Fanography 3-Folds¶
This section records the 105 smooth Fano threefold families from Fanography. Entries with supported flag-variety inputs include computed ambient matrices; entries marked Catalog only have reference descriptions without a computed matrix.
Realization types. Entries labelled direct flag variety or complete intersection can be entered in gwflags. Homogeneous-bundle entries require the corresponding bundle constructor. Catalog-only entries link to reference information; no executable input or computed matrix is provided here.
Fano Variety 1-1¶
1-1
Description: Double cover / weighted degree-6 hypersurface (Fanography). Realization type: Catalog only — not represented by the current gwflags flag-variety/ordinary complete-intersection input model.
Fano Variety 1-2¶
1-2
- Ambient Space:
$A4$ - Keep Nodes:
1 - Bundle K:
4 - Ambient Index:
1 - Dimension:
3 - Basis Rank:
4 - Description: Hypersurface of degree 4 in \(\mathbb{P}^4\); double cover of 1-16 with branch locus a divisor of degree 8.
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 1-3¶
1-3
- Ambient Space:
$A5$ - Keep Nodes:
1 - Bundle K:
2;3 - Ambient Index:
1 - Dimension:
3 - Basis Rank:
4 - Description: Complete intersection of a quadric and a cubic in \(\mathbb{P}^5\).
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 1-4¶
1-4
- Ambient Space:
$A6$ - Keep Nodes:
1 - Bundle K:
2;2;2 - Ambient Index:
1 - Dimension:
3 - Basis Rank:
4 - Description: Complete intersection of three quadrics in \(\mathbb{P}^6\).
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 1-5¶
1-5
- Ambient Space:
$A4$ - Keep Nodes:
2 - Bundle K:
1;1;2 - Ambient Index:
1 - Dimension:
3 - Basis Rank:
4 - Description: Gushel–Mukai threefold: section of Gr(2,5) by two hyperplanes and a quadric.
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 1-6¶
1-6
- Ambient Space:
$D5$ - Keep Nodes:
5 - Bundle K:
1;1;1;1;1;1;1 - Ambient Index:
1 - Dimension:
3 - Basis Rank:
4 - Description: Section of the half-spinor embedding of a connected component of \(\mathrm{OGr}_+(5,10)\) by codimension 7 subspace.
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 1-7¶
1-7
- Ambient Space:
$A5$ - Keep Nodes:
2 - Bundle K:
1;1;1;1;1 - Ambient Index:
1 - Dimension:
3 - Basis Rank:
4 - Description: Section of the Plücker embedding of \(\mathrm{Gr}(2,6)\) by codimension 5 subspace.
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 1-8¶
1-8
- Ambient Space:
$C3$ - Keep Nodes:
3 - Bundle K:
1;1;1 - Ambient Index:
1 - Dimension:
3 - Basis Rank:
4 - Description: Section of the Plücker embedding of \(\mathrm{SGr}(3,6)\) by codimension 3 subspace.
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 1-9¶
1-9
- Ambient Space:
$G2$ - Keep Nodes:
2 - Bundle K:
1;1 - Ambient Index:
1 - Dimension:
3 - Basis Rank:
4 - Description: Section of the adjoint \(\mathrm{G}_2\)-Grassmannian \(\mathrm{G}_2\mathrm{Gr}(2,7)\) by codimension 2 subspace.
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 1-10¶
1-10
- Ambient Space:
$A6$ - Keep Nodes:
3 - Bundle K:
osum(wedge(2, dual(taut_sub(X, 3))), wedge(2, dual(taut_sub(X, 3))), wedge(2, dual(taut_sub(X, 3)))) - Ambient Index:
1 - Dimension:
3 - Basis Rank:
4 - Description: Zero locus of \((\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}\) on \(\mathrm{Gr}(3,7)\).
- Realization type: Homogeneous-bundle zero locus; independently recomputed with
gwflags. - Matrix status: Published after an exact software run and an independent regularized-period check against the smooth-Fano database entry for \(V(3,22)\).
Quantum Matrix (\(y=1\))
Fano Variety 1-11¶
1-11
Description: Double Veronese cone / weighted degree-6 hypersurface (Fanography). Realization type: Catalog only — not represented by the current gwflags flag-variety/ordinary complete-intersection input model.
Fano Variety 1-12¶
1-12
Description: Quartic double solid / weighted degree-4 hypersurface (Fanography). Realization type: Catalog only — not represented by the current gwflags flag-variety/ordinary complete-intersection input model.
Fano Variety 1-13¶
1-13
- Ambient Space:
$A4$ - Keep Nodes:
1 - Bundle K:
3 - Ambient Index:
2 - Dimension:
3 - Basis Rank:
4 - Description: Hypersurface of degree 3 in \(\mathbb{P}^4\) (smooth cubic threefold).
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 1-14¶
1-14
- Ambient Space:
$A5$ - Keep Nodes:
1 - Bundle K:
2;2 - Ambient Index:
2 - Dimension:
3 - Basis Rank:
4 - Description: Complete intersection of two quadrics in P⁵.
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 1-15¶
1-15
- Ambient Space:
$A4$ - Keep Nodes:
2 - Bundle K:
1;1;1 - Ambient Index:
2 - Dimension:
3 - Basis Rank:
4 - Description: Quintic del Pezzo threefold; section of Plücker embedding of \(\mathrm{Gr}(2,5)\) by codimension 3 subspace.
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 1-16¶
1-16
- Ambient Space:
$A4$ - Keep Nodes:
1 - Bundle K:
2 - Ambient Index:
3 - Dimension:
3 - Basis Rank:
4 - Description: Hypersurface of degree 2 in \(\mathbb{P}^4\) (smooth quadric threefold).
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 1-17¶
1-17
- Ambient Space:
$A3$ - Keep Nodes:
1 - Bundle K: (none — projective space itself)
- Ambient Index:
4 - Dimension:
3 - Basis Rank:
4 - Description: Projective space \(\mathbb{P}^3\).
- Realization type: Direct flag variety.
Quantum Matrix (\(y=1\))
Fano Variety 2-1¶
2-1
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-1.
Fano Variety 2-2¶
2-2
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-2.
Fano Variety 2-3¶
2-3
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-3.
Fano Variety 2-4¶
2-4
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-4.
Fano Variety 2-5¶
2-5
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-5.
Fano Variety 2-6¶
2-6
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-6.
Fano Variety 2-7¶
2-7
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-7.
Fano Variety 2-8¶
2-8
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-8.
Fano Variety 2-9¶
2-9
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-9.
Fano Variety 2-10¶
2-10
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-10.
Fano Variety 2-11¶
2-11
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-11.
Fano Variety 2-12¶
2-12
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-12.
Fano Variety 2-13¶
2-13
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-13.
Fano Variety 2-14¶
2-14
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-14.
Fano Variety 2-15¶
2-15
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-15.
Fano Variety 2-16¶
2-16
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-16.
Fano Variety 2-17¶
2-17
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-17.
Fano Variety 2-18¶
2-18
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-18.
Fano Variety 2-19¶
2-19
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-19.
Fano Variety 2-20¶
2-20
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-20.
Fano Variety 2-21¶
2-21
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-21.
Fano Variety 2-22¶
2-22
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-22.
Fano Variety 2-23¶
2-23
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-23.
Fano Variety 2-24¶
2-24
- Ambient Space:
A2xA2 - Keep Nodes:
1,3 - Bundle K:
1,2 - Ambient Index:
1 - Dimension:
3 - Basis Rank:
6 - Description: Divisor of type (1,2) in P² × P².
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 2-25¶
2-25
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-25.
Fano Variety 2-26¶
2-26
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-26.
Fano Variety 2-27¶
2-27
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-27.
Fano Variety 2-28¶
2-28
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-28.
Fano Variety 2-29¶
2-29
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-29.
Fano Variety 2-30¶
2-30
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-30.
Fano Variety 2-31¶
2-31
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-31.
Fano Variety 2-32¶
2-32
- Ambient Space:
A2 - Keep Nodes:
1,2 - Bundle K:
— - Ambient Index:
2 - Dimension:
3 - Basis Rank:
6 - Description: Full flag variety \(Fl(1,2;\mathbb{C}^3)\); divisor of bidegree \((1,1)\) in \(\mathbb{P}^2 \times \mathbb{P}^2\).
- Realization type: Direct flag variety (also a divisor in the product embedding).
Quantum Matrix (symbolic in y1, y2)
Fano Variety 2-33¶
2-33
Description: Blow-up of P³ along a line (Fanography; outside the current flag/ordinary-CI input model). Realization type: Catalog only — not represented by the current gwflags flag-variety/ordinary complete-intersection input model.
Fano Variety 2-34¶
2-34
- Ambient Space:
A1xA2 - Keep Nodes:
1,2 - Bundle K:
— - Ambient Index:
1 - Dimension:
3 - Basis Rank:
6 - Description: P¹ × P².
- Realization type: Direct product of flag varieties.
Quantum Matrix (\(y=1\))
Fano Variety 2-35¶
2-35
Description: Blow-up of P³ at a point (Fanography; outside the current flag/ordinary-CI input model). Realization type: Catalog only — not represented by the current gwflags flag-variety/ordinary complete-intersection input model.
Fano Variety 2-36¶
2-36
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 2-36.
Fano Variety 3-1¶
3-1
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-1.
Fano Variety 3-2¶
3-2
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-2.
Fano Variety 3-3¶
3-3
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-3.
Fano Variety 3-4¶
3-4
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-4.
Fano Variety 3-5¶
3-5
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-5.
Fano Variety 3-6¶
3-6
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-6.
Fano Variety 3-7¶
3-7
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-7.
Fano Variety 3-8¶
3-8
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-8.
Fano Variety 3-9¶
3-9
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-9.
Fano Variety 3-10¶
3-10
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-10.
Fano Variety 3-11¶
3-11
- Ambient Space:
A1xA1xA2 - Keep Nodes:
1,2,3 - Bundle K:
1,1,1 - Ambient Index:
1 - Dimension:
3 - Basis Rank:
8 - Description: Divisor of tridegree \((1,1,1)\) in \(\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^2\); a blow-up of the quadric \(Q^3\) along a conic.
- Realization type: Line-bundle complete intersection.
Quantum Matrix (\(y=1\))
Fano Variety 3-12¶
3-12
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-12.
Fano Variety 3-13¶
3-13
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-13.
Fano Variety 3-14¶
3-14
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-14.
Fano Variety 3-15¶
3-15
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-15.
Fano Variety 3-16¶
3-16
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-16.
Fano Variety 3-17¶
3-17
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-17.
Fano Variety 3-18¶
3-18
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-18.
Fano Variety 3-19¶
3-19
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-19.
Fano Variety 3-20¶
3-20
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-20.
Fano Variety 3-21¶
3-21
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-21.
Fano Variety 3-22¶
3-22
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-22.
Fano Variety 3-23¶
3-23
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-23.
Fano Variety 3-24¶
3-24
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-24.
Fano Variety 3-25¶
3-25
Description: Blow-up of P³ along two lines (Fanography; outside the current flag/ordinary-CI input model). Realization type: Catalog only — not represented by the current gwflags flag-variety/ordinary complete-intersection input model.
Fano Variety 3-26¶
3-26
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-26.
Fano Variety 3-27¶
3-27
- Ambient Space:
A1xA1xA1 - Keep Nodes:
1,2,3 - Bundle K: (none — product flag variety)
- Ambient Index:
2 - Dimension:
3 - Basis Rank:
8 - Description: Product \((\mathbb{P}^1)^3\).
- Realization type: Direct product of flag varieties.
Quantum Matrix (\(y=1\))
Fano Variety 3-28¶
3-28
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-28.
Fano Variety 3-29¶
3-29
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-29.
Fano Variety 3-30¶
3-30
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-30.
Fano Variety 3-31¶
3-31
Realization type: Catalog only — no computed matrix is provided here.
Reference: Fanography 3-31.
Fano Variety 4-1¶
4-1
- Ambient Space:
A1xA1xA1xA1 - Keep Nodes:
1,2,3,4 - Bundle K:
1,1,1,1 - Ambient Index:
1 - Dimension:
3 - Basis Rank:
10 - Description: Smooth (1,1,1,1) hypersurface in (P¹)⁴.
- Realization type: Line-bundle complete intersection on a product of flag varieties.
Quantum Matrix (\(y=1\))
Fano Variety 4-2¶
4-2
Description: 3-31
Fano Variety 4-3¶
4-3
Description: 3-17 , 3-27 , 3-28
Fano Variety 4-4¶
4-4
Description: 3-18 , 3-19 , 3-30
Fano Variety 4-5¶
4-5
Description: 3-21 , 3-28 , 3-31
Fano Variety 4-6¶
4-6
Description: 3-25 , 3-27
Fano Variety 4-7¶
4-7
Description: 3-24 , 3-28
Fano Variety 4-8¶
4-8
Description: 3-27 , 3-31
Fano Variety 4-9¶
4-9
Description: 3-25 , 3-26 , 3-28 , 3-30
Fano Variety 4-10¶
4-10
Description: 3-27 , 3-28
Fano Variety 4-11¶
4-11
Description: 3-28 , 3-31
Fano Variety 4-12¶
4-12
Description: 3-30
Fano Variety 4-13¶
4-13
Description: 3-27 , 3-31
Fano Variety 5-1¶
5-1
Description: 4-4 , 4-12
Fano Variety 5-2¶
5-2
Description: 4-9 , 4-11 , 4-12
Fano Variety 5-3¶
5-3
Description: 4-10
Fano Variety 6-1¶
6-1
Description: 5-3
Fano Variety 7-1¶
7-1
Description: 6-1
Fano Variety 8-1¶
8-1
Description: 7-1
Fano Variety 9-1¶
9-1
Description: 8-1
Fano Variety 10-1¶
10-1
Description: yes