# Questions tagged [fano-varieties]

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