# Questions tagged [fano-varieties]

The fano-varieties tag has no usage guidance.

77
questions

2
votes

1
answer

182
views

### $K3$ surfaces in $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$

I am considering $K3$ surfaces in $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$ with an automorphism that preserves an ample divisor class.
For an automorphism $\rho$ of a $K3$ surface, let ${\...

1
vote

0
answers

119
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?

1
vote

1
answer

129
views

### Markov triples and Newton-Okounkov bodies of $\mathbb{P}^2$

I am working on symplectic geometry and I have some questions about a degeneration of $\mathbb{P}^2$.
Question: Can we obtain the moment polytope (or the polytope associated with the anti-canonical ...

1
vote

0
answers

97
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
vote

0
answers

77
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),...

2
votes

0
answers

127
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 ...

2
votes

0
answers

143
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 $\...

3
votes

0
answers

115
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
vote

0
answers

20
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 \...

1
vote

1
answer

111
views

### A short exact sequence on del Pezzo threefold and Gushel-Mukai

Let $Y$ be degree 5 index two prime Fano threefold. Let $\mathcal{E}$ and $\mathcal{Q}$ be the tautological sub and quotient bundle on $Y$. It is not hard to show that there is a short exact sequence:
...

4
votes

1
answer

291
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 ...

2
votes

0
answers

192
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
...

4
votes

1
answer

246
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\...

2
votes

1
answer

428
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
vote

1
answer

154
views

### Dual of stable vector bundle on a Fano threefold

Let $E$ be a rank $2$ stable vector bundle on a prime Fano threefold of genus $8$, with Chern numbers $c_1=1, c_2=6, c_3=0$.
Question. Is it true that $E(-1)=E^*$?
What I am able to show is that ...

2
votes

1
answer

179
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
vote

0
answers

65
views

### 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. ...

1
vote

0
answers

138
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. $\...

3
votes

0
answers

239
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)...

2
votes

1
answer

191
views

### The locus of lines intersecting with another fixed line on a Fano threefold

Let $Y$ be an index $2$, degree $5$, Picard number $1$ Fano threefold, i.e $Y$ is a linear section of Grassmannian $\operatorname{Gr}(2,5)$. Let $\Sigma(Y)$ be the Hilbert scheme of lines on $Y$, it ...

2
votes

0
answers

83
views

### arithmetic del Pezzo surfaces in comparison with del Pezzo surfaces over a field

A del Pezzo surface is a smooth, 2-dimensional projective variety $X$ with ample anticanonical divisor, i.e. a 2-dimensional Fano variety.
I am interested in the arithmetic analogue, a 2-dimensional ...

2
votes

0
answers

131
views

### Structure of the big cone and Seshadri constant on Fano manifolds

I would like to know something about the following two questions.
Given $X$ Fano manifold and $L$ an ample line bundle on $X$, we define
\begin{gather}
\sigma(L,x):=\sup\{t>0\, :\, \mu^{*}L-tE \,\,...

2
votes

1
answer

233
views

### Do non-compact Fano manifolds exist?

Suppose $(M,g, \omega)$ is a Kähler manifold with $\text{Ric}(g) = g$, i.e., $M$ is a Fano manifold. Is $M$ necessarily compact? If not, perhaps complete and Fano implies compact? I'd like to build a ...

10
votes

1
answer

458
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 ...

0
votes

1
answer

158
views

### Negative Definite Fano Manifolds

A complex manifold $M$ is said to be Fano if the Chern curvature $2$-form is a positive definite $(1,1)$-form. What happens if the Chern curvature $2$-form is a negative definite $(1,1)$-form? What ...

3
votes

1
answer

188
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 ...

3
votes

1
answer

147
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 $D^b(Coh(X))$....

6
votes

1
answer

439
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 $C$?
...

2
votes

0
answers

152
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 ...

3
votes

0
answers

168
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 ...

0
votes

0
answers

154
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 ...

5
votes

0
answers

158
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 ...

1
vote

2
answers

155
views

### Minimal embeddings of certain Fano varieties with Picard number one

Let $X$ and $Y$ be two Fano varieties of the same dimension embedded into a same projective space $\mathbb P^N$, assume $Pic X= \mathbb Z\mathcal O_X(1)$ and $Pic Y=\mathbb Z\mathcal O_Y(1)$, where $\...

9
votes

1
answer

425
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 ...

4
votes

1
answer

411
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 ...

15
votes

1
answer

725
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 ...

4
votes

0
answers

135
views

### Integrable systems with Fano phase space?

What are some known examples of finite-dimensional integrable systems with symplectic Fano phase space?
Here by integrable system we mean a symplectic manifold $(X,
\omega)$ of dimension $2n$ with $...

4
votes

1
answer

456
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\...

3
votes

1
answer

217
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 $H\to X^n$?
...

5
votes

0
answers

151
views

### McLean theorem for Fano varieties?

Well-known McLean theorem states that deformations of special Lagrangian $L$ submanifolds in Calabi-Yau manifold are unobstructed and in bijection with harmonic 1-forms on $L$. The proof relies on the ...

11
votes

1
answer

552
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 ...

3
votes

1
answer

262
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?
...

2
votes

0
answers

311
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 ...

7
votes

1
answer

260
views

### Fiberwise compactification of a LG model

It is believed that a mirror of $\mathbb{CP}^2$ is a fiberwise compactification of the family $$W \colon (\mathbb{C}^\times)^2 \rightarrow \mathbb{C}, \quad (x,y)\mapsto x+y+\frac{1}{xy}.$$ Is it ...

1
vote

1
answer

222
views

### Analogy of a Fano manifold with anticanonical divisor

Some people say that a Fano manifold with anticanonical divisor is an analogue of a manifold with boundary. Where does this intuition come from?

4
votes

1
answer

151
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 ...

1
vote

1
answer

189
views

### Does a moving family of lines through a fixed point produce a singularity?

This is just a feeling that I had and I am curious if it is totally wrong or true to some extent.
Let $X\subseteq \mathbb{P}^r$ be an integral hypersurface of degree $r-1$, which is not a cone. In ...

1
vote

1
answer

222
views

### simple normal crossing divisors on Fano manifold

Let $M$ be a Fano manifold. And $D=\mathop\sum\limits_{i=1}^r\tau_iD_i\in|-\lambda K_M|$ is a simple normal crossing $\mathbb{R}$-divisor where $\tau_i\in(0,1)$. Can we know that $D_i$s are ample (or $...

2
votes

0
answers

228
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 ...

3
votes

1
answer

376
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 ...