All Questions
Tagged with fano-varieties ag.algebraic-geometry
33 questions with no upvoted or accepted answers
11
votes
0
answers
684
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Big tangent bundle
Let $X$ be a nonsingular complex algebraic variety whose tangent bundle is $T_X$. I use Lazarsfeld's book for the definition of a big vector bundle. A line bundle $L$ is big if it has Itaka dimension ...
- 121
6
votes
0
answers
184
views
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
6
votes
0
answers
378
views
Bound for the Picard number of a Fano 3-fold
Let $X$ be a Fano 3-fold with terminal singularities. Is there some bound (possibly explicit) for the Picard rank of $X$ ?
If $X$ is smooth, it is well-known that the bound is $10$, obtained by del ...
- 7,722
5
votes
0
answers
260
views
Equations for conic del Pezzo surfaces of degree one
Let $X$ be a del Pezzo surface of degree one over a field $k$ of characteristic not $2$ equipped with a conic bundle $\pi: X \rightarrow \mathbb{P}^1$. By Theorem 5.6 of this paper, $X$ admits a ...
- 241
4
votes
0
answers
150
views
Kodaira vanishing + simple connectedness implies Fano
To avoid monkey business, let $X$ be a smooth complex projective variety of dimension $m$. As usual, $K_X$ is its canonical line bundle. Let us further assume that $X$ is simply-connected and the ...
- 12.4k
4
votes
0
answers
95
views
Formula for bound on number of smooth projective toric Fano varieties of dimension n
In dimension 1, the only smooth projective toric Fano variety is $\mathbb{P}^1$. In dimension $2$, there are 5: $\mathbb{P}^1\times \mathbb{P}^1$, and then successive blow-ups of $\mathbb{P}^2$ at up ...
- 511
3
votes
0
answers
322
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K3 surfaces in Fano threefolds
By K3 surfaces and Fano threefolds, I mean smooth ones.
If a K3 surface $S$ is an anticanonical section of a Fano threefold $V$ of Picard rank one (hence, $Pic(V)=\mathbb Z H_V$ for some ample divisor ...
- 1,841
3
votes
0
answers
286
views
A K3 cover over a Del Pezzo surface
Let $V \rightarrow \mathbb P^2$ be the blow-up at two distinct points. ($V$ is a Del Pezzo surface of degree 7.)
Choose a smooth curve $C$ from the linear system $|-2K_V|$ and let $S \rightarrow V$ be ...
- 1,841
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
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 $$D^b(X)=\langle\mathrm{Ku}(X)...
- 1,982
3
votes
0
answers
244
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Is the action of braid group on the set of full exceptional collections always transitive?
Let $X$ be a smooth complex projective variety and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. Then the braid group on $\mathrm{dim}\:K_0(X)$ strands acts by mutations on ...
user74900
3
votes
0
answers
219
views
Cohomology of the Hilbert square of a degree 14 K3 surface [Beauville-Donagi]
I have a question about the article by Beauville-Donagi called La variété des droites d'une hypersurface cubique de dimension 4 (C. R. Acad. Sc. Paris, t. 301, Série I, n° 14, 1985).
Their ...
3
votes
0
answers
418
views
Lines on Fano complete intersections
Let $X \subset \mathbb{P}^n$ be a non-singular complete intersection of $s$ hypersurfaces
of degrees $d_1,\dots,d_s$ over an algebraically closed field $k$ of characteristic zero. Let $d=d_1 + \dots + ...
- 21.4k
3
votes
0
answers
486
views
Arithmetic of Fano varieties of lines
Let $k$ be a number field and let $X \subset \mathbb{P}^n$ be a non-singular hypersurface of degree $n-1$. Let $F(X)$ denote the Fano variety of lines of $X$. Then it is known that for general $X$ the ...
- 21.4k
2
votes
0
answers
154
views
Non-triviality of a morphism
Let $X$ be a smooth Gushel–Mukai fourfold and $Y$ a smooth hyperplane section, which is a Gushel–Mukai threefold. I consider semi-orthogonal decomposition of $X$ and $Y$:
$$D^b(X)=\langle\mathcal{O}_X(...
- 1,982
2
votes
0
answers
144
views
Fundamental group of the moduli space of parabolic bundles with fixed determinant
I am looking for the fundamental group of the moduli space of parabolic bundles with fixed determinant over a smooth projective curve.
I know that the fundamental group of the moduli space of vector ...
- 195
2
votes
0
answers
214
views
Number of lines on a weak del Pezzo surface
By a line I mean a (-1)-curve. Given a weak del Pezzo surface $X$ of degree $d$, how many lines would $X$ contain?
- 481
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 ...
- 1,982
2
votes
0
answers
164
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 $\...
- 1,982
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
2
votes
0
answers
337
views
Stability notion to smoothing varieties under a flat deformation with a smooth total space
Is there any stability notion that led to an algebraic variety be smoothable in general for Fano varieties or for Calabi-Yau varieties?
Note that Friedman found a nesessary condition that $X$ to be ...
user21574
2
votes
0
answers
276
views
Components of Kontsevich moduli space of stable maps and reducible curves
Let $X\subset \mathbb P^n$ be a smooth projective variety which is Fano and $M_{0,0}(X,e)$ the (projective) Kontsevich moduli space of rational curves of degree $e>1$. Is it possible (or are there ...
- 69
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 $Y$ ...
- 1,982
1
vote
0
answers
197
views
Intersection of three quadrics: associating something geometric to these analogously to intersection of two quadrics
Consider the smooth intersection of two $4$-dimensional quadrics $Y = Q \cap Q' \subset P^5$. To the Fano threefold $Y$ we can associate a genus $2$ curve as follows. Take the pencil of quadrics $\{ ...
- 123
1
vote
0
answers
154
views
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 ...
- 1,982
1
vote
0
answers
94
views
Invariant category and coinvariant category under group action
Let $\mathcal{C}$ be a category with a finite group action $G$, There is a notion called G-equivariant category, denoted by $\mathcal{C}^G$. In the paper Kuznetsov's Fano threefold conjecture via K3 ...
- 1,982
1
vote
0
answers
132
views
The geography of models of Fano varieties
This question aims to compute ${\rm Vol}(-K_X-tD)$ where $X$ is a $\mathbb{Q}$-factorial Fano variety of dimension $n$ and $D$ is a nonzero effective divisor on $X$. This volume is positive when $0\le ...
- 498
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 ...
- 1,982
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
171
views
Fano surface of conics on Gushel-Mukai threefolds
Let $X$ be a smooth Gushel-Mukai threefold, there are following four cases:
$X_1$ is a special Gushel-Mukai with branch locus $\mathcal{B}$ on $Y_5$ general, i.e, it does contain any line or conic. $\...
- 1,982
0
votes
0
answers
98
views
$h^0(X, 4H-5E)$ on weak Fano threefold
Let $X$ be a smooth weak fano threesfold arising as the blowup of a smooth curve $C$ of degree and genus $(d,g)=(10,2)$ in a rank 1 smooth fano threefold $Y$, $-K_Y^3 = 22$. Let $H$ be a hyperplane in ...
- 71
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 ...
- 3,779