Fanography

A tool to visually study the geography of Fano 3-folds.

Identification

Fano variety 3-10

blowup of 1-16 in the disjoint union of 2 conics

Alternative description:

  • complete intersection of degree $(1,0,1)$, $(0,1,1)$ and $(0,0,2)$ in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4$
Picard rank
3 (others)
$-\mathrm{K}_X^3$
26
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond and polyvector parallelogram
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
1
0 0*
0 2* 5
0 0 0 16
0 0 0
0 0
0
* indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1, 2
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
16
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is rational.


This variety is the blowup of

  • 2-29, in a curve of genus 0
Deformation theory
number of moduli
2
Bott vanishing
does not hold
$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathbb{G}_{\mathrm{m}}^2$ 2 0
$\mathbb{G}_{\mathrm{m}}$ 1 1
$0$ 0 2
Period sequence

The following period sequences are associated to this Fano 3-fold:

GRDB
#99
Fanosearch
#67
Extremal contractions
Semiorthogonal decompositions

A full exceptional collection can be constructed using Orlov's blowup formula.

Structure of quantum cohomology

Generic semisimplicity of:

  • small quantum cohomology, proved by Ciolli in 2005, see [MR2168069] , using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
Zero section description

Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):

variety
$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^4$
bundle
$\mathcal{O}(1,0,1) \oplus \mathcal{O}(0,1,1) \oplus \mathcal{O}(0,0,2)$

See the big table for more information.

K-stability
  • general member is K‑stable but there exists one that is not
  • general member is K‑polystable but there exists one that is not
  • every member is K‑semistable
See and the big table for more information.