All Questions
10
questions
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
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
85
views
semi-orthogonal decomposition of Fano fourfold associated to threefold
Let $Y$ be Gushel-Mukai threefold and $X$ a Gushel-Mukai fourfold containing $Y$ as its hyperplane section, the semi-orthogonal decomposition of $X$ and $Y$ are both known. Also, for cubic fourfold ...
- 1,962
2
votes
0
answers
150
views
Normal bundle of a Fano threefold as Brill-Noether loci
Let $X$ be a degree 12 or degree 16 index one prime Fano threefold. In the paper of Mukai https://arxiv.org/pdf/math/0304303.pdf page 500, Theorem 4 and Theorem 5. He said $X_{12}$ has two ambient ...
- 1,962
2
votes
0
answers
163
views
Conics on Gushel-Mukai fourfold
Let $X$ be a very general Gushel-Mukai fourfold, let $\mathcal{U}$ be the tautological sub-bundle and $\mathcal{Q}$ be the tautological quotient bundle. Let $C\subset X$ be a $\rho$-conic, then $\...
- 1,962
2
votes
1
answer
328
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 ...
- 1,962
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
21
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 ...
3
votes
0
answers
297
views
Resolving structure sheaf of diagonal via universal bundle on moduli space
To study the geometry of the moduli space $\mathcal M$ of semi-stable sheaves on a variety $X$ with fixed Hilbert polynomial, it is useful to have a locally free resolution of the structure sheaf $\...
- 1,990