# Questions tagged [fano-varieties]

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### Number of vertices of a lattice reflexive polytope

Casagrande's paper "The number of vertices of a Fano polytope" says that for $P$ a simplicial reflexive polytope of dimension $n$ has no more than $3n$ vertices. The polymake database ...
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### 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 ...
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### How does the Torelli theorem behave with respect to cyclic covering?

Let $Y\xrightarrow{2:1}\mathbb{P}^3$ be the double cover, branched over a quartic K3 surface $S$, known as quartic double solid. Assume $S$ is generic, we know that there is a Torelli theorem for $Y$ ...
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### Geometry of destabilizing centers in $K$-stability

In $K$-stability destabilizing centers are, roughly speaking, centers of valuations computing the stability thresholds. It is known that if $X$ is non $K$-semistable Fano variety then there exists a ...
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### Torelli theorem for veronese double cone(reference needed)

Let $Y$ be a smooth Veronese double cone, which is a smooth del Pezzo threefold of degree one, which can be regarded as a weighted hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. I was wondering ...
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### 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 ...
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### liftability of isomorphism of curves in $P^3$

It is well known that the isomorphism between smooth curves $C$ and $C'$ in $\mathbb{P}^2$ can be lifted to an automorphism of $\mathbb{P}^2$ if degree of $C$ and $C'\geq 4$. Now I am considering an ...
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### Projection from point on line on quintic del Pezzo surface

Let $X\subset \mathbb{P}^5$ be a quintic del Pezzo surface embedded anti-canonically and suppose $X$ is smooth. Suppose further we are given a line $L\subset X$. After a suitable change of variables ...
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### 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 ...
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### 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 ...
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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 \,\,...
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 ...