All Questions
Tagged with derived-categories fano-varieties
20 questions with no upvoted or accepted answers
6
votes
0
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184
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Mirror of the autoequivalences of the derived category of del Pezzo surface?
One version of the homological mirror symmetry conjecture states that for every Fano variety $X$ there exists a Landau--Ginzburg model $W$ such that the category of B-branes on $X$ (i.e. the bounded ...
user74900
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)=\...
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3
votes
0
answers
313
views
Are two versions of Kuznetsov components equivalent?
Let $X$ be a degree $10$, genus $6$ index $1$ and Picard number $1$ smooth Fano threefold. There are two versions of Semi-orthogonal decompositions. The First version is $$D^b(X)=\langle\mathrm{Ku}(X)...
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3
votes
0
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244
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Is the action of braid group on the set of full exceptional collections always transitive?
Let $X$ be a smooth complex projective variety and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. Then the braid group on $\mathrm{dim}\:K_0(X)$ strands acts by mutations on ...
user74900
2
votes
0
answers
45
views
Torelli theorem for veronese double cone(reference needed)
Let $Y$ be a smooth Veronese double cone, which is a smooth del Pezzo threefold of degree one, which can be regarded as a weighted hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. I was wondering ...
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2
votes
0
answers
75
views
What is happening on the second step of left mutation?
Let $X$ be a smooth Gushel-Mukai fourfold, whose semi-orthogonal decomposition is given by
$$D^b(X)=\langle\mathcal{K}u(X),\mathcal{O}_X,\mathcal{U}^{\vee}_X,\mathcal{O}_X(H),\mathcal{U}^{\vee}(H)\...
- 1,982
2
votes
0
answers
154
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Non-triviality of a morphism
Let $X$ be a smooth Gushel–Mukai fourfold and $Y$ a smooth hyperplane section, which is a Gushel–Mukai threefold. I consider semi-orthogonal decomposition of $X$ and $Y$:
$$D^b(X)=\langle\mathcal{O}_X(...
- 1,982
2
votes
0
answers
154
views
Normal bundle of a Fano threefold as Brill-Noether loci
Let $X$ be a degree 12 or degree 16 index one prime Fano threefold. In the paper of Mukai https://arxiv.org/pdf/math/0304303.pdf page 500, Theorem 4 and Theorem 5. He said $X_{12}$ has two ambient ...
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2
votes
0
answers
164
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Conics on Gushel-Mukai fourfold
Let $X$ be a very general Gushel-Mukai fourfold, let $\mathcal{U}$ be the tautological sub-bundle and $\mathcal{Q}$ be the tautological quotient bundle. Let $C\subset X$ be a $\rho$-conic, then $\...
- 1,982
2
votes
0
answers
282
views
Global Torelli and local Torelli for Fano threefolds (need reference)
It is known that in general Globally Torelli does not imply the local Torelli theorem, see
Is the Torelli map an immersion?
Globally Torelli means that the period map $\mathcal{P}$ is injective and
...
- 1,982
1
vote
0
answers
155
views
How does the Torelli theorem behave with respect to cyclic covering?
Let $Y\xrightarrow{2:1}\mathbb{P}^3$ be the double cover, branched over a quartic K3 surface $S$, known as quartic double solid. Assume $S$ is generic, we know that there is a Torelli theorem for $Y$ ...
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1
vote
0
answers
154
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Intermediate Jacobian for small resolution of a singular Fano threefold?
I am mainly interested in the nodal Gushel-Mukai threefold. Let $X$ be a Gushel-Mukai threefold with one node, then by page 21 of the paper https://arxiv.org/pdf/1004.4724.pdf there is a short exact ...
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1
vote
0
answers
101
views
Intersection of two quadrics as moduli space
Let $Y:=Q_1\cap Q_2\subset\mathbb{P}^{n-1}$ be smooth complete intersection of two quadrics. If $n$ is even, then it admits a semi-orthogonal decomposition:
$$D^b(Y)=\langle D^b(C),\mathcal{O}_Y,\...
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1
vote
0
answers
73
views
Action of involution on instanton bundle
Let $Y$ be a quartic double solid and $E$ be an rank two instanton bundle on $Y$. By Serre's correspondence, it is not hard to show that $E$ fits into the following short exact sequence $0\rightarrow\...
- 1,982
1
vote
0
answers
93
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
1
vote
0
answers
149
views
Dimension of Hilbert scheme of curves on Gushel-Mukai varieties
I have several questions on Hilbert scheme of Gushel-Mukai varieties. Let $X$ be a Gushel-Mukai fourfold and let $\mathcal{H}_3$ be Hilbert scheme of twisted cubics. I was wondering what is the ...
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1
vote
0
answers
125
views
Adjunctions of residue categories of Gushel-Mukai threefolds and Gushel-Mukai fourfolds
Let $X$ be an ordinary Gushel-Mukai fourfold and $Y$ its hyperplane section, which is a Gushel-Mukai threefold. I consider semi-orthogonal decompositions of $X$ and $Y$:
$D^b(X)=\langle\mathcal{K}u(X),...
- 1,982
1
vote
0
answers
25
views
The uniqueness of some semistable torsion free sheaves on Fano threefold
Let $X$ be a prime Fano threefold of index one and even genus $g\geq 6$, one can show that the moduli space of torsion free semistable sheaves $M(2,1,m_g)$ with $m_g=\left \lceil{\frac{g+2}{2}}\right \...
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1
vote
0
answers
88
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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. ...
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0
votes
0
answers
187
views
Derived Category of the Fano 4fold variety of lines
Let $X\subset P^5$ be a smooth cubic fourfold. It is well known that its variety of lines $F(X)$ is a smooth fourfold Fano variety. Hence its derived category should have a semi-orthogonal ...
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