Fanography

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Fano variety 1-13

hypersurface of degree 3 in $\mathbb{P}^4$

Picard rank

1 (others)

$-\mathrm{K}_X^3$

24

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

5

Mori–Mukai $d$

6

The holomorphic Poisson structures form the following irreducible components of $\mathbb{P}\mathrm{H}^0(X,\wedge^2\mathrm{T}_X)$:

See Loray–Pereira–Touzet.

index

2

del Pezzo of degree 3

$X\hookrightarrow\mathbb{P}^{4}$

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

15

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is not rational but unirational.

A very general member is not stably rational.

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

This variety is primitive.

This variety can be blown up (in a curve) to

• 2-5, in a curve of genus 1
• 2-11, 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

10

Bott vanishing

does not hold

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

GRDB

#106

Fanosearch

#0

The naive atomic decomposition has atom dimensions $2+1+1$.

A standard semiorthogonal decomposition is $\langle \operatorname{Ku}(X), \mathcal{O}_X, \mathcal{O}_X(1) \rangle$.

Both decompositions are recorded in Böhning–Graf von Bothmer–Su'a.

By Hertling–Manin–Teleman we have that quantum cohomology cannot be generically semisimple, as $\mathrm{h}^{1,2}\neq 0$.

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

variety

$\mathbb{P}^4$

bundle

$\mathcal{O}(3)$

See the big table for more information.

• every member is K‑stable
• every member is K‑polystable
• every member is K‑semistable

• the book

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 Futaki invariant vanishes identically on the entire Kähler cone.

This is the computation of Sektnan–Tipler.

The Hilbert scheme of lines is a Fano surface, a minimal surface of general type.

Its Hodge diamond is