# Questions tagged [fano-varieties]

The fano-varieties tag has no usage guidance.

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

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### 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 $C$?
...

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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 ...

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### 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 ...

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114 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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### 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 ...

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120 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 $\...

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274 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 ...

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308 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 ...

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565 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 ...

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### 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 $...

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300 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\...

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

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### 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 ...

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383 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 ...

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199 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?
...

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### Do line bundles which divide the canonical bundle lift

Let $S $ be the spectrum of a complete dvr with algebraically closed residue field. Let $\eta$ be its generic point and let $s$ be its closed point with $k(s)$ of positive characteristic. Let $X\to S$ ...

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259 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 ...

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225 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 ...

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### 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?

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### 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 ...

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183 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 ...

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### 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 $...

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### 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 ...

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### 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 ...

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### Why is the Fano variety of lines on a smooth three-dimensional quadric isomorphic to $\mathbb{P}^3$?

Let $Q \subset \mathbb{P}^4$ be a smooth three-dimensional quadric over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$) and let $F$ be the Fano variety of lines on $Q$.
In "Iskovskikh ...

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### The canonical bundle of an infinitesimal deformation

Let $X_0$ be a smooth projective variety over the complex numbers and let $X$ be an infinitesimal deformation of $X_0$ over the ring of dual numbers.
If the canonical bundle of $X_0$ is ample (resp. ...

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### 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 \...

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### Reference request: log Fano varieties

I need a reference for a proof of the following fact: let $X$ be a toric variety then $X$ is log Fano.
Thanks a lot.

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### 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 ...

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### 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 $\mathbb{P}^n$, ...

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### 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 ...

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### Is being fano preserved under flat base change [closed]

Let $K$ be a field of characteristic zero and $X$ be a projective variety over $\mbox{Spec} K$. Denote by $\bar{K}$ the algebraic closure of $K$. Let $\bar{X}$ be the fiber product $X \times_K \bar{K}$...

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### 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 -\...

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363 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 ...

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### 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 ...

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### 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 ...

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### Toric Fano Kahler manifolds and Delzant polytopes

Let $P$ be a Delzant polytope in $\mathbb R^n$, given by a set of inequalities $\ell_i(x) > 0$ where $\ell_i(x) = \sum_k \mu_k^i x_k - \lambda_i$.
In his paper http://arxiv.org/abs/0803.0985 ...

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### 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 + ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Derived category of varieties and derived category of quiver algebras

I have heard that derived category of coherent sheaves $\mathrm{Coh}(X)$ on any Fano varieties $X$ may be realized as derived category $\mathrm{Coh}(\mathrm{Rep}(Q,W))$ of representation of quiver $Q$ ...

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### 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 ...

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### 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 ...