Fano fourfolds of type A beta

A ZL62 label is a compact, case-sensitive encoding of an ambient product of type A flag varieties and a defining homogeneous vector bundle. It identifies a zero-locus presentation; several presentations can belong to the same deformation family.

The part before the dot encodes the ambient variety. When present, the part after the dot encodes the defining bundle. A presentation with zero defining bundle is the ambient variety itself.

For example, 40 presents projective four-space with zero defining bundle. Keep ZL62 labels as text: letter case matters, and even an entirely numerical label is an identifier.

The defining data on each family page displays the ambient variety and bundle explicitly. See the bundle notation, or use the ZL62 encoder and decoder to inspect an encoding.

Family and presentation

A family groups fourfolds related by smooth deformation; a presentation constructs a general zero locus in a specified ambient variety. Known matches are grouped together; distinct catalogue entries can still describe the same family.

Family label

The entries are Picard rank, anticanonical volume, and anticanonical section count, followed by a distinguishing letter. Thus 5-120-31-a has $\rho=5$, $\mathrm{c}_1^4=120$, and $\mathrm{h}^0(X,\omega_X^\vee)=31$.

Preferred model and presentation list

The preferred model is the catalogue's chosen representative presentation, displayed by default when opening a family page. Selecting another presentation changes its defining data and polyvector/Hochschild data; the family invariants stay fixed.

Connected components

When a zero locus has several isomorphic connected components, the displayed invariants describe one component.

FanoK3 reference

A recorded case label links to the corresponding example in Bernardara–Fatighenti–Manivel–Tanturri, Fano fourfolds of K3 type.

Defining bundle

The fourfold is the zero locus of a general section of a globally generated homogeneous vector bundle $\mathcal{E}$ on the ambient variety $Y$.

Strongly Fano

The ambient line bundle $L=\omega_Y^\vee\otimes(\det\mathcal{E})^\vee$ is ample. Adjunction gives $\omega_X^\vee\cong L|_X$. The catalogue's enumeration is bounded, so it is not an exhaustive classification.

Weights and Dynkin diagram

$\mathcal{E}^{\mu}$ denotes an irreducible homogeneous bundle of highest weight $\mu$. The fundamental weights $\omega_i$ number the diagram's nodes consecutively across ambient factors. Filled nodes mark the subspace dimensions defining each flag.

Tautological notation

$\mathcal{U}_i$ and $\mathcal{Q}_i$ are the tautological subbundle and quotient bundle on factor $i$. Pullback to the ambient product is understood. The two-index versions on flags denote successive quotient bundles, ordered from the ambient quotient down to the smallest subspace.

Line bundles

On a Grassmannian, $\mathcal{O}(1)=\det\mathcal{U}^{\vee}$ is the Plücker line bundle. Determinant factors in Grassmannian bundle expressions are combined into these twists. On $\mathbb{P}^2$, this simplifies $\mathcal{U}^{\vee}\otimes\mathcal{Q}^{\vee}=\mathcal{Q}^{\vee}(1)$ to $\mathcal{Q}$. On a flag variety, $\mathcal{O}(d_1,\ldots,d_s)$ gives the twists in the Plücker line bundles of its marked dimensions.

Projections and their indices

Each marked dimension of a flag gives a Grassmannian coordinate, recording the subspace of that dimension. Number these coordinates from left to right across the ambient factors, with increasing subspace dimensions within each flag. The projection $\pi_i$ forgets coordinate $i$ and keeps the others. A Grassmannian factor contributes one index; a flag $\operatorname{Fl}(k_1,\ldots,k_s;n)$ contributes $s$. Thus an elementary projection of a flag forgets one subspace, retaining the rest of the flag. The notation $\pi_{ij}$ means that both coordinates $i$ and $j$ are forgotten. These indices count marked nodes, whereas the fundamental-weight indices count all Dynkin nodes.

Product and flag examples

On $Y=\mathbb{P}^2\times\mathbb{P}^4$, the convention is $$\begin{equation} \notag \pi_1(x,y)=y,\qquad \pi_2(x,y)=x. \end{equation}$$ For family 2-513-105-a, their restrictions to $X$ are the blow-up map to $\mathbb{P}^4$ along a line and a $\mathbb{P}^2$-bundle over $\mathbb{P}^2$, respectively. On $\operatorname{Fl}(1,3;5)$, a point is a flag $L\subset W\subset\mathbb{C}^5$ with $\dim L=1$ and $\dim W=3$. Here $\pi_1$ forgets $L$ and maps to $\operatorname{Gr}(3,5)$; $\pi_2$ forgets $W$ and maps to $\mathbb{P}^4$.

Images and fibres on the fourfold

In a family description, the same notation denotes the restriction of an ambient projection to $X$. Its image can be a proper subvariety of the ambient target; the description identifies that image and describes the fibres or exceptional locus. A projection can be an isomorphism onto its image, a fibration, or a birational contraction such as a blow-up map. Connected fibres are not automatic: a stated Stein factorization separates the map into one with connected fibres followed by a finite map. Projection indices refer to the model used in the description; another presentation of the same family can give different projections.

ZL62 label

A compact encoding of the ambient variety and defining bundle. See ZL62 labels.

Hodge diamond

Total degree $p+q$ increases downwards. The independent entries are $\mathrm{h}^{1,1}$, $\mathrm{h}^{1,2}$, $\mathrm{h}^{1,3}$, and $\mathrm{h}^{2,2}$.

Picard rank $\rho$

For these fourfolds, $\rho=\mathrm{h}^{1,1}$.

Volume and Chern numbers

Write $\mathrm{c}_i=\mathrm{c}_i(\mathrm{T}_X)$. The volume is $\operatorname{vol}(X)=\int_X\mathrm{c}_1^4$. The entries $\mathrm{c}_1^4$, $\mathrm{c}_1^2\mathrm{c}_2$, $\mathrm{c}_2^2$, $\mathrm{c}_1\mathrm{c}_3$, and $\mathrm{c}_4$ mean the integrals of those products over $X$. The last equals the topological Euler characteristic.

Tangent Euler characteristic $\chi(X,\mathrm{T}_X)$

On these Fano fourfolds, $\chi(X,\mathrm{T}_X)=\mathrm{h}^0(X,\mathrm{T}_X)-\mathrm{h}^1(X,\mathrm{T}_X)$.

Polyvector parallelogram

Each entry is $\mathrm{h}^q(X,\bigwedge^p\mathrm{T}_X)$. Row $k$ has $p+q=k$, with $p$ increasing from left to right. The Hochschild–Kostant–Rosenberg decomposition identifies its row sum with $$\begin{equation} \notag \dim\mathrm{HH}^k(X)=\sum_{p+q=k}\mathrm{h}^q(X,\bigwedge^p\mathrm{T}_X). \end{equation}$$

Vanishing entries

Akizuki–Nakano vanishing gives zero when $q>p$; Kodaira vanishing also gives $\mathrm{H}^q(X,\omega_X^\vee)=0$ for $q>0$.

Dependence on the model

These dimensions can jump in a deformation family, so the displayed values belong to the selected presentation. For the general definitions, see Polyvector fields for Fano 3-folds, §2.

Polarization and Fano index

The primitive ample line bundle used here is $\mathcal{O}_X(1)$. The Fano index $r$ is the largest positive integer for which $\omega_X^\vee\cong\mathcal{O}_X(r)$. This does not require the Picard rank to be one.

Hilbert series

$$\begin{equation} \notag \operatorname{Hilb}_X(t)=\sum_{m\geq0}\mathrm{h}^0(X,\mathcal{O}_X(m))t^m =\frac{P_X(t)}{(1-t)^5}. \end{equation}$$ For $r>1$, these degrees differ from anticanonical degrees.

Showing more terms

The expansion initially shows degrees $0$ through $5$. Click the expansion to add four coefficients computed from the rational form; the rational expression itself is static.

Quantum period and regularization

The ordinary quantum period $\mathrm{G}_X$ encodes genus-zero, one-point descendant Gromov–Witten invariants. The hat denotes factorial regularization.

$$\begin{equation} \notag \begin{gathered} \mathrm{G}_X(t)=\sum_{d\geq0}a_dt^d, \qquad \widehat{\mathrm{G}}_X(t)=\sum_{d\geq0}c_dt^d, \qquad c_d=d!\,a_d, \\ a_0=1,\quad a_1=0, \qquad a_d=\sum_{\substack{\beta\in \mathrm{H}_2(X,\mathbb{Z})\\ \int_\beta\mathrm{c}_1=d}} \left\langle[\mathrm{pt}]\,\psi^{d-2}\right\rangle^X_{0,1,\beta} \quad(d\geq2). \end{gathered} \end{equation}$$

It is a deformation invariant of $X$. See Coates–Corti–Galkin–Kasprzyk, §B for the definition.

Degrees and coefficients

$d$ is the curve's anticanonical degree, with $\mathrm{c}_1=\mathrm{c}_1(\mathrm{T}_X)$.

Showing more terms

Click the period to add six coefficients. Where a quantum differential system is available, its recurrence computes further terms exactly; otherwise expansion stops at the recorded prefix.

Quantum differential equation

The scalar operator $L_X=\sum_m p_m(t)D^m$, with $D=t\frac{\mathrm{d}}{\mathrm{d}t}$, annihilates the regularized period: $L_X\widehat{\mathrm{G}}_X=0$. Its polynomial coefficients are printed to the left of powers of $D$, as in Coates–Galkin–Kasprzyk–Strangeway, Appendix B.

Period table and search

The period table lists the same twelve initial coefficients for each family with recorded data. The search accepts individual coefficient constraints; the overview displays them in degree order.

Previously recorded quantum periods

Each family page compares its regularized quantum period with three published catalogues:

• Kalashnikov (2019), Four-dimensional Fano quiver flag zero loci: 749 entries, numbered CKK.
• Coates–Galkin–Kasprzyk–Strangeway (2020), Quantum periods for certain four-dimensional Fano manifolds: 272 named table entries.
• Coates–Kasprzyk–Prince (2015), Four-dimensional Fano toric complete intersections: 738 entries, numbered CKP.

The comparison uses exact coefficients in anticanonical degree, beginning at degree zero, with no rescaling. Named paper entries are cross-checked against the authors' longer coefficient lists in the database of regularized quantum periods. A match requires at least 16 coefficients and agreement of every overlapping available coefficient; hovering over an entry shows how many were compared.

“No match” means that every entry in that catalogue disagrees with an available coefficient. “Insufficient coefficients” means a compatible prefix exists but fewer than 16 coefficients can be compared. Agreement of finite prefixes does not prove equality of infinite series or an identification of deformation families. Absence from these three catalogues does not establish absence from all literature.

Quantum spectrum and ambient sector

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\omega_X^\vee)$ after setting every Novikov variable to $1$. The calculation uses the flag-ambient sector associated with the named ambient variety; this sector can be smaller than the full quantum cohomology of $X$.

Quantum multiplication matrix

The exact matrix of the same operator, in the recorded ambient-sector basis.

Plot and spectral radius

The axes use equal scales; dot area and optional labels encode algebraic multiplicity. The dashed circle has radius $\max_\lambda|\lambda|$, the spectral radius.

Exact and numerical data

The matrix, characteristic polynomial, and minimal polynomial are exact; the eigenvalue coordinates are numerical approximations. “At 1” refers to the Novikov specialization.

Dependence on the presentation

The plot stays attached to a fixed reference presentation, whose ambient is named above it. In the current data, eigenvalue sets and minimal polynomials agree within families, while ambient-sector dimensions and multiplicities can differ.