Căldăraru, Perry and Ottem, Rennemo that what is known as the birational Torelli problem is false in general for Calabi-Yau threefolds, but I would like to know if there are ...
- 1,482
4
votes
0
answers
173
views
Bondal-Orlov conjecture on Calabi-Yau varieties
Recently, I am trying to study the various progress made on the Bondal-Orlov conjecture: Birational Calabi-Yau varieties ⟹ Equivalent derived categories.
I have started reading the paper by Bridgeland ...
- 345
3
votes
1
answer
664
views
Is there a blow-up formula for the derived category of a singular ambient variety?
For a nonsingular variety sitting inside a nonsingular ambient variety there is a semi-orthogonal decomposition of the derived category of the blow-up (with center that subvariety).
What can be said ...
- 1,162
3
votes
1
answer
296
views
Confusion about the (Grothendieck–Poincaré) double dual of reflexive differentials vs usual differentials on a normal Cohen–Macaulay scheme
$\DeclareMathOperator\Hom{Hom}$Let $\mathcal{A}$ be an abelian category, my question is about the case when $\mathcal{A}$ is the category of quasi-coherent sheaves on a scheme $X$. There is a fully ...
- 61
3
votes
0
answers
150
views
How to distinguish the singularities on moduli space?
Let me start with concrete examples. Let $X$ be a smooth special Gushel-Mukai threefold and $\mathcal{C}(X)$ be its honest Fano surface of conics, it has two irreducible components $\mathcal{C}(X)=\...
- 1,982
3
votes
0
answers
398
views
What is the most useful rationality criterion of surfaces?
The motivation for this question is that I would like to extract some information from derived category of surfaces to conclude the rationality of surface. There is a well known rationality criterion ...
- 1,982
2
votes
1
answer
353
views
Cohomology of normal bundle and tangent bundle on Gushel-Mukai threefold
Let $X$ be a smooth general ordinary Gushel-Mukai threefold. There is an embedding $X\rightarrow\mathrm{Gr}(2,5):=G$. Consider the normal bundle $\mathcal{N}_{X|G}$, how to compute cohomology of this ...
- 1,982
2
votes
1
answer
271
views
Does rational surface have exceptional collection of maximal length but not full?
Let $X$ be a rational surface. Let $\mathbb{E}:=(E_1,\ldots,E_n)$ be strong exceptional collection of line bundles of maximal length $l=rk Pic(X)+2$ in $D^b(coh(X))$, Is there any example that such ...
- 1,982
2
votes
1
answer
175
views
Explicit functor from Kuznetsov component to derived category of K3 for rational cubic fourfolds
Let $X \subset \mathbb{P}^5$ be a Pfaffian cubic fourfold (or one of the other known rational cubic fourfolds). It is known by Kuznetsov's Homological Projective Duality that $\mathcal{K}u(X) \simeq D^...
- 305
2
votes
1
answer
138
views
A Fourier-Mukai kernel locally given by a graph of a birational map and compatibility with extension
Let $X$ and $Y$ be smooth projective complex varieties. Suppose we have a Fourier-Mukai equivalence
$$
\Phi_\mathcal P :Perf X \to Perf Y
$$
with kernel $\mathcal P$. Moreover, suppose $\mathcal P$ ...
- 256
2
votes
0
answers
165
views
Euler form on three-fold
Let $X$ be a smooth projective $3$-fold over $\mathbb{C}$. Let $K_0(X)$ be its Grothendieck group, consider the Euler form defined as: $\chi(M,N): K_0(X)\times K_0(X)\rightarrow\mathbb{Z}$ by $(M,N)\...
- 1,982
1
vote
0
answers
88
views
Is there a direct way to show Fano surface of lines and conics on the pairs of Fano threefolds isomorphic?
I am considering the following setting:
Let $(Y_d, X_{4d+2})$ be the pair of degree $d$ and index 2 Fano threefold $Y_d$ and degree $4d+2$ index 1 Fano threefold and both of them are Picard number 1. ...
- 1,982