All Questions
Tagged with fano-varieties ag.algebraic-geometry
91 questions
15
votes
2
answers
1k
views
How does one prove that the complete intersection of a quadric and a cubic of $\mathbb P^5$ is unirational?
The question is stated in the title, but I would like to add some motivation.
I've been teaching a course on complex tori and abelian varieties this semester and I would like to end it by showing ...
- 6,253
15
votes
1
answer
828
views
symplectic form on an algebraic family
I know that smooth Fano varieties over $\mathbb{C}$ may be classified into a finite number of families in each dimension (1 in dimension 1, 10 in dimension 2, 105 in dimension 3 ...).
I am ...
- 6,995
14
votes
1
answer
1k
views
Frobenius splitting of Fano varieties
Dear MO,
Question 1. Do you know of an example of a Fano variety which is not Frobenius split?
Background
(1) A variety $X$ in characteristic $p$ is called Frobenius split if there is a "$p$-th ...
- 16.2k
12
votes
1
answer
669
views
Deformation invariance of Fano varieties
Let $f:X \to T$ be a flat, projective morphism of noetherian schemes with $T$ an irreducible curve. Suppose that there exists a point $0 \in T$ such that the fiber $f^{-1}(0)$ is Fano.
Q. Is it ...
- 1,593
11
votes
2
answers
760
views
Families of Fano varieties over non-hyperbolic curves
Let $C$ be a non-hyperbolic (smooth quasi-projective connected complex algebraic) curve. That is, $C$ is isomorphic to $\mathbb P^1, \mathbb A^1, \mathbb G_m$, or an elliptic curve.
Let $f:X\to C$ be ...
- 9,497
11
votes
0
answers
684
views
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
10
votes
1
answer
572
views
Torsion in the cohomology of Fano varieties of lines
Let $\mathrm{X}$ be a cubic $d$-fold, and $\mathrm{F}(\mathrm{X})$ its Fano variety of lines. Is the integral cohomology of $\mathrm{F}(\mathrm{X})$ torsion-free? For $d=3$ A. Collino (`The ...
- 2,818
9
votes
1
answer
593
views
Do all Fano threefolds have effective $c_2$?
Let $X$ be a smooth complex projective Fano threefold. Then the class $c_1(X)$ can be realised as an effective divisor in $X$. It is it true that the class $c_2(X)$ can be realised as an effective ...
- 14.3k
8
votes
2
answers
771
views
How to compute the periodic cyclic homology of this algebra
Let $k=\mathbb{C}$ be the field of complex numbers. I consider the (DG) algebra $A:=k[x]/(x^2)$ such that $\deg(x)=-1$. My question is how to compute the periodic cyclic homology, Hochschild homology ...
- 1,982
8
votes
2
answers
467
views
Necessary condition on Calabi-Yau manfiold to be a hypersurface in a Fano manifold
Let $X$ be a smooth projective Calabi-Yau threefold. Are there any known obstructions to it
being a member of a base-point-free linear system in a nef-Fano fourfold?
What, in anything, is known ...
- 5,186
7
votes
1
answer
609
views
Is there an Enriques–Kodaira-like classification of Fano threefolds?
I am mainly interested in varieties over an algebraic closed field $k$ (or $\mathbb{C}$). The classification of complex surface is established in the last century and known as Enriques–Kodaira ...
- 73
6
votes
3
answers
868
views
Pseudo-automorphisms on Fano varieties
Is every pseudo-automorphism (self-birational map which does not contract any hypersurface) of a smooth Fano variety of Picard rank $1$ equal to a biregular automorphism?
Remark: For $\mathbb{P}^n$, ...
- 7,722
6
votes
1
answer
375
views
Cohomology of twisted holomorphic forms on Fano threefolds
Given a Fano threefold $X$, its index $ind(X)$ is the largest integer $r$ such that there exists a divisor $H$ such that $rH \cong -K_X$. Let $\mathcal{L}$ be the associated (ample) line bundle and ...
- 1,298
6
votes
1
answer
552
views
Can free rational curves lift to ramified covers of Fano varieties?
Does there exist $X$ a smooth Fano manifold, $f: Y \to X$ a nontrivial ramified finite cover, $C \subseteq X$ a smooth very free rational curve, such that $f$ is étale over a neighborhood of $C$?
...
- 149k
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
1
answer
585
views
Anti-canonical divisor of a Fano variety
Let $X$ be a normal projective Fano variety, that is the anti-canonical divisor $-K_X$ is ample.
For any $m>0$ let us consider the complete linear system $|-mK_X|$ and the map
$$f_{|-mK_X|}:X\...
user101890
5
votes
1
answer
162
views
Rationality of $V_1$ fano threefold
In the book of Iskovskikh and Prokhorov it seems not known wether the $V_1$, an hypersurface of degree $6$ in the weighted projective space $\mathbb{P}(3,2,1,1,1)$, is rational or not. Is there any ...
- 382
5
votes
3
answers
3k
views
Weak Fano and Log fano varieties
A projective smooth variety $X$ is weak Fano if $-K_X$ is nef and big. We say that $X$ is log Fano is there exists a divisor $D$ such that $-(K_X+D)$ is ample and $(X,D)$ is Kawamata log terminal.
Is ...
user49214
5
votes
3
answers
1k
views
Toric Fano manifolds with Picard number 1
As far as I know, toric Fano manifolds are classified only up to dimension 4. In dimension one the projective line is the only example. In dimension two we have five examples: $\mathbb P ^2$, $\mathbb ...
- 1,058
5
votes
1
answer
858
views
Fano varieties of cubic threefolds
Let $X$ be a smooth cubic threefold over $\mathbb{C}$. Let $F(X)$ denote the Fano variety of lines in $X$, which is a smooth surface of general type.
Is this class of surfaces distingushed ...
- 21.4k
5
votes
1
answer
344
views
Fano blow ups of $\mathbb CP^n$
Let $X$ be a smooth complex variety. Is it always possible to find an embedding $\varphi: X\to \mathbb CP^n$ for some $n$, such that the blow up of $\mathbb CP^n$ at $\varphi(X)$ is a Fano variety?
...
- 14.3k
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
1
answer
382
views
Formula for genus of a Fano variety
This is a simple question, just looking for a reference for a formula.
As far I understand the genus of a prime Fano $n$-fold is defined to be the genus of a complete intersection of $n-1$ smooth ...
- 6,995
4
votes
2
answers
1k
views
Varieties with big anti-canonical divisor
I recently heard about the following problem:
Let $X$ be a projective variety with klt singularities and such that $-K_X$ is big. Is $X$ a Mori Dream Space ?
Now, $-K_X$ big if and only if $-K_X -\...
user47036
4
votes
1
answer
458
views
Mirror symmetry for blowups of the projective plane
Let $S$ be a blowup of the projective plane $\mathbb{CP}^2$ at $n$ points. When $n\le 9$, Auroux, Katzarkov and Orlov showed that them a mirror Landau-Ginzburg model is given by a certain elliptic ...
- 41
4
votes
1
answer
819
views
Picard groups of Fano varieties in positive characteristic
Let $k$ be an algebraically closed field of characteristic $p \geq 0$. Let $X$ be a smooth Fano variety over $k$ and let $\ell \neq p$ be a prime.
Is the natural morphism $\mathrm{Pic}(X) \otimes \...
- 21.4k
4
votes
1
answer
311
views
Irrational Fano threefold whose intermediate Jacobian is Jacobian of curve
Clemens-Griffiths criterion for 3-fold says that if a smooth projective threefold $X/\mathbb C$ is rational, then the intermediate Jacobian $J(X)$ is isomorphic to product of Jacobians $J(C_1)\times \...
- 1,803
4
votes
1
answer
680
views
Blow-up of $\mathbb{P}^4$ along a quadric surface
Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of ...
user49214
4
votes
1
answer
299
views
Auto-equivalences of non-trivial components of derived category of $X_{18}$
Let $X:=X_{18}$ be an index one smooth prime Fano threefold of degree 18.
Consider its semi-orthogonal decomposition: $D^b(X)=\langle\mathcal{O}_X,\mathcal{E}^{\vee},\mathcal{A}_X\rangle=\langle\...
- 1,982
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
1
answer
308
views
Automorphism of moduli space of stable vector bundles over a curve
Let $C$ be a smooth genus two hyperelliptic curve and $\mathcal{M}_C$ be a moduli space of stable rank two vector bundles of fixed degree(or fixed determinant). Then is $\mathrm{Aut}(\mathcal{M}_C)\...
- 1,982
3
votes
1
answer
609
views
Semiorthogonal decompositions for Fano 3-folds and 4folds
Let $X$ be a projective Fano 3-fold or 4-fold and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. For what $X$ is it known a semi orthogonal decomposition into indecomposable ...
- 31
3
votes
1
answer
188
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(...
- 1,982
3
votes
1
answer
893
views
Existence of constant scalar curvature Kahler metrics on projective manifolds
It is well known that the blow-up of $\mathbb P^2$ in one or two points does not accept a Kahler-Einstein metric. Kahler-Einstein metrics are particular cases of constant scalar curvature Kahler ...
- 2,215
3
votes
1
answer
310
views
Are weak Fano 4-folds with canonical Gorenstein singularities bounded?
A Fano variety over $\mathbb{C}$ with Gorenstein singularity is called weak Fano if the anti-canonical divisor is nef and big.
Are there finite families of weak Fano 4-folds with canonical ...
- 3,472
3
votes
1
answer
254
views
Quotient of a Fano variety by a torus
We work over an algebraically closed field of characteristic zero. Let $X$ be a Fano variety, and $T\cong \mathbb{G}_m^r$ a torus acting faithfully on $X$.
I think we can canonically linearize the ...
- 2,384
3
votes
1
answer
185
views
How to check that exceptional sequence of vector bundles on Fano variety is helix foundation
Let $X$ be smooth Fano variety with $\operatorname{Pic}(X) = \mathbb{Z}$ of dimension $m$ with canonical class $K$, and $E_0,...,E_n$ is exceptional sequence of $(n+1)$ vector bundles in $D^b(Coh(X))$....
- 1,307
3
votes
1
answer
449
views
Components of Kontsevich moduli space of stable maps and multiple covers
Let $X\subset \mathbb P^n$ be a smooth projective variety over $\mathbb C$ which is Fano and $M_{0,0}(X,e)$ the (projective) Kontsevich moduli space of rational cuves of degree $e>1$. Is it ...
- 69
3
votes
1
answer
273
views
Are varieties with negative Kodaira dimension covered by hyperkahlers
Let $X$ be a smooth projective variety with negative Kodaira dimension over $\mathbb{C}$.
Is there an integer $n\geq 1$, a smooth projective hyperkahler variety $H$, and a finite morphism $H\to X^n$?
...
- 31
3
votes
0
answers
322
views
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
views
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
1
answer
556
views
A Fourier-Mukai equivalence between non trivial component of cubic threefold and degree 14 prime Fano threefold
Consider a cubic threefold $Y$ and its associated degree $14$ prime Fano threefold $X$, we have the equivalences of non-trivial components of $D^b(Y)$ and $D^b(X)$, i.e, $\mathcal{A}_X\cong\mathcal{B}...
- 1,982