All Questions
Tagged with ag.algebraic-geometry derived-categories
402
questions
1
vote
1
answer
124
views
About the category $\mathbf{Coh}(\mathbb{P}^2,\mathcal{B}_0)$
In the paper A Categorical Invariant for Cubic Threefolds, Bernardara, Macrì, Mehrotra, and Stellari consider the category $\mathbf{Coh}(\mathbb{P}^2,\mathcal{B}_0)$ where $\mathcal{B}_0$ is a rank $4$...
anon
3
votes
1
answer
415
views
Derived $\ell$-completion of $\mathbf{Q}_\ell$ sheaf?
I came across some notation that I’m having trouble understanding in Hansen-Scholze’s preprint ‘Relative Perversity.’ In the last paragraph of Proposition 3.4 there is the notation
$A\widehat{\otimes^{...
- 1,187
6
votes
1
answer
465
views
How does the Balmer spectrum fail to describe the algebraic geometry of categories of non-compact objects?
In trying to understand the higher algebraic geometry of the stable homotopy category, one thing I've come across repeatedly is the claim that one should only consider the Balmer spectrum of a tt-...
- 1,837
2
votes
1
answer
118
views
Right adjoint of subcollection of semi-orthogonal decomposition
Suppose $X$ is a prime Fano threefold of index 1 such that $H = -K_X$ is ample. There is a full classification of the derived category of such threefolds depending on the genus of $X$; in the case ...
- 275
2
votes
1
answer
299
views
Fourier-Mukai functors and autoequivalence groups of $G$-equivariant derived categories
I have a few questions about $G$-equivariant derived categories. For my question, I'm assuming $G$ is cyclic. Also, in my case $G$ does not act on $X$, only on $D^b(X)$.
Q1: Orlov's Representability ...
- 305
1
vote
0
answers
92
views
Fourier-Mukai kernels for Fano threefolds
Let $Y_1$ and $Y_1'$ be index two degree one Fano threefolds. Suppose we have a Fourier-Mukai equivalence $\Phi_P : \mathrm{D}^b(Y_1) \to \mathrm{D}^b(Y_1')$. Can anything be said about the kernel $P$,...
- 305
3
votes
1
answer
180
views
Semi-orthogonal decomposition for maximally non-factorial Fano threefolds
Let $X$ be a nodal maximally non-factorial Fano threefold. If there is $1$-node and no other singularities, they by the work of Kuznetsov-Shinder https://arxiv.org/pdf/2207.06477.pdf Lemma 6.18, $D^b(...
- 1,962
3
votes
1
answer
170
views
Left adjoint for nested admissible categories
This question is motivated by the construction of the Kuznetsov component on a prime Fano threefold $X$ of index 1 (say genus $g \geq 6$, $g \neq 7, 9$):
$$
D^b(X) = \langle Ku(X), E, \mathcal{O}_X \...
- 275
3
votes
1
answer
318
views
Existence of rigid objects in the derived category of a smooth projective variety
Let $X$ be a smooth projective variety (say over $\mathbb{C}$). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a ...
- 7,250
3
votes
0
answers
119
views
Existence of different Jordan-Holder filtrations
Assume $X/\mathbb{C}$ is a projective K3 surface. Let $\sigma$ be a (geometric) Bridgeland stability condition for $\rm{D}^b(X)$.
My questions are :
Is there any nontrivial example of $E \in \rm{D}^...
- 31
1
vote
0
answers
109
views
Does there exist other known pair of Fano threefolds/fourfolds with residue categories being K3/Enriques?
Let $Y$ be a Gushel-Mukai threefold, we can either consider an ordinary Gushel-Mukai fourfold $X$ containing $Y$ as a hyperplane section, or we consider a special Gushel-Mukai fourfold $X'$ as double ...
- 1,962
2
votes
1
answer
227
views
How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?
Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...
- 8,767
1
vote
1
answer
204
views
Semi-orthogonal decomposition of Verra threefold
Let $X$ be a Verre-threefold, which is by definition a $(2,2)$ hypersurface in $\mathbb{P}^2\times\mathbb{P}^2$, it is a Fano threefold. What is the semi-orthogonal decomposition of $D^b(X)$? It ...
- 1,962
1
vote
0
answers
216
views
Do we have a left adjoint of $i^*$ for a closed immersion $i$?
Let $i: X\hookrightarrow Y$ be a closed immersion of varieties. We have the derived pullback functor $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$.
My questions is: can we construct a left adjoint of $i^*$ in ...
- 8,767
0
votes
1
answer
148
views
What is the cone of $\mathcal{F}\to i_*i^*\mathcal{F}$ for a divisor $i: D\hookrightarrow X$?
Let $X$ be a smooth projective variety of dimension $n$ and $i: D\hookrightarrow X$ be a smooth, anti-canonical divisor in $X$. In other words,$[D]+[\omega_X]=[0]$ where $\omega_X$ is the canonical ...
- 8,767