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3
votes
2answers
213 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 ...
-1
votes
1answer
129 views

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 ...
4
votes
2answers
216 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 ...
2
votes
1answer
196 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 ...
4
votes
3answers
310 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 ...
7
votes
2answers
278 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 ...
4
votes
1answer
176 views

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 ...
3
votes
0answers
103 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 + ...
3
votes
1answer
142 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 ...
2
votes
1answer
220 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
votes
0answers
169 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 ...
5
votes
1answer
205 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 ...
7
votes
0answers
343 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 ...
3
votes
3answers
360 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 ...
5
votes
1answer
235 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 ...
13
votes
2answers
332 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 ...
5
votes
2answers
580 views

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 $Q$ ...
10
votes
1answer
513 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 ...
3
votes
1answer
349 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 ...
3
votes
0answers
205 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 ...
2
votes
0answers
136 views

Controlling singularities on log mmp

Suppose all my varieties are complex threefolds $X\rightarrow Y$ over some smooth base curve germ $Y$. We can assume the fibres are Del Pezzo surfaces with generic smooth fibre. If I do (relative) ...