Fano Varieties
Legend for period sequences:
n/a – known, and not one of 165 from Minkowski list, [7 entries]
# – known to be number # from Minkowski list, [91 entries]
A# – known to be number # from Minkowski list, computed with A/NA corr., [5 entries]
?*# – unknown, but STD ansatz predicts it is number #, [1 entry]
? – unknown and no guess. [0 entries]
Rank 1 Fano 3-folds
Name |
Fano index |
Period sequence |
Construction |
![\PP^3 \PP^3](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/805/80578734ac32d4a21437d3b40ee514e4-ffffff-000000-0.png) |
4 |
12 |
projective 3-space |
![Q^3 Q^3](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/6ae/6ae9bd742a0132f9165500729bc9d1a4-ffffff-000000-0.png) |
3 |
4 |
the quadric 3-fold |
![B_5 B_5](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/1cb/1cbab575f605089f277cae2dc59a6d6c-ffffff-000000-0.png) |
2 |
14 |
a linear section of ![\mathrm{Gr}(2,5) \mathrm{Gr}(2,5)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/fa8/fa8e3c9e5c227b9c2f400d64403619fc-ffffff-000000-0.png) |
![B_4 B_4](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/936/936647334bd1602528177fc2bb33eeee-ffffff-000000-0.png) |
2 |
2 |
a (2,2) complete intersection in ![\PP^5 \PP^5](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/3be/3be740d46be2b20f9c40f59f7dcd99aa-ffffff-000000-0.png) |
![B_3 B_3](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/7fc/7fc13d7af04367f93852375411286af2-ffffff-000000-0.png) |
2 |
0 |
the cubic 3-fold |
![B_2 B_2](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/6f5/6f5ef944a2d6b5db7b0f5eb7664fbe8d-ffffff-000000-0.png) |
2 |
1 |
a quartic in ![\PP(1,1,1,1,2) \PP(1,1,1,1,2)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/1ec/1ec38706bb0b3c23232d99d23d9f3d8f-ffffff-000000-0.png) |
![B_1 B_1](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/262/262e0afc75c8a9fc536a7dce57e6ebe1-ffffff-000000-0.png) |
2 |
n/a |
a sextic in ![\PP(1,1,1,2,3) \PP(1,1,1,2,3)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/a5a/a5a872433bd2ec8a665f61293503d6de-ffffff-000000-0.png) |
![V_2 V_2](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/81e/81ed5ef3779e6b081b22740d7399b22f-ffffff-000000-0.png) |
1 |
n/a |
a sextic in ![\PP(1,1,1,1,3) \PP(1,1,1,1,3)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/62b/62b7d9896761acdf2f763e0b2b19ba39-ffffff-000000-0.png) |
![V_4 V_4](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/92d/92d4a97d2a7b5a862a18804f41085d99-ffffff-000000-0.png) |
1 |
15 |
a quartic in ![\PP^4 \PP^4](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/e68/e68461e5992f8ae959bc527dfa5f8294-ffffff-000000-0.png) |
![V_6 V_6](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/9f0/9f054e777730e7d9d7007d94d5c97111-ffffff-000000-0.png) |
1 |
19 |
a (2,3) complete intersection in ![\PP^5 \PP^5](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/3be/3be740d46be2b20f9c40f59f7dcd99aa-ffffff-000000-0.png) |
![V_8 V_8](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/0c9/0c98a8e49466e6248cade59218464e0f-ffffff-000000-0.png) |
1 |
5 |
a (2,2,2) complete intersection in ![\PP^6 \PP^6](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/0b7/0b71fce7b4e799bbc1522f282fab24a2-ffffff-000000-0.png) |
![V_{10} V_{10}](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/144/1441e8d4d5b22a15a5fa8fe72244aa26-ffffff-000000-0.png) |
1 |
9 |
section of by intersection of quadric and linear space |
![V_{12} V_{12}](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/cd2/cd2fac040c703d003fbbd854c156651e-ffffff-000000-0.png) |
1 |
7 |
linear section of spinor variety ![OGr(5,10) OGr(5,10)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/3b8/3b82f21a030dbeb5cb03d8f52cdfbc72-ffffff-000000-0.png) |
![V_{14} V_{14}](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/9f3/9f397bde4d2cdf00fef987ffe670ebc5-ffffff-000000-0.png) |
1 |
18 |
linear section of ![Gr(2,6) Gr(2,6)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/69f/69f3fcd91d4d1e9710c4a5dcdbf418c3-ffffff-000000-0.png) |
![V_{16} V_{16}](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/e6b/e6b44397de76c59e1c071ad0f4e6cc62-ffffff-000000-0.png) |
1 |
20 |
linear section of ![LG(3,6) LG(3,6)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/714/7143278b451b8bd14e4be8ac3953873d-ffffff-000000-0.png) |
![V_{18} V_{18}](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/f47/f473bb004f2be9382f7bdf934e09c5bc-ffffff-000000-0.png) |
1 |
10 |
linear section of ![G_2/P G_2/P](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/7ef/7ef00e02f5dd70bc197374a7796a0cb2-ffffff-000000-0.png) |
![V_{22} V_{22}](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/fd6/fd6f23903d6a3b125f4ce1e96f8df4c7-ffffff-000000-0.png) |
1 |
17 |
section of triple universal bundle on ![Gr(3,7) Gr(3,7)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/eb3/eb36f06d04e88ae6f1c6b279c9b26197-ffffff-000000-0.png) |
Rank 2 Fano 3-folds
Number |
![(-K_X)^3 (-K_X)^3](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/ee8/ee8d8114bdb7490e5a081a201c990952-ffffff-000000-0.png) |
Period sequence |
Description |
1 |
4 |
n/a |
the blow-up of with centre an elliptic curve which is the intersection of two members of ![|{-{1/2}} K_{V_1}| |{-{1/2}} K_{V_1}|](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/48a/48a998c60274690cb962f965ad4dccf5-ffffff-000000-0.png) |
2 |
6 |
n/a |
a double cover of branched along a divisor of bidegree (2,4) |
3 |
8 |
n/a |
the blow up of with centre an elliptic curve which is the intersection of two elements of ![|-1/2K_{V_2}| |-1/2K_{V_2}|](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/a11/a11ff940b282bc3ba6e0f70a907504f6-ffffff-000000-0.png) |
4 |
10 |
49 |
the blow up of with centre an intersection of two cubics |
5 |
12 |
34 |
the blow up of with centre a plane cubic on it |
6 |
12 |
11 |
a divisor of bidegree (2,2) in ![\PP^2 \times \PP^2 \PP^2 \times \PP^2](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/698/698f7ce91c5f1ceea3c7e9252fb21411-ffffff-000000-0.png) |
7 |
14 |
51 |
the blow up of a quadric with centre the intersection of two members of ![|\cO_Q(2)| |\cO_Q(2)|](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/d87/d87e2f599f23869b960a0700a3756a62-ffffff-000000-0.png) |
8 |
14 |
26 |
a double cover of with branch locus a member of such that is smooth, where is the exceptional divisor of the blow-up ![V_7 \to \PP^3 V_7 \to \PP^3](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/761/76135f4757fe1ba17ad2cc8aa90fa8ef-ffffff-000000-0.png) |
9 |
16 |
62 |
the blow up of in a curve of degree 7 and genus 5 |
10 |
16 |
40 |
the blow up of with centre an elliptic curve which is the intersection of two hyperplane sections |
11 |
18 |
56 |
the blow up of with centre a line on it |
12 |
20 |
13 |
the blow up of in a curve of degree 6 and genus 3 |
13 |
20 |
52 |
the blow-up of a 3-dimensional quadric in a curve of genus 2 and degree 6 |
14 |
20 |
A 39 |
the blow-up of with center an elliptic curve which is an intersection of two hyperplane sections |
15 |
22 |
35 |
the blow-up of with center the intersection of a quadric and a cubic |
16 |
22 |
59 |
the blow-up of with center a conic on it |
17 |
24 |
A 38 |
the blow-up of a 3-dimensional quadric with center an elliptic curve of degree 5 on it |
18 |
24 |
60 |
a double cover of with branch locus a divisor of bidegree (2,2) |
19 |
26 |
55 |
the blow-up of with center a line on it |
20 |
26 |
A 46 |
the blow-up of with center a twisted cubic on it |
21 |
28 |
A 8 |
the blow-up of with center a twisted quartic on it (a smooth rational curve of degree 4 which spans ) |
22 |
30 |
A 50 |
the blow-up of with center a conic on it |
23 |
30 |
29 |
the blow-up of a quadric with center an intersection of and ![B \in |\cO(2)| B \in |\cO(2)|](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/029/02935e4931b096ae72ec6dfbf21c82c2-ffffff-000000-0.png) |
24 |
30 |
66 |
A divisor of bidegree (1,2) on ![\PP^2 \times \PP^2 \PP^2 \times \PP^2](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/698/698f7ce91c5f1ceea3c7e9252fb21411-ffffff-000000-0.png) |
25 |
32 |
28 |
The blow up of with centre an elliptic curve which is the complete intersection of two quadrics |
26 |
34 |
?*47 |
the blow up of with center a line on it |
27 |
38 |
61 |
the blow up of with center a twisted cubic |
28 |
40 |
33 |
the blow-up of with centre a plane cubic |
29 |
40 |
42 |
the blow-up of a quadric 3-fold with centre a conic on it |
30 |
46 |
70 |
the blow-up of with center a conic |
31 |
46 |
48 |
the blow-up of a quadric 3-fold with center a line on it |
32 |
48 |
6 |
a divisor on of bidegree ![(1,1) (1,1)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/fb0/fb0ce7c2864d45cd277575f863f6af1c-ffffff-000000-0.png) |
33 |
54 |
54 |
the blow-up of with center a line |
34 |
54 |
44 |
![X = \PP^1 \times \PP^2 X = \PP^1 \times \PP^2](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/814/814713fb6789c9335accca71715a4d5d-ffffff-000000-0.png) |
35 |
56 |
30 |
, which is the blow-up of at a point |
36 |
62 |
58 |
the scroll over ![\PP^2 \PP^2](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/fc4/fc40778b711617ef146a3ec76339a0d5-ffffff-000000-0.png) |
Rank 3 Fano 3-folds
Number |
![(-K_X)^3 (-K_X)^3](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/ee8/ee8d8114bdb7490e5a081a201c990952-ffffff-000000-0.png) |
Period sequence |
Description |
1 |
12 |
22 |
a double cover of with branch locus a divisor of tridegree (2,2,2) |
2 |
14 |
97 |
a member of on the -bundle over , where is the tautological line bundle |
3 |
18 |
31 |
a divisor on of tridegree ![(1,1,2) (1,1,2)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/fb7/fb77dab149322a6594a5573501d4531e-ffffff-000000-0.png) |
4 |
18 |
151 |
the blow-up of (rank 2 table, number 18) with center a smooth fiber of , where the first map is the double cover and the second map is the projection |
5 |
20 |
109 |
the blow-up of with center a curve of bidegree (5,2) that projects isomorphically to a conic in ![\PP^2 \PP^2](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/fc4/fc40778b711617ef146a3ec76339a0d5-ffffff-000000-0.png) |
6 |
22 |
146 |
the blow-up of with center a disjoint union of a line and an elliptic curve of degree 4 |
7 |
24 |
36 |
the blow-up of with center an elliptic curve which is a complete intersection of two members of ![|-{1 \over 2} K_W| |-{1 \over 2} K_W|](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/193/193a9424706038b625364f1c5e0ba647-ffffff-000000-0.png) |
8 |
24 |
85 |
a member of the linear system on , where are the projections and is the blow-up |
9 |
26 |
68 |
the blow-up of the cone over the Veronese surface with center a disjoint union of the vertex and a quartic in ![R_4 \cong \PP^2 R_4 \cong \PP^2](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/b1c/b1cf8f0e29a0f5775a9f74af8076bb36-ffffff-000000-0.png) |
10 |
26 |
67 |
the blow-up of with center a disjoint union of two conics on it |
11 |
28 |
107 |
the blow-up of with center a complete intersection of two general members of ![{-1/2} K_{V_7} {-1/2} K_{V_7}](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/dba/dbad7edc589f67a9cf2ef0f60405eaed-ffffff-000000-0.png) |
12 |
28 |
144 |
the blow-up of with centre a disjoint union of a twisted cubic and a line |
13 |
30 |
16 |
the blow-up |
14 |
32 |
148 |
the blow-up |
15 |
32 |
112 |
the blow-up of a quadric with center the disjoint union of a line on and a conic on ![Q Q](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/f09/f09564c9ca56850d4cd6b3319e541aee-ffffff-000000-0.png) |
16 |
34 |
119 |
the blow-up |
17 |
36 |
37 |
a smooth divisor on of tridegree ![(1,1,1) (1,1,1)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/eef/eef1707b922c6a822e838999cdc5a6c2-ffffff-000000-0.png) |
18 |
36 |
160 |
the blow-up |
19 |
38 |
57 |
the blow-up |
20 |
38 |
63 |
the blow-up |
21 |
38 |
98 |
the blow-up of along a curve of bidegree (2,1) |
22 |
40 |
152 |
the blow-up of along a curve of bidegree (0,2), that is a conic in ![\{t\}\times \PP^2 \{t\}\times \PP^2](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/fc9/fc9568542843f1dfbf3614b24a50544d-ffffff-000000-0.png) |
23 |
42 |
158 |
the blow-up |
24 |
42 |
86 |
the fiber product where is a (1,1) hypersurface in ![\PP^2 \times \PP^2 \PP^2 \times \PP^2](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/698/698f7ce91c5f1ceea3c7e9252fb21411-ffffff-000000-0.png) |
25 |
44 |
41 |
the blow-up |
26 |
46 |
113 |
the blow-up of with center a disjoint union of a point and a line |
27 |
48 |
21 |
![X = \PP^1 \times \PP^1 \times \PP^1 X = \PP^1 \times \PP^1 \times \PP^1](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/d45/d45d31b92d98131fe3f5b679fd7ba778-ffffff-000000-0.png) |
28 |
48 |
90 |
![X = \PP^1 \times \mathbb{F}_1 X = \PP^1 \times \mathbb{F}_1](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/e96/e96d5d0ea8334f2786dd04caa2eb562f-ffffff-000000-0.png) |
29 |
50 |
163 |
the blow-up |
30 |
50 |
84 |
the blow-up |
31 |
52 |
53 |
the blow-up |
Rank 4 Fano 3-folds
Number |
![(-K_X)^3 (-K_X)^3](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/ee8/ee8d8114bdb7490e5a081a201c990952-ffffff-000000-0.png) |
Period sequence |
Description |
1 |
24 |
3 |
a smooth divisor on of polydegree ![(1,1,1,1) (1,1,1,1)](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/2dc/2dcf1dc514b76711a995aac23685e39a-ffffff-000000-0.png) |
2 |
28 |
32 |
|
3 |
30 |
122 |
|
4 |
32 |
103 |
|
5 |
32 |
147 |
|
6 |
34 |
65 |
|
7 |
36 |
69 |
|
8 |
38 |
105 |
|
9 |
40 |
102 |
|
10 |
42 |
142 |
|
11 |
44 |
93 |
|
12 |
46 |
150 |
|
13 |
26 |
88 |
blowup of curve of tridegree on ![\PP^1 \times \PP^1 \times \PP^1 \PP^1 \times \PP^1 \times \PP^1](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/b78/b78ed4e7c12f09f99a9576792e29affd-ffffff-000000-0.png) |
Rank 5 Fano 3-folds
Number |
![(-K_X)^3 (-K_X)^3](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/ee8/ee8d8114bdb7490e5a081a201c990952-ffffff-000000-0.png) |
Period sequence |
Description |
1 |
28 |
114 |
|
2 |
36 |
87 |
“smooth toric, but not 5-3” |
3 |
36 |
43 |
![\PP^1 \times S_6 \PP^1 \times S_6](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/25d/25d5ad51234778679aee86c007a56d2b-ffffff-000000-0.png) |
Fano 3-folds with ![B_2 > 5 B_2 > 5](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/513/513b29198e40716b392a34323ce5092a-ffffff-000000-0.png)
Picard number |
![(-K_X)^3 (-K_X)^3](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/ee8/ee8d8114bdb7490e5a081a201c990952-ffffff-000000-0.png) |
Period sequence |
Description |
6 |
30 |
64 |
![\PP^1 \times S_5 \PP^1 \times S_5](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/da8/da8d04bc8a6139b2cee017cccc2ee6b0-ffffff-000000-0.png) |
7 |
24 |
71 |
![\PP^1 \times S_4 \PP^1 \times S_4](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/3b1/3b1131f47c06fa155e180383b772800a-ffffff-000000-0.png) |
8 |
18 |
45 |
![\PP^1 \times S_3 \PP^1 \times S_3](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/d8e/d8e95dd2fadd1d23ed8f896d147f393e-ffffff-000000-0.png) |
9 |
12 |
n/a |
![\PP^1 \times S_2 \PP^1 \times S_2](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/0bc/0bcceb82f71a5a7b2b8d869fe1fe8679-ffffff-000000-0.png) |
10 |
6 |
n/a |
![\PP^1 \times S_1 \PP^1 \times S_1](http://coates.ma.ic.ac.uk/fanosearch/wp-content/latex/753/753ee698235fe12b1c2ff8ac93657785-ffffff-000000-0.png) |
There is a discrepancy between tables 1 and 2:
in table 2 everywhere
should be replaced with
.
There are 49 period sequences with N=4.
45 of them correspond to Fano threefolds and (and all these threefolds have expected N equal to 4, except for one case – 4.7 and period sequence 69).
Fano variety 4.7 (period sequence 69) contradicts to expected N (from Matsuki).
symmetry, so expected N is 5, while for this period sequence expected N is 4.
Matsuki says it has just
And This comment explains that remaining 4 period sequences (23, 24, 25 and 27) are parasitic.