Fanography

3-12 3-14

Fano variety 3-13

blowup of 2-32 in a curve $C$ of bidegree $(2,2)$ such that the composition $C\hookrightarrow W\hookrightarrow\mathbb{P}^2\times\mathbb{P}^2\overset{p_i}{\to}\mathbb{P}^2$ is an embedding for $i=1,2$

Picard rank

3 (others)

$-\mathrm{K}_X^3$

30

$\mathrm{h}^{1,2}(X)$

0

Mori–Mukai $d$

40

Still open.

index

1

$\dim\mathrm{H}^0(X,\omega_X^\vee)$

18

$-\mathrm{K}_X$ very ample?

yes

$-\mathrm{K}_X$ basepoint free?

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is stably rational.

The Weyl group associated to the KKMR decomposition is $\mathrm{A}_1$; see the Weyl-group table for sources and conventions.

This variety is the blowup of

• 2-32, in a curve of genus 0

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

number of moduli

1

Bott vanishing

does not hold

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

GRDB

#70

Fanosearch

#16

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

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

variety

$(\mathbb{P}^2)^3$

bundle

$\mathcal{O}(1,1,0) \oplus \mathcal{O}(1,0,1) \oplus \mathcal{O}(0,1,1)$

See the big table for more information.

• none are K-stable
• general member is K‑polystable but there exist members that are not
• every member is K‑semistable

• the book
• 2309.12518

For this deformation family, every smooth member has coregularity 0.

The generic and uniform statements are due to Avilov–Loginov–Przyjalkowski; the existence of a smooth member of coregularity zero in every family is due to Zhakupov.

See coregularity for terminology.

For every K-polystable member, the zero locus of the Futaki invariant contains a 2-dimensional family of Kähler classes including the anticanonical class.

This is the computation of Sektnan–Tipler.

The Hilbert scheme of conics is the disjoint union of 3 projective planes.

Its Hodge diamond is