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 4 12 projective 3-space
Q^3 3 4 the quadric 3-fold
B_5 2 14 a linear section of \mathrm{Gr}(2,5)
B_4 2 2 a (2,2) complete intersection in \PP^5
B_3 2 0 the cubic 3-fold
B_2 2 1 a quartic in \PP(1,1,1,1,2)
B_1 2 n/a a sextic in \PP(1,1,1,2,3)
V_2 1 n/a a sextic in \PP(1,1,1,1,3)
V_4 1 15 a quartic in \PP^4
V_6 1 19 a (2,3) complete intersection in \PP^5
V_8 1 5 a (2,2,2) complete intersection in \PP^6
V_{10} 1 9 section of Gr(2,5) by intersection of quadric and linear space
V_{12} 1 7 linear section of spinor variety OGr(5,10)
V_{14} 1 18 linear section of Gr(2,6)
V_{16} 1 20 linear section of LG(3,6)
V_{18} 1 10 linear section of G_2/P
V_{22} 1 17 section of triple universal bundle on Gr(3,7)

Rank 2 Fano 3-folds

Number (-K_X)^3 Period sequence Description
1 4 n/a the blow-up of V_1 with centre an elliptic curve which is the intersection of two members of |{-{1/2}} K_{V_1}|
2 6 n/a a double cover of \PP^1 \times \PP^2 branched along a divisor of bidegree (2,4)
3 8 n/a the blow up of V_2 with centre an elliptic curve which is the intersection of two elements of |-1/2K_{V_2}|
4 10 49 the blow up of \PP^3 with centre an intersection of two cubics
5 12 34 the blow up of V_3 with centre a plane cubic on it
6 12 11 a divisor of bidegree (2,2) in \PP^2 \times \PP^2
7 14 51 the blow up of a quadric Q with centre the intersection of two members of |\cO_Q(2)|
8 14 26 a double cover of V_7 with branch locus a member B of |-K_{V_7}| such that B\cap D is smooth, where D is the exceptional divisor of the blow-up V_7 \to \PP^3
9 16 62 the blow up of \PP^3 in a curve \Gamma of degree 7 and genus 5
10 16 40 the blow up of V_4 with centre an elliptic curve which is the intersection of two hyperplane sections
11 18 56 the blow up of V_3 with centre a line on it
12 20 13 the blow up of \PP^3 in a curve \Gamma of degree 6 and genus 3
13 20 52 the blow-up of a 3-dimensional quadric Q in a curve \Gamma of genus 2 and degree 6
14 20 A 39 the blow-up of V_5 with center an elliptic curve which is an intersection of two hyperplane sections
15 22 35 the blow-up of \PP^3 with center the intersection of a quadric and a cubic
16 22 59 the blow-up of V_4 \subset \PP^5 with center a conic on it
17 24 A 38 the blow-up of a 3-dimensional quadric Q with center an elliptic curve of degree 5 on it
18 24 60 a double cover of \PP^1 \times \PP^2 with branch locus a divisor of bidegree (2,2)
19 26 55 the blow-up of V_4 with center a line on it
20 26 A 46 the blow-up of V_5 \subset \PP^6 with center a twisted cubic on it
21 28 A 8 the blow-up of Q \subset \PP^4 with center a twisted quartic on it (a smooth rational curve of degree 4 which spans \PP^4)
22 30 A 50 the blow-up of V_5 with center a conic on it
23 30 29 the blow-up of a quadric with center an intersection of A \in |\cO(1)| and B \in |\cO(2)|
24 30 66 A divisor of bidegree (1,2) on \PP^2 \times \PP^2
25 32 28 The blow up of \PP^3 with centre an elliptic curve which is the complete intersection of two quadrics
26 34 ?*47 the blow up of V_5 with center a line on it
27 38 61 the blow up of \PP^3 with center a twisted cubic
28 40 33 the blow-up of \PP^3 with centre a plane cubic
29 40 42 the blow-up of a quadric 3-fold Q with centre a conic on it
30 46 70 the blow-up of \PP^3 with center a conic
31 46 48 the blow-up of a quadric 3-fold Q with center a line on it
32 48 6 a divisor W on \PP^2 \times \PP^2 of bidegree (1,1)
33 54 54 the blow-up of \PP^3 with center a line
34 54 44 X = \PP^1 \times \PP^2
35 56 30 V_7, which is the blow-up of \PP^3 at a point
36 62 58 the scroll \PP(\cO \oplus \cO(2)) over \PP^2

Rank 3 Fano 3-folds

Number (-K_X)^3 Period sequence Description
1 12 22 a double cover of \PP^1 \times \PP^1 \times \PP^1 with branch locus a divisor of tridegree (2,2,2)
2 14 97 a member of |L^{\otimes 2} \otimes \cO(2,3)| on the \PP^2-bundle \PP(\cO\oplus\cO(-1,-1)^{\oplus 2}) over \PP^1 \times \PP^1, where L is the tautological line bundle
3 18 31 a divisor on \PP^1 \times \PP^1 \times \PP^2 of tridegree (1,1,2)
4 18 151 the blow-up of Y (rank 2 table, number 18) with center a smooth fiber of Y \to \PP^1 \times \PP^2 \to \PP^2, where the first map is the double cover and the second map is the projection
5 20 109 the blow-up of \PP^1 \times \PP^2 with center a curve of bidegree (5,2) that projects isomorphically to a conic in \PP^2
6 22 146 the blow-up of \PP^3 with center a disjoint union of a line and an elliptic curve of degree 4
7 24 36 the blow-up X of W with center an elliptic curve which is a complete intersection of two members of |-{1 \over 2} K_W|
8 24 85 a member of the linear system |p_1^\star g^\star \cO(1) \otimes p_2^\star \cO(2)| on \mathbb{F}_1 \times \PP^2, where p_1, p_2 are the projections and g:\mathbb{F}_1 \to \PP^2 is the blow-up
9 26 68 the blow-up of the cone W_4 \subset \PP^6 over the Veronese surface R_4 \subset \PP^5 with center a disjoint union of the vertex and a quartic in R_4 \cong \PP^2
10 26 67 the blow-up of Q \subset \PP^4 with center a disjoint union of two conics on it
11 28 107 the blow-up X of V_7 with center a complete intersection of two general members of {-1/2} K_{V_7}
12 28 144 the blow-up of \PP^3 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 X of a quadric Q \subset \PP^4 with center the disjoint union of a line on Q and a conic on Q
16 34 119 the blow-up
17 36 37 a smooth divisor on \PP^1 \times \PP^1 \times \PP^2 of tridegree (1,1,1)
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 \PP^1 \times \PP^2 along a curve of bidegree (2,1)
22 40 152 the blow-up of \PP^1 \times \PP^2 along a curve of bidegree (0,2), that is a conic in \{t\}\times \PP^2
23 42 158 the blow-up
24 42 86 the fiber product X = W \times_{\PP^2} \mathbb{F}_1 where W is a (1,1) hypersurface in \PP^2 \times \PP^2
25 44 41 the blow-up
26 46 113 the blow-up X of \PP^3 with center a disjoint union of a point and a line
27 48 21 X = \PP^1 \times \PP^1 \times \PP^1
28 48 90 X = \PP^1 \times \mathbb{F}_1
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 Period sequence Description
1 24 3 a smooth divisor on \PP^1 \times \PP^1 \times \PP^1 \times \PP^1 of polydegree (1,1,1,1)
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 (1,1,3) on \PP^1 \times \PP^1 \times \PP^1

Rank 5 Fano 3-folds

Number (-K_X)^3 Period sequence Description
1 28 114
2 36 87 “smooth toric, but not 5-3”
3 36 43 \PP^1 \times S_6

Fano 3-folds with B_2 > 5

Picard number (-K_X)^3 Period sequence Description
6 30 64 \PP^1 \times S_5
7 24 71 \PP^1 \times S_4
8 18 45 \PP^1 \times S_3
9 12 n/a \PP^1 \times S_2
10 6 n/a \PP^1 \times S_1
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2 Comments

  1. Sergey says:
    September 13, 2010 at 10:50 am

    There is a discrepancy between tables 1 and 2:

    in table 2 everywhere V_n should be replaced with B_n.

  2. Sergey says:
    September 17, 2010 at 4:46 pm

    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 A_1 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.

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