# Fanography

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

## Description of Fano 3-folds as zero loci in a product of (weighted) Grassmannians

In Fano 3-folds from homogeneous vector bundles over Grassmannians De Biase–Fatighenti–Tanturri obtained a description of a general member of each deformation family of Fano 3-folds as the zero locus of a homogeneous vector bundle in a product of Grassmannians and weighted projective spaces. Such description gives new tools to compute cohomological invariants of Fano 3-folds (by virtue of the Borel–Weil–Bott theorem), and gives a representation-theoretic flavour to the classification.

Some of these descriptions are classical (and in fact agree with the usual description of the Fano 3-folds), others were obtained for the Fanosearch project in Quantum periods for 3-dimensional Fano manifolds by Coates–Corti–Galkin–Kasprzyk. In the table below we list the description of De Biase–Fatighenti–Tanturri, using their notation.

ID description ambient variety homogeneous vector bundle origin
1-1

double cover of $\mathbb{P}^3$ with branch locus a divisor of degree 6

1. $\mathbb{P}(1^4,3)$
2. $\mathbb{P}^3 \times \mathbb{P}^{20}$
1. $\mathcal{O}(6)$
2. $\mathcal{O}(0,2) \oplus K(0,1)$
$K \in \operatorname{Ext}^2_{\mathbb{P}^3}(\operatorname{Sym}^3 \mathcal{Q}, \mathcal{Q}(-2))$
1. classical
2. [DBFT]
1-2
1. hypersurface of degree 4 in $\mathbb{P}^4$
2. double cover of 1-16 with branch locus a divisor of degree 8
$\mathbb{P}^4$ $\mathcal{O}(4)$ classical
1-3

complete intersection of quadric and cubic in $\mathbb{P}^5$

$\mathbb{P}^5$ $\mathcal{O}(2) \oplus \mathcal{O}(3)$ classical
1-4

complete intersection of 3 quadrics in $\mathbb{P}^6$

$\mathbb{P}^6$ $\mathcal{O}(2)^{\oplus 3}$ classical
1-5 Gushel–Mukai 3-fold
1. section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 2 subspace and a quadric
2. double cover of 1-15 with branch locus an anticanonical divisor
$\operatorname{Gr}(2,5)$ $\mathcal{O}(2) \oplus \mathcal{O}(1)^{\oplus2}$ classical
1-6

section of half-spinor embedding of a connected component of $\mathrm{OGr}_+(5,10)$ by codimension 7 subspace

1. $\operatorname{OGr}^+(5,10)$
2. $\operatorname{Gr}(2,5)$
1. $\mathcal{O}(\frac{1}{2})^{\oplus7}$
2. $\mathcal{U}^{\vee}(1)\oplus \mathcal{O}(1)$
1. classical
2. [CCGK]
1-7

section of Plücker embedding of $\mathrm{Gr}(2,6)$ by codimension 5 subspace

$\operatorname{Gr}(2,6)$ $\mathcal{O}(1)^{\oplus5}$ classical
1-8

section of Plücker embedding of $\mathrm{SGr}(3,6)$ by codimension 3 subspace

$\operatorname{Gr}(3,6)$ $\bigwedge^2\mathcal{U}^{\vee} \oplus \mathcal{O}(1)^{\oplus3}$ classical
1-9

section of the adjoint $\mathrm{G}_2$-Grassmannian $\mathrm{G}_2\mathrm{Gr}(2,7)$ by codimension 2 subspace

$\operatorname{Gr}(2,7)$ $\mathcal{Q}^{\vee}(1) \oplus \mathcal{O}(1)^{\oplus2}$ classical
1-10

zero locus of $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$ on $\mathrm{Gr}(3,7)$

$\operatorname{Gr}(3,7)$ $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$ classical
1-11

hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$

$\mathbb{P}(1^3,2,3)$ $\mathcal{O}(6)$ classical
1-12

double cover of $\mathbb{P}^3$ with branch locus a smooth quartic surface

1. $\mathbb{P}(1^4,2)$
2. $\mathbb{P}^3 \times \mathbb{P}^{10}$
1. $\mathcal{O}(4)$
2. $\Lambda(0,1) \oplus \mathcal{O}(0,2)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^3}(\operatorname{Sym}^2 \mathcal{Q}, \mathcal{Q}(-1))$
1. classical
2. [DBFT]
1-13

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

$\mathbb{P}^4$ $\mathcal{O}(3)$ classical
1-14

complete intersection of 2 quadrics in $\mathbb{P}^5$

$\mathbb{P}^5$ $\mathcal{O}(2)^{\oplus 2}$ classical
1-15

section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 3 subspace

$\operatorname{Gr}(2,5)$ $\mathcal{O}(1)^{\oplus 3}$ classical
1-16

hypersurface of degree 2 in $\mathbb{P}^4$

$\mathbb{P}^4$ $\mathcal{O}(2)$ classical
1-17

projective space $\mathbb{P}^3$

$\mathbb{P}^3$ classical
2-1

blowup of 1-11 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system

$\mathbb{P}(1^3,2,3) \times \mathbb{P}^1$ $\mathcal{O}(6,0) \oplus \mathcal{O}(1,1)$ [CCGK]
2-2

double cover of 2-34 with branch locus a $(2,4)$-divisor

$\mathbb{P}^1\times \mathbb{P}^2 \times \mathbb{P}^{12}$ $\mathcal{O}(0,0,2) \oplus K(0,0,1)$
$K \in \operatorname{Ext}_{\mathbb{P}^1 \times \mathbb{P}^2}^2(\mathcal{O}(1,0)^{\oplus 6}, \mathcal{Q}_{\mathbb{P}^2}(-1,-1))$
[DBFT]
2-3

blowup of 1-12 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system

1. $\mathbb{P}(1^4,2) \times \mathbb{P}^1$
2. $\mathbb{P}^3 \times \mathbb{P}^{10} \times \mathbb{P}^1$
1. $\mathcal{O}(4,0) \oplus \mathcal{O}(1,1)$
2. $\Lambda(0,1,0) \oplus \mathcal{O}(0,2,0) \oplus \mathcal{O}(1,0,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^3}(\operatorname{Sym}^2 \mathcal{Q}, \mathcal{Q}(-1))$
1. [CCGK]
2. [DBFT]
2-4

blowup of 1-17 in the intersection of two cubics

$\mathbb{P}^1 \times \mathbb{P}^3$ $\mathcal{O}(1,3)$ [CCGK]
2-5

blowup of 1-13 in a plane cubic

$\mathbb{P}^1 \times \mathbb{P}^4$ $\mathcal{O}(0,3) \oplus \mathcal{O}(1,1)$ [DBFT]
2-6 Verra 3-fold
1. $(2,2)$-divisor on $\mathbb{P}^2\times\mathbb{P}^2$
2. double cover of 2-32 with branch locus an anticanonical divisor
$\mathbb{P}^2 \times \mathbb{P}^2$ $\mathcal{O}(2,2)$ [CCGK]
2-7

blowup of 1-16 in the intersection of two divisors from $|\mathcal{O}_Q(2)|$

$\mathbb{P}^1 \times \mathbb{P}^4$ $\mathcal{O}(0,2) \oplus \mathcal{O}(1,2)$ [CCGK]
2-8
1. double cover of 2-35 with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is smooth
2. double cover of 2-35 with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is singular but reduced
$\mathbb{P}^2 \times \mathbb{P}^3 \times \mathbb{P}^{12}$ $\Lambda(0,0,1) \oplus \mathcal{O}(0,0,2)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^2 \times \mathbb{P}^3}(\mathcal{Q}_{\mathbb{P}^3}^{\oplus 3}, \mathcal{Q}_{\mathbb{P}^2}(0, -1))$
[DBFT]
2-9

complete intersection of degree $(1,1)$ and $(2,1)$ in $\mathbb{P}^3\times\mathbb{P}^2$

$\mathbb{P}^2 \times \mathbb{P}^3$ $\mathcal{O}(1,1) \oplus \mathcal{O}(1,2)$ [CCGK]
2-10

blowup of 1-14 in an elliptic curve which is an intersection of 2 hyperplanes

$\operatorname{Gr}(2,4) \times \mathbb{P}^1$ $\mathcal{O}(2,0) \oplus \mathcal{O}(1,1)$ [DBFT]
2-11

blowup of 1-13 in a line

1. $\mathbb{P}^2 \times \mathbb{P}^4$
2. $\operatorname{Fl}(1,3,5)$
1. $\mathcal{Q}_{\mathbb{P}^2}(0,1) \oplus \mathcal{O}(1,2)$
2. $\mathcal{Q}_2^{\oplus 2} \oplus \mathcal{O}(2,1)$
1. [DBFT]
2. [DBFT]
2-12

intersection of 3 $(1,1)$-divisors in $\mathbb{P}^3\times\mathbb{P}^3$

$\mathbb{P}^3 \times \mathbb{P}^3$ $\mathcal{O}(1,1)^{\oplus 3}$ [CCGK]
2-13

blowup of 1-16 in a curve of degree 6 and genus 2

$\mathbb{P}^2 \times \mathbb{P}^4$ $\mathcal{O}(1,1)^{\oplus 2} \oplus \mathcal{O}(0,2)$ [CCGK]
2-14

blowup of 1-15 in an elliptic curve which is an intersection of 2 hyperplanes

$\operatorname{Gr}(2,5) \times \mathbb{P}^1$ $\mathcal{O}(1,0)^{\oplus 3} \oplus \mathcal{O}(1,1)$ [CCGK]
2-15
1. blowup of 1-17 in the intersection of a quadric and a cubic where the quadric is smooth
2. blowup of 1-17 in the intersection of a quadric and a cubic where the quadric is singular but reduced
1. $\mathbb{P}^3 \times \mathbb{P}^4$
2. $\operatorname{Fl}(1,2,5)$
1. $\mathcal{Q}_{\mathbb{P}^3}(0,1) \oplus \mathcal{O}(2,1)$
2. $\mathcal{Q}_2 \oplus \mathcal{O}(1,2)$
1. [DBFT]
2. [DBFT]
2-16

blowup of 1-14 in a conic

1. $\mathbb{P}^2 \times \operatorname{Gr}(2,4)$
2. $\operatorname{Fl}(1,2,4)$
1. $\mathcal{U}^{\vee}_{\operatorname{Gr}(2,4)}(1,0) \oplus \mathcal{O}(0,2)$
2. $\mathcal{O}(1,0) \oplus \mathcal{O}(0,2)$
1. [DBFT]
2. [DBFT]
2-17

blowup of 1-16 in an elliptic curve of degree 5

1. $\operatorname{Gr}(2,4) \times \mathbb{P}^3$
2. $\operatorname{Fl}(1,2,4)$
1. $\mathcal{U}_{\operatorname{Gr}(2,4)}^{\vee}(0,1) \oplus \mathcal{O}(1,1) \oplus \mathcal{O}(1,0)$
2. $\mathcal{O}(0,1) \oplus \mathcal{O}(1,1)$
1. [CCGK]
2. [DBFT]
2-18

double cover of 2-34 with branch locus a divisor of degree $(2,2)$

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^6$ $\Lambda(0,0,1) \oplus \mathcal{O}(0,0,2)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^1 \times \mathbb{P}^2}(\mathcal{Q}_{\mathbb{P}^2}^{\oplus 2}, \mathcal{O}(1,-1))$
[DBFT]
2-19

blowup of 1-14 in a line

1. $\mathbb{P}^3 \times \mathbb{P}^5$
2. $\operatorname{Fl}(1,3,6)$
1. $\mathcal{Q}_{\mathbb{P}^3}(0,1) \oplus \mathcal{O}(1,1)^{\oplus 2}$
2. $\mathcal{Q}_2^{\oplus 2} \oplus \mathcal{O}(1,1)^{\oplus 2}$
1. [DBFT]
2. [DBFT]
2-20

blowup of 1-15 in a twisted cubic

$\operatorname{Gr}(2,5) \times \mathbb{P}^2$ $\mathcal{U}^{\vee}_{\operatorname{Gr}(2,5)}(0,1) \oplus \mathcal{O}(1,0)^{\oplus 3}$ [CCGK]
2-21

blowup of 1-16 in a twisted quartic

$\operatorname{Gr}(2,4) \times \mathbb{P}^4$ $\mathcal{U}^{\vee}_{\operatorname{Gr}(2,4)}(0,1)^{\oplus 2} \oplus \mathcal{O}(1,0)$ [CCGK]
2-22

blowup of 1-15 in a conic

1. $\mathbb{P}^3 \times \operatorname{Gr}(2,5)$
2. $\operatorname{Fl}(1,2,5)$
1. $\mathcal{Q}_{\operatorname{Gr}(2,5)}(1,0) \oplus \mathcal{O}(0,1)^{\oplus 3}$
2. $\mathcal{O}(1,0) \oplus \mathcal{O}(0,1)^{\oplus 3}$
1. [DBFT]
2. [CCGK]
2-23
1. blowup of 1-16 in an intersection of $A\in|\mathcal{O}_Q(1)|$ and $B\in|\mathcal{O}_Q(2)|$ such that $A$ is smooth
2. blowup of 1-16 in an intersection of $A\in|\mathcal{O}_Q(1)|$ and $B\in|\mathcal{O}_Q(2)|$ such that $A$ is singular
1. $\mathbb{P}^4 \times \mathbb{P}^5$
2. $\operatorname{Fl}(1,2,6)$
1. $\mathcal{Q}_{\mathbb{P}^4}(0,1) \oplus \mathcal{O}(2,0) \oplus \mathcal{O}(1,1)$
2. $\mathcal{Q}_2 \oplus \mathcal{O}(0,2) \oplus \mathcal{O}(1,1)$
1. [DBFT]
2. [DBFT]
2-24

divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,2)$

$\mathbb{P}^2 \times \mathbb{P}^2$ $\mathcal{O}(1,2)$ classical
2-25

blowup of 1-17 in an elliptic curve which is an intersection of 2 quadrics

$\mathbb{P}^1 \times \mathbb{P}^3$ $\mathcal{O}(1,2)$ [CCGK]
2-26

blowup of 1-15 in a line

1. $\operatorname{Gr}(2,4) \times \operatorname{Gr}(2,5)$
2. $\operatorname{Fl}(2,3,5)$
1. $\mathcal{Q}_{\operatorname{Gr}(2,4)} \boxtimes \mathcal{U}_{\operatorname{Gr}(2,5)}^{\vee} \oplus \mathcal{O}(1,0) \oplus \mathcal{O}(0,1)^{\oplus 2}$
2. $\mathcal{U}_1^{\vee} \oplus \mathcal{O}(1,0) \oplus \mathcal{O}(0,1)^{\oplus 2}$
1. [DBFT]
2. [DBFT]
2-27

blowup of 1-17 in a twisted cubic

$\mathbb{P}^3 \times \mathbb{P}^2$ $\mathcal{O}(1,1)^{\oplus 2}$ [CCGK]
2-28

blowup of 1-17 in a plane cubic

$\mathbb{P}^3 \times \mathbb{P}^{10}$ $\Lambda(0,1) \oplus \mathcal{O}(1,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^3}(\operatorname{Sym}^2\mathcal{Q}, \mathcal{Q}(-1))$
[DBFT]
2-29

blowup of 1-16 in a conic

$\mathbb{P}^1 \times \mathbb{P}^4$ $\mathcal{O}(0,2) \oplus \mathcal{O}(1,1)$ [DBFT]
2-30

blowup of 1-17 in a conic

1. $\mathbb{P}^3 \times \mathbb{P}^4$
2. $\operatorname{Fl}(1,2,5)$
1. $\mathcal{Q}_{\mathbb{P}^3}(0,1) \oplus \mathcal{O}(1,1)$
2. $\mathcal{Q}_2 \oplus \mathcal{O}(1,1)$
1. [DBFT]
2. [DBFT]
2-31

blowup of 1-16 in a line

$\mathbb{P}^2 \times \operatorname{Gr}(2,4)$ $\mathcal{U}^{\vee}_{\operatorname{Gr}(2,4)}(1,0) \oplus \mathcal{O}(0,1)$ [DBFT]
2-32

divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,1)$

1. $\mathbb{P}^2 \times \mathbb{P}^2$
2. $\operatorname{Fl}(1,2,3)$
1. $\mathcal{O}(1,1)$
1. classical
2. classical
2-33

blowup of 1-17 in a line

$\mathbb{P}^1 \times \mathbb{P}^3$ $\mathcal{O}(1,1)$ [DBFT]
2-34

$\mathbb{P}^1\times\mathbb{P}^2$

$\mathbb{P}^1 \times \mathbb{P}^2$ classical
2-35

$\mathrm{Bl}_p\mathbb{P}^3$

1. $\mathbb{P}^2 \times \mathbb{P}^3$
2. $\operatorname{Fl}(1,2,4)$
1. $\mathcal{Q}_{\mathbb{P}^2}(0,1)$
2. $\mathcal{Q}_2$
1. [DBFT]
2. [DBFT]
2-36

$\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$

$\mathbb{P}^2 \times \mathbb{P}^6$ $\Lambda(0,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^2}(\operatorname{Sym}^2\mathcal{Q}, \mathcal{Q}(-1))$
[DBFT]
3-1

double cover of 3-27 with branch locus a divisor of degree $(2,2,2)$

$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^8$ $K(0,0,0,1) \oplus \mathcal{O}(0,0,0,2)$
$K \in \operatorname{Ext}^2_{(\mathbb{P}^1)^3}(\mathcal{O}(0,0,1)^{\oplus 4},\mathcal{O}(1,-1,-1))$
[DBFT]
3-2

divisor from $|\mathcal{L}^{\otimes 2}\otimes\mathcal{O}(2,3)|$ on the $\mathbb{P}^2$-bundle $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1,-1)^{\oplus 2})$ over $\mathbb{P}^1\times\mathbb{P}^1$ such that $X\cap Y$ is irreducible, and $\mathcal{L}$ is the tautological bundle, and $Y\in|\mathcal{L}|$

$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^5$ $\Lambda(0,0,1) \oplus \mathcal{O}(0,1,2)$
$\Lambda \in \operatorname{Ext}^1_{(\mathbb{P}^1)^2}(\mathcal{O}(0,1)^{\oplus 2}, \mathcal{O}(1, -1))$
[DBFT]
3-3

divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,2)$

$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^2$ $\mathcal{O}(1,1,2)$ [CCGK]
3-4

blowup of 2-18 in a smooth fiber of the composition of the projection to $\mathbb{P}^1\times\mathbb{P}^2$ with the projection to $\mathbb{P}^2$ of the double cover with the projection

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^6 \times \mathbb{P}^1$ $\Lambda(0,0,1,0) \oplus \mathcal{O}(0,0,2,0) \oplus \mathcal{O}(0,1,0,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^1 \times \mathbb{P}^2}(\mathcal{Q}_{\mathbb{P}^2}^{\oplus 2}, \mathcal{O}(1,-1))$
[DBFT]
3-5

blowup of 2-34 in a curve $C$ of degree $(5,2)$ such that $C\hookrightarrow\mathbb{P}^1\times\mathbb{P}^2\to\mathbb{P}^2$ is an embedding

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^7$ $\Lambda(0,0,1) \oplus \mathcal{O}(0,1,1)^{\oplus 2}$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^1 \times \mathbb{P}^2}(\mathcal{Q}_{\mathbb{P}^2}^{\oplus 2}, \mathcal{O}(1,-1))$
[DBFT]
3-6

blowup of 1-17 in the disjoint union of a line and an elliptic curve of degree 4

$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^3$ $\mathcal{O}(1,0,2) \oplus \mathcal{O}(0,1,1)$ [DBFT]
3-7

blowup of 2-32 in an elliptic curve which is the intersection of two divisors from $|-\frac{1}{2}\mathrm{K}_W|$

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^2$ $\mathcal{O}(0,1,1) \oplus \mathcal{O}(1,1,1)$ [CCGK]
3-8

divisor from the linear system $|(\alpha\circ\pi_1)^*(\mathcal{O}_{\mathbb{P}^2}(1)\otimes\pi_2^*(\mathcal{O}_{\mathbb{P}^2}(2))|$ where $\pi_i\colon\mathrm{Bl}_1\times\mathbb{P}^2$ are the projections, and $\alpha\colon\mathrm{Bl}_1\mathbb{P}^2\to\mathbb{P}^2$ is the blowup

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^2$ $\mathcal{O}(0,1,2) \oplus \mathcal{O}(1,1,0)$ [DBFT]
3-9

blowup of the cone over the Veronese of $\mathbb{P}^2$ in $\mathbb{P}^5$ with center the disjoint union of the vertex and a quartic curve on $\mathbb{P}^2$

$\mathbb{P}^2 \times \mathbb{P}^6 \times \mathbb{P}^{20}$ $\Lambda(0,1,0) \oplus \mathcal{Q}_{\mathbb{P}^6}(0,0,1) \oplus K(0,0,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^2}(\operatorname{Sym}^2\mathcal{Q}, \mathcal{Q}(-1))$, $K \in \operatorname{Ext}^3_{\mathbb{P}^2}(\operatorname{Sym}^4\mathcal{Q}, \mathcal{Q}(-3))$
[DBFT]
3-10

blowup of 1-16 in the disjoint union of 2 conics

$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^4$ $\mathcal{O}(1,0,1) \oplus \mathcal{O}(0,1,1) \oplus \mathcal{O}(0,0,2)$ [DBFT]
3-11

blowup of 2-35 in an elliptic curve which is the intersection of two divisors from $|-\frac{1}{2}\mathrm{K}_{V_7}|$

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^3$ $\mathcal{O}(1,1,1) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,1)$ [DBFT]
3-12

blowup of 1-17 in the disjoint union of a line and a twisted cubic

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^3$ $\mathcal{O}(0,1,1) \oplus \mathcal{O}(0,1,1) \oplus \mathcal{O}(1,0,1)$ [DBFT]
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 $i=1,2$

$(\mathbb{P}^2)^3$ $\mathcal{O}(1,1,0) \oplus \mathcal{O}(1,0,1) \oplus \mathcal{O}(0,1,1)$ [CCGK]
3-14

blowup of 1-17 in the disjoint union of a plane cubic curve and a point outside the plane

$\mathbb{P}^3 \times \mathbb{P}^{10} \times \mathbb{P}^2$ $\Lambda(0,1,0) \oplus \mathcal{Q}_{\mathbb{P}^2}(1,0,0) \oplus \mathcal{O}(1,1,0)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^3}(\operatorname{Sym}^2\mathcal{Q}, \mathcal{Q}(-1))$
[DBFT]
3-15

blowup of 1-16 in the disjoint union of a line and a conic

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^4$ $\mathcal{O}(1,0,1) \oplus \mathcal{O}(0,1,1) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,1)$ [DBFT]
3-16

blowup of 2-35 in the proper transform of a twisted cubic containing the center of the blowup

$\mathbb{P}^2_1 \times \mathbb{P}^2_2 \times \mathbb{P}^3$ $\mathcal{O}(0,1,1) \oplus \mathcal{O}(1,1,0) \oplus \mathcal{Q}_{\mathbb{P}^2_1}(0,0,1)$ [DBFT]
3-17

divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$

$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^2$ $\mathcal{O}(1,1,1)$ [CCGK]
3-18

blowup of 1-17 in the disjoint union of a line and a conic

1. $\mathbb{P}^1\times \mathbb{P}^3 \times \mathbb{P}^4$
2. $\mathbb{P}^1 \times \operatorname{Fl}(1,2,5)$
1. $\mathcal{O}(1,1,0) \oplus \mathcal{O}(0,1,1) \oplus \mathcal{Q}_{\mathbb{P}^3}(0,0,1)$
2. $\mathcal{Q}_2(0;0,0) \oplus \mathcal{O}(0;1,1) \oplus \mathcal{O}(1;0,1)$
1. [DBFT]
2. [DBFT]
3-19

blowup of 1-16 in two non-collinear points

$\mathbb{P}^2 \times \mathbb{P}^4$ $\mathcal{O}(0,2) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,1)$ [DBFT]
3-20

blowup of 1-16 in the disjoint union of two lines

$\mathbb{P}^2_1 \times \mathbb{P}^2_2 \times \mathbb{P}^4$ $\mathcal{O}(1,1,0) \oplus \mathcal{Q}_{\mathbb{P}^2_1}(0,0,1) \oplus \mathcal{Q}_{\mathbb{P}^2_2}(0,0,1)$ [DBFT]
3-21

blowup of 2-34 in a curve of degree $(2,1)$

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^6$ $\mathcal{O}(0,1,1) \oplus \Lambda(0,0,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^1 \times \mathbb{P}^2}(\mathcal{Q}_{\mathbb{P}^2}^{\oplus 2}, \mathcal{O}(1,-1))$
[DBFT]
3-22

blowup of 2-34 in a conic on $\{x\}\times\mathbb{P}^2$, $x\in\mathbb{P}^1$

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^6$ $\mathcal{O}(1,0,1) \oplus \Lambda(0,0,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^2}(\operatorname{Sym}^2 \mathcal{Q}, \mathcal{Q}(-1))$
[DBFT]
3-23

blowup of 2-35 in the proper transform of a conic containing the center of the blowup

$\mathbb{P}^2 \times \mathbb{P}^3 \times \mathbb{P}^4$ $\mathcal{Q}_{\mathbb{P}^2}(0,1,0) \oplus \mathcal{O}(1,0,1) \oplus \mathcal{Q}_{\mathbb{P}^3}(0,0,1)$ [DBFT]
3-24

the fiber product of 2-32 with $\mathrm{Bl}_p\mathbb{P}^2$ over $\mathbb{P}^2$

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^2$ $\mathcal{O}(1,1,0) \oplus \mathcal{O}(0,1,1)$ [CCGK]
3-25

blowup of 1-17 in the disjoint union of two lines

1. $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^3$
2. $\operatorname{Fl}(1,2,4)$
1. $\mathcal{O}(1,0,1) \oplus \mathcal{O}(0,1,1)$
2. $\mathcal{O}(0,1)^{\oplus 2}$
1. [DBFT]
2. [DBFT]
3-26

blowup of 1-17 in the disjoint union of a point and a line

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^3$ $\mathcal{O}(1,0,1) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,1)$ [DBFT]
3-27

$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$

$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ classical
3-28

$\mathbb{P}^1\times\mathrm{Bl}_p\mathbb{P}^2$

$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^2$ $\mathcal{O}(1,0,1)$ [DBFT]
3-29

blowup of 2-35 in a line on the exceptional divisor

$\mathbb{P}^3 \times \mathbb{P}^2 \times \mathbb{P}^9$ $\mathcal{Q}_{\mathbb{P}^2}(1,0,0) \oplus \mathcal{Q}_{\mathbb{P}^3}(0,0,1) \oplus\Lambda(0,0,1) \oplus \mathcal{O}(0,-1,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^2}(\operatorname{Sym}^2\mathcal{Q}, \mathcal{Q}(-1))$
[DBFT]
3-30

blowup of 2-35 in the proper transform of a line containing the center of the blowup

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^3$ $\mathcal{O}(1,1,0) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,1)$ [DBFT]
3-31

blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex

1. $\mathbb{P}^3 \times \mathbb{P}^4$
2. $\operatorname{Fl}(1,2,5)$
1. $\mathcal{Q}_{\mathbb{P}^3}(0,1) \oplus \mathcal{O}(2,0)$
2. $\mathcal{Q}_2 \oplus \mathcal{O}(0,2)$
1. [DBFT]
2. [DBFT]
4-1

divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of degree $(1,1,1,1)$

$(\mathbb{P}^1)^4$ $\mathcal{O}(1,1,1,1)$ classical
4-2

blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the disjoint union of the vertex and an elliptic curve on the quadric

$\mathbb{P}^3 \times \mathbb{P}^4 \times \mathbb{P}^5$ $\mathcal{Q}_{\mathbb{P}^3}(0,1,0) \oplus \mathcal{Q}_{\mathbb{P}^4}(0,0,1) \oplus \mathcal{O}(2,0,0) \oplus \mathcal{O}(0,1,1)$ [DBFT]
4-3

blowup of 3-27 in a curve of degree $(1,1,2)$

$(\mathbb{P}^1)^3 \times \mathbb{P}^2$ $\mathcal{O}(1,1,0,1) \oplus \mathcal{O}(0,0,1,1)$ [DBFT]
4-4

blowup of 3-19 in the proper transform of a conic through the points

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^4$ $\mathcal{O}(1,1,0) \oplus \mathcal{O}(0,0,2) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,1)$ [DBFT]
4-5

blowup of 2-34 in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^6 \times \mathbb{P}^1$ $\mathcal{O}(0,1,1,0) \oplus \Lambda(0,0,1,0) \oplus \mathcal{O}(0,1,0,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^1 \times \mathbb{P}^2}(\mathcal{Q}_{\mathbb{P}^2}^{\oplus 2},\mathcal{O}(1,-1))$
[DBFT]
4-6

blowup of 1-17 in the disjoint union of 3 lines

$(\mathbb{P}^1)^3 \times \mathbb{P}^3$ $\mathcal{O}(1,0,0,1) \oplus \mathcal{O}(0,1,0,1) \oplus \mathcal{O}(0,0,1,1)$ [DBFT]
4-7

blowup of 2-32 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$

1. $(\mathbb{P}^1)^2 \times (\mathbb{P}^2)^2$
2. $(\mathbb{P}^1)^2 \times \operatorname{Fl}(1,2,3)$
1. $\mathcal{O}(0,0,1,1) \oplus \mathcal{O}(1,0,1,0) \oplus \mathcal{O}(0,1,0,1)$
2. $\mathcal{O}(1,0; 1,0) \oplus \mathcal{O}(0,1;0,1)$
1. [DBFT]
2. [DBFT]
4-8

blowup of 3-27 in a curve of degree $(0,1,1)$

1. $\mathbb{P}^1 \times \mathbb{P}^3 \times \mathbb{P}^4$
2. $\mathbb{P}^1 \times \operatorname{Fl}(1,2,5)$
1. $\mathcal{O}(1,0,1) \oplus \mathcal{O}(0,2,0) \oplus \mathcal{Q}_{\mathbb{P}^3}(0,0,1)$
2. $\mathcal{O}(1;1,0) \oplus \mathcal{O}(0;0,2) \oplus \mathcal{Q}_2$
1. [DBFT]
2. [DBFT]
4-9

blowup of 3-25 in an exceptional curve of the blowup

$(\mathbb{P}^1)^2 \times \mathbb{P}^2 \times \mathbb{P}^3$ $\mathcal{O}(1,0,1,0) \oplus \mathcal{O}(0,1,0,1) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,0,1)$ [DBFT]
4-10

$\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$

1. $(\mathbb{P}^1)^3 \times \mathbb{P}^2$
2. $\mathbb{P}^1\times \mathbb{P}^2 \times \mathbb{P}^3$
1. $\mathcal{O}(1,0,0,1) \oplus \mathcal{O}(0,1,0,1)$
2. $\mathcal{O}(0,1,1) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,1)$
1. [DBFT]
2. [DBFT]
4-11

blowup of 3-28 in $\{x\}\times E$, $x\in\mathbb{P}^1$ and $E$ the $(-1)$-curve

$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^1 \times \mathbb{P}^6$ $\mathcal{O}(0,1,1,0) \oplus \mathcal{Q}_{\mathbb{P}^2}(0,0,0,1) \oplus\Lambda(0,0,0,1) \oplus \mathcal{O}(0,0,-1,1)$
$\Lambda \in \operatorname{Ext}^1_{(\mathbb{P}^1)^2}(\mathcal{O}(0,1)^{\oplus 2}, \mathcal{O}(1, -1))$
[DBFT]
4-12

blowup of 2-33 in the disjoint union of two exceptional lines of the blowup

$\mathbb{P}^1 \times \mathbb{P}^3 \times \mathbb{P}^{8}$ $\mathcal{O}(1,1,0) \oplus \Lambda(0,0,1) \oplus \mathcal{O}(-1,1,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^1 \times \mathbb{P}^3}(\mathcal{Q}_{\mathbb{P}^3}^{\oplus 2},\mathcal{O}(1,-1))$
[DBFT]
4-13

blowup of 3-27 in a curve of degree $(1,1,3)$

$\mathbb{P}^1_1 \times \mathbb{P}^1_2 \times \mathbb{P}^1_3 \times \mathbb{P}^4$ $\Lambda(0,0,0,1) \oplus \mathcal{O}(1,0,1,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^1_1 \times \mathbb{P}^1_2}(\mathcal{O}(0,1)^{\oplus 2}, \mathcal{O}(1, -1))$
[DBFT]
5-1

blowup of 2-29 in the disjoint union of three exceptional lines of the blowup

$\mathbb{P}^1 \times \mathbb{P}^3 \times \mathbb{P}^{8} \times \mathbb{P}^{11}$ $\mathcal{O}(1,1,0,0) \oplus \Lambda(0,0,1,0) \oplus \mathcal{O}(-1,1,1,0) \oplus \mathcal{Q}_{\mathbb{P}^3}(0,0,0,1) \oplus \mathcal{Q}_{\mathbb{P}^8}(0,0,0,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^1 \times \mathbb{P}^3}(\mathcal{Q}_{\mathbb{P}^3}^{\oplus 2},\mathcal{O}(1,-1))$
[DBFT]
5-2

blowup of 3-25 in the disjoint union of two exceptional lines on the same irreducible component

$\mathbb{P}^1 \times \mathbb{P}^3 \times \mathbb{P}^{8} \times \mathbb{P}^1$ $\mathcal{O}(1,1,0,0) \oplus \Lambda(0,0,1,0) \oplus \mathcal{O}(-1,1,1,0) \oplus \mathcal{O}(0,1,0,1)$
$\Lambda \in \operatorname{Ext}^1_{\mathbb{P}^1 \times \mathbb{P}^3}(\mathcal{Q}_{\mathbb{P}^3}^{\oplus 2},\mathcal{O}(1,-1))$
[DBFT]
5-3

$\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$

1. $\mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^1$
2. $(\mathbb{P}^1)^4$
1. $\mathcal{O}(1,1,0)^{\oplus 2}$
2. $\mathcal{O}(1,1,1,0)$
1. classical
2. classical
6-1

$\mathbb{P}^1\times\mathrm{Bl}_4\mathbb{P}^2$

$\operatorname{Gr}(2,5) \times \mathbb{P}^1$ $\mathcal{O}(1,0)^{\oplus 4}$ classical
7-1

$\mathbb{P}^1\times\mathrm{Bl}_5\mathbb{P}^2$

$\mathbb{P}^4 \times \mathbb{P}^1$ $\mathcal{O}(2,0)^{\oplus 2}$ classical
8-1

$\mathbb{P}^1\times\mathrm{Bl}_6\mathbb{P}^2$

$\mathbb{P}^3 \times \mathbb{P}^1$ $\mathcal{O}(3,0)$ classical
9-1

$\mathbb{P}^1\times\mathrm{Bl}_7\mathbb{P}^2$

1. $\mathbb{P}(1^3,2) \times \mathbb{P}^1$
2. $\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^1$
1. $\mathcal{O}(4,0)$
2. $\mathcal{O}(2,2,0)$
1. classical
2. classical
10-1

$\mathbb{P}^1\times\mathrm{Bl}_8\mathbb{P}^2$

$\mathbb{P}(1^2,2,3) \times \mathbb{P}^1$ $\mathcal{O}(6,0)$ classical