Fanography

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

Identification

Fano variety 2-21

blowup of 1-16 in a twisted quartic

Picard rank
2 (others)
$-\mathrm{K}_X^3$
28
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond and polyvector parallelogram
1
0 0
0 2 0
0 0 0 0
0 2 0
0 0
1
1
0 0*
0 2* 8
0 0 0 17
0 0 0
0 0
0
* indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1, 3
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
17
$-\mathrm{K}_X$ 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

  • 1-16, in a curve of genus 0

This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

Deformation theory
number of moduli
2
Bott vanishing
does not hold
$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathrm{PGL}_2$ 3 0
$\mathbb{G}_{\mathrm{a}}$ 1 0
$\mathbb{G}_{\mathrm{m}}$ 1 1
$0$ 0 2
Period sequence

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

GRDB
#84
Fanosearch
#8
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
$\operatorname{Gr}(2,4) \times \mathbb{P}^4$
bundle
$\mathcal{U}^{\vee}_{\operatorname{Gr}(2,4)}(0,1)^{\oplus 2} \oplus \mathcal{O}(1,0)$

See the big table for more information.

Hilbert schemes of curves

The Hilbert scheme of conics is a quadric surface.

Its Hodge diamond is

1
0 0
0 2 0
0 0
1