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 3folds
Name 
Fano index 
Period sequence 
Construction 

4 
12 
projective 3space 

3 
4 
the quadric 3fold 

2 
14 
a linear section of 

2 
2 
a (2,2) complete intersection in 

2 
0 
the cubic 3fold 

2 
1 
a quartic in 

2 
n/a 
a sextic in 

1 
n/a 
a sextic in 

1 
15 
a quartic in 

1 
19 
a (2,3) complete intersection in 

1 
5 
a (2,2,2) complete intersection in 

1 
9 
section of by intersection of quadric and linear space 

1 
7 
linear section of spinor variety 

1 
18 
linear section of 

1 
20 
linear section of 

1 
10 
linear section of 

1 
17 
section of triple universal bundle on 
Rank 2 Fano 3folds
Number 

Period sequence 
Description 
1 
4 
n/a 
the blowup of with centre an elliptic curve which is the intersection of two members of 
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 
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 
7 
14 
51 
the blow up of a quadric with centre the intersection of two members of 
8 
14 
26 
a double cover of with branch locus a member of such that is smooth, where is the exceptional divisor of the blowup 
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 blowup of a 3dimensional quadric in a curve of genus 2 and degree 6 
14 
20 
A 39 
the blowup of with center an elliptic curve which is an intersection of two hyperplane sections 
15 
22 
35 
the blowup of with center the intersection of a quadric and a cubic 
16 
22 
59 
the blowup of with center a conic on it 
17 
24 
A 38 
the blowup of a 3dimensional 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 blowup of with center a line on it 
20 
26 
A 46 
the blowup of with center a twisted cubic on it 
21 
28 
A 8 
the blowup of with center a twisted quartic on it (a smooth rational curve of degree 4 which spans ) 
22 
30 
A 50 
the blowup of with center a conic on it 
23 
30 
29 
the blowup of a quadric with center an intersection of and 
24 
30 
66 
A divisor of bidegree (1,2) on 
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 blowup of with centre a plane cubic 
29 
40 
42 
the blowup of a quadric 3fold with centre a conic on it 
30 
46 
70 
the blowup of with center a conic 
31 
46 
48 
the blowup of a quadric 3fold with center a line on it 
32 
48 
6 
a divisor on of bidegree 
33 
54 
54 
the blowup of with center a line 
34 
54 
44 

35 
56 
30 
, which is the blowup of at a point 
36 
62 
58 
the scroll over 
Rank 3 Fano 3folds
Number 

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 
4 
18 
151 
the blowup 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 blowup of with center a curve of bidegree (5,2) that projects isomorphically to a conic in 
6 
22 
146 
the blowup of with center a disjoint union of a line and an elliptic curve of degree 4 
7 
24 
36 
the blowup of with center an elliptic curve which is a complete intersection of two members of 
8 
24 
85 
a member of the linear system on , where are the projections and is the blowup 
9 
26 
68 
the blowup of the cone over the Veronese surface with center a disjoint union of the vertex and a quartic in 
10 
26 
67 
the blowup of with center a disjoint union of two conics on it 
11 
28 
107 
the blowup of with center a complete intersection of two general members of 
12 
28 
144 
the blowup of with centre a disjoint union of a twisted cubic and a line 
13 
30 
16 
the blowup 
14 
32 
148 
the blowup 
15 
32 
112 
the blowup of a quadric with center the disjoint union of a line on and a conic on 
16 
34 
119 
the blowup 
17 
36 
37 
a smooth divisor on of tridegree 
18 
36 
160 
the blowup 
19 
38 
57 
the blowup 
20 
38 
63 
the blowup 
21 
38 
98 
the blowup of along a curve of bidegree (2,1) 
22 
40 
152 
the blowup of along a curve of bidegree (0,2), that is a conic in 
23 
42 
158 
the blowup 
24 
42 
86 
the fiber product where is a (1,1) hypersurface in 
25 
44 
41 
the blowup 
26 
46 
113 
the blowup of with center a disjoint union of a point and a line 
27 
48 
21 

28 
48 
90 

29 
50 
163 
the blowup 
30 
50 
84 
the blowup 
31 
52 
53 
the blowup 
Rank 4 Fano 3folds
Number 

Period sequence 
Description 
1 
24 
3 
a smooth divisor on of polydegree 
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 
Rank 5 Fano 3folds
Number 

Period sequence 
Description 
1 
28 
114 

2 
36 
87 
“smooth toric, but not 53” 
3 
36 
43 

Fano 3folds with
Picard number 

Period sequence 
Description 
6 
30 
64 

7 
24 
71 

8 
18 
45 

9 
12 
n/a 

10 
6 
n/a 

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).
Matsuki says it has just symmetry, so expected N is 5, while for this period sequence expected N is 4.
And This comment explains that remaining 4 period sequences (23, 24, 25 and 27) are parasitic.