Ambient varieties and bundles
- 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.