Appendix

We collect in this chapter numerical data about the constructed or studied surfaces and give references to the points in the thesis where the corresponding additional information can be found.

1. Degree 10

10.A.

Invariants d = 10, π = 9, HK = 6, K^{2} = −3, χ = 2, q = 0, N_{6} = 3

Classification S = S_{min}(p_{1}, p_{2}, p_{3}) non-minimal K3 surface H = H_{min} − 4E_{1} − E_{2} − E_{3}

Syzygies of I_{S}

O(−4)

⊕
0←I_{S}←9O(−5) 15O(−6) 7O(−7) O(−8)
⊕ ↖ ⊕ ⟵ ⊕ ⟵ ⊕ ←0
O(−6)  3O(−7) 3O(−8) O(−9)

Cohomology

↑i

1

1  h^{i}(I_{S}(p))
1 3 1

1 ⟶ p

Construction

E = O(−1) ⊕ Ω^{3}(3), F = ker(5O ⊕ 2O(1)⟶^{ψ}O(2))

where ψ is a special morphism. see (2.2) for details and (2.4) for a liaison construction, Family properties irreducible, unirational, of dimension 45 which differs from χ(N_{S|IP^{4}}) = 44

2. Degree 11

11.A.

Invariants d = 11, π = 10, HK = 7, K^{2} = −6, χ = 1, q = 0, N_{6} = 10, [DES]

Classification S = S_{min}(p_{1}, ..., p_{6}) non-minimal Enriques surface H = H_{min} − 2E_{1} − ∑_{2}^{6} E_{i}

Syzygies of I_{S}

15O(−5) 3O(−6)
0←I_{S}← ⊕ ⟵ ⊕
10O(−6) 26O(−7) ↖ 20O(−8) ⟵ 5O(−9) ← 0

Cohomology

↑ i

2  h^{i}(I_{S}(p))
1 5 5

⟶ p

Construction

E = 2Ω^{3}(3), F = ker(10O⟶^{ψ}O(2))

where ψ is a special morphism. see [DES] and (3.7) for details see also (3.8) for a liaison construction Family properties irreducible, unirational, of dimension 34 which differs from χ(N_{S|IP^{4}}) = 36 11.B1.

Invariants d = 11, π = 11, HK = 9, K^{2} = −11, χ = 1, q = 0, N_{6} = 7, [DES]

Classification S = IP^{2}(p_{0}, ..., p_{19}) rational surface H = 10l − 4E_{0} − 3 ∑_{1}^{3} E_{i} − 2 ∑_{4}^{13} E_{j} − ∑_{14}^{19} E_{k}

Syzygies of I_{S}

10O(−5) 13O(−6) 4O(−7)
0←I_{S}← ⊕ ⟵ ⊕ ⟵ ⊕
O(−6) 3O(−7) 3O(−8) ↖ O(−9) ← 0

Cohomology

↑ i

3 1  h^{i}(I_{S}(p))
2 1

⟶ p

Construction

E = 3Ω^{3}(3), F = Ω^{2}(2) ⊕ ker(9O⟶^{ψ}2O(1))

where ψ is a general morphism. see (3.23) for details Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 41

11.B2.

Invariants d = 11, π = 11, HK = 9, K^{2} = −11, χ = 1, q = 0, N_{6} = 7

Classification S = IP^{2}(p_{0}, ..., p_{19}) rational surface H = 11l − 5E_{0} − 3 ∑_{1}^{6} E_{i} − 2 ∑_{7}^{12} E_{j} − ∑_{13}^{19} E_{k}

Syzygies of I_{S}

0←I_{S}←10O(−5) 12O(−6) 3O(−7)
↖ ⊕ ⟵ ⊕
2O(−7) 3O(−8) ↖ O(−9) ← 0

Cohomology

↑ i

3 1  h^{i}(I_{S}(p))
2 1

⟶ p

Construction

E = 3Ω^{3}(3), F = ker(Ω^{2}(2) ⊕ 2Ω^{1}(1)⟶^{ψ}O)

where ψ is a general morphism. see (3.23) for details Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 41

11.B3.

Invariants d = 11, π = 11, HK = 9, K^{2} = −11, χ = 1, q = 0, N_{6} = 7

Classification S = IP^{2}(p_{0}, ..., p_{19}) rational surface H = 13l − 5E_{0} − 4 ∑_{1}^{7} E_{i} − 2 ∑_{8}^{10} E_{j} − ∑_{11}^{19} E_{k}

Syzygies of I_{S}

10O(−5) 14O(−6) 6O(−7) O(−8)
0←I_{S}← ⊕ ⟵ ⊕ ⟵ ⊕ ⟵ ⊕ ←0
2O(−6) 5O(−7) 4O(−8) O(−9)

Cohomology

↑ i

3 1  h^{i}(I_{S}(p))
2 1

⟶ p

Construction

E = 3Ω^{3}(3), F = ker(Ω^{2}(2) ⊕ 2Ω^{1}(1)⟶^{ψ}O)

where ψ is a special morphism. see (3.23) for details Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 41 11.C1.

Invariants d = 11, π = 11, HK = 9, K^{2} = −5, χ = 2, q = 0, N_{6} = 4, [DES]

Classification S = S_{min}(p_{1}, ..., p_{5}) non-minimal K3 surface H = H_{min} − 5E_{1} − ∑_{2}^{5} E_{i}

Syzygies of I_{S}

0←I_{S}←9O(−5) 8O(−6)
↖ ⊕
5O(−7) ↖ 7O(−8) ⟵ 2O(−9) ← 0

Cohomology

↑ i

1

2  h^{i}(I_{S}(p))
3 2

⟶ p

Construction

E = O ⊕ 2Ω^{3}(3), F = ker(13O⟶^{ψ}3O(1))

where ψ is a general morphism. see [DES] or (3.30) for details Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 43

11.C2.

Invariants d = 11, π = 11, HK = 9, K^{2} = −5, χ = 2, q = 0, N_{6} = 4

Classification S = S_{min}(p_{1}, ..., p_{5}) non-minimal K3 surface H = H_{min} − 4E_{1} − 2E_{2} − ∑_{3}^{5} E_{i}

Syzygies of I_{S}

9O(−5) 9O(−6) O(−7)
0←I_{S}← ⊕ ⟵ ⊕ ⟵ ⊕
O(−6) 6O(−7) 7O(−8) ↖ 2O(−9) ← 0

Cohomology

↑ i

1

2  h^{i}(I_{S}(p))
3 2

⟶ p

Construction

E = O ⊕ 2Ω^{3}(3), F = ker(13O⟶^{ψ}3O(1))

where ψ is a special morphism. see (3.30) for more details Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 43

11.C3.

Invariants d = 11, π = 11, HK = 9, K^{2} = −5, χ = 2, q = 0, N_{6} = 4,

Classification S = S_{min}(p_{1}, ..., p_{5}) non-minimal K3 surface H = H_{min} − 3E_{1} − 2 ∑_{2}^{3} E_{i} − ∑_{4}^{5} E_{j}

Syzygies of I_{S}

9O(−5) 10O(−6) 2O(−7)
0←I_{S}← ⊕ ⟵ ⊕ ⟵ ⊕
2O(−6) 7O(−7) 7O(−8) ↖ 2O(−9) ← 0

Cohomology

↑ i

1

2  h^{i}(I_{S}(p))
3 2

⟶ p

Construction

E = O ⊕ 2Ω^{3}(3), F = ker(13O⟶^{ψ}3O(1))

where ψ is a special morphism. see (3.30) for more details Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 43 11.C4.

Invariants d = 11, π = 11, HK = 9, K^{2} = −5, χ = 2, q = 0, N_{6} = 4,

Classification S = S_{min}(p_{1}, ..., p_{5}) non-minimal K3 surface H = H_{min} − 2 ∑_{1}^{4} E_{i} − E_{5}

Syzygies of I_{S}

9O(−5) 11O(−6) 3O(−7)
0←I_{S}← ⊕ ⟵ ⊕ ⟵ ⊕
3O(−6) 8O(−7) 7O(−8) ↖ 2O(−9) ← 0

Cohomology

↑ i

1

2  h^{i}(I_{S}(p))
3 2

⟶ p

Construction

E = O ⊕ 2Ω^{3}(3), F = ker(13O⟶^{ψ}3O(1))

where ψ is a special morphism. see (3.30) for more details Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 43

11.D.

Invariants d = 11, π = 11, HK = 9, K^{2} = 1, χ = 3, q = 0, N_{6} = 5,

Classification S = S_{min}(p_{1}) non-minimal general type surface H = H_{min} − E_{1}

Syzygies of I_{S}

8O(−5) 8O(−6) O(−7)
0←I_{S}← ⊕ ⟵ ⊕ ⟵ ⊕
4O(−6) 12O(−7) 11O(−8) ↖ 3O(−9) ← 0

Cohomology

↑i

2

1  h^{i}(I_{S}(p))
1 4 3

⟶ p

Construction

E = 2O(−1) ⊕ Ω^{3}(3), F = ker(7O ⊕ O(1)⟶^{ψ}O(2))

where ψ is a special morphism. see (3.32) for more details Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 45

11.E.

Invariants d = 11, π = 12, HK = 11, K^{2} = −10, χ = 2, q = 0, N_{6} = 9

Classification S = S_{min}(p_{1}, ..., p_{10}) non-minimal K3 surface H = H_{min} − 2E_{1} − ∑_{2}^{10} E_{i}

Syzygies of I_{S}

2O(−4)

0←I_{S}← ⊕
4O(−5) ↖ 7O(−6) ⟵ 2O(−7) ← 0

Cohomology

↑ i

1

3 2  h^{i}(I_{S}(p))

2 ⟶ p

Construction E = O(−1) ⊕ 2Ω^{3}(3), F = 2Ω^{2}(2) ⊕ 2O see (3.38) for more details see (3.39) and (3.41) for liaison Family properties irreducible, unirational, of dimension 48 χ(N_{S|IP^{4}}) = 48 11.F1.

Invariants d = 11, π = 12, HK = 11, K^{2} = −4, χ = 3, q = 0, N_{6} = 3

Classification S = S_{min}(p_{0}, ..., p_{3}) non-minimal proper elliptic surface H = H_{min} − 2E_{0} − ∑_{1}^{3} E_{i}

Syzygies of I_{S}

O(−4)

0←I_{S}← ⊕
8O(−5) ↖ 13O(−6) ← 6O(−7) ← O(−8) ← 0

Cohomology

↑ i

2

2 1  h^{i}(I_{S}(p))

1

1 ⟶ p

Construction E = 2O(−1) ⊕ 2Ω^{3}(3), F = Ω^{2}(2) ⊕ Ω^{1}(1) ⊕ O see (3.43) for more details see also (3.44) for a liaison construction Family properties irreducible, unirational, of dimension 50 which equals χ(N_{S|IP^{4}}) = 50

11.F2.

Invariants d = 11, π = 12, HK = 11, K^{2} = −4, χ = 3, q = 0, N_{6} = 3

Classification S = S_{min}(p_{1}, ..., p_{5}) blown-up, general type Horikawa surface H = H_{min} − ∑_{1}^{5} E_{i}

Syzygies of I_{S}

2O(−4)

⊕
0←I_{S}←3O(−5) 5O(−6) O(−7)
⊕ ↖ ⊕ ⟵ ⊕
2O(−6) 5O(−7) 4O(−8) ↖ O(−9) ← 0

Cohomology

↑ i

2

2 1  h^{i}(I_{S}(p))
1 1

2 ⟶ p

Construction

E = 2O(−1) ⊕ 2Ω^{3}(3), F = 2O ⊕ ker(Ω^{2}(2) ⊕ Ω^{1}(1)⟶^{ψ}O)

where ψ is a special morphism; see (3.43) for more details see also (3.43) for a discussion of liaison Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 50

11.G1.

Invariants d = 11, π = 12, HK = 11, K^{2} = 2, χ = 4, q = 0, N_{6} = 1

Classification S = S_{min}(p_{1}) non-minimal, general type Horikawa surface H = H_{min} − E_{1}

Syzygies of I_{S}

0←I_{S}←12O(−5) ⟵ 19O(−6) ⟵ 10O(−7) ⟵ 2O(−8) ← 0

Cohomology

↑ i

3

1  h^{i}(I_{S}(p))

2

⟶ p

Construction E = 3O(−1) ⊕ Ω^{3}(3), F = 2Ω^{1}(1) see (3.49) for more details Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 52 11.G2.

Invariants d = 11, π = 12, HK = 11, K^{2} = 2, χ = 4, q = 0, N_{6} = 1

Classification S = S_{min} minimal, general type surface H = H_{min}

Syzygies of I_{S}

2O(−4)

⊕
0←I_{S}←7O(−5) 10O(−6) 3O(−7)
⊕ ↖ ⊕ ⟵ ⊕
O(−6)  3O(−7) 3O(−8) ↖ O(−9) ← 0

Cohomology

↑ i

3

1  h^{i}(I_{S}(p))
2 1

1 ⟶ p

Construction

E = 3O(−1) ⊕ Ω^{3}(3), F = O ⊕ ker(9O⟶^{ψ}2O(1))

where ψ is a general morphism. see (3.49) for more details Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 52

11.G3.

Invariants d = 11, π = 12, HK = 11, K^{2} = 2, χ = 4, q = 0, N_{6} = 1

Classification S = S_{min}(p_{1}, p_{2}) non-minimal, general type surface H = H_{min} − E_{1} − E_{2}

Syzygies of I_{S}

2O(−4)

⊕
0←I_{S}←2O(−5) 3O(−6)
⊕ ↖ ⊕
4O(−6) 10O(−7) ↖ 8O(−8) ⟵ 2O(−9) ← 0

Cohomology

↑ i

3

1  h^{i}(I_{S}(p))
2 2

2 ⟶ p

Construction

E = 3O(−1) ⊕ Ω^{3}(3), F = 2O ⊕ ker(8O⟶^{ψ}2O(1))

where ψ is a general morphism; see (3.49) for more details see also (3.50) for a discussion of liaison Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 52

11.H.

Invariants d = 11, π = 12, HK = 11, K^{2} = 8, χ = 5, q = 0, N_{6} = 3

Classification S = S_{min} minimal, general type surface H = H_{min}

Syzygies of I_{S}

O(−4)

⊕
0←I_{S}←6O(−5) 8O(−6) 2O(−7)
⊕ ↖ ⊕ ⟵ ⊕
3O(−6) 8O(−7) 7O(−8) ↖ 2O(−9) ← 0

Cohomology

↑ i

4

h^{i}(I_{S}(p))
1 3 2

1 ⟶ p

Construction

E = 4O(−1), F = ker(4O(−1) ⊕ 2O⟶^{ψ}O(1))

where ψ is a general morphism. see (3.52) for more details Family properties irreducible, unirational

χ(N_{S|IP^{4}}) = 54 112

11.J.

Invariants d = 11, π = 13, HK = 13, K^{2} = 3, χ = 5, q = 0, N_{6} = 1

Classification S = S_{min}(p_{1}) non-minimal, general type surface H = H_{min} − E_{1}

Syzygies of I_{S}

3O(−4) O(−5)
0←I_{S}← ⊕ ⟵ ⊕
2O(−5) 4O(−6) ↖ O(−7) ← 0

Cohomology

↑ i

4

1 1  h^{i}(I_{S}(p))

3 ⟶ p

Construction E = 4O(−1) ⊕ Ω^{3}(3), F = 3O ⊕ Ω^{2}(2) see (3.56) for more details see also (3.59) for a liaison construction Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 59

11.L.

Invariants d = 11, π = 13, HK = 13, K^{2} = 9, χ = 6, q = 0, N_{6} = 0

Classification S = S_{min} minimal, general type surface H = H_{min}

Syzygies of I_{S}

2O(−4)

0←I_{S}← ⊕
5O(−5) ↖ 10O(−6) ⟵ 5O(−7) ⟵ O(−8) ← 0

Cohomology

↑ i

5

h^{i}(I_{S}(p))

1

2 ⟶ p

Construction E = 5O(−1), F = 2O ⊕ Ω^{1}(1) see (3.60) for more details see also (3.60) for the (4, 4) liaison with an elliptic quintic scroll Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 61

11.M.

Invariants d = 11, π = 14, HK = 15, K^{2} = 10, χ = 7, q = 0, N_{6} = 0

Classification S = S_{min} minimal, general type surface H = H_{min}

Syzygies of I_{S}

O(−3)

0←I_{S}← ⊕
4O(−5) ↖ 5O(−6) ⟵ O(−7) ← 0

Cohomology

↑ i

6

1  h^{i}(I_{S}(p))
1 5 ⟶ p

Construction E = 6O(−1), F = Ω^{2}(2) ⊕ O(1) see (3.68) for more details see also (3.68) for the (3, 5) liaison with a Veronese surface Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 68 11.N.

Invariants d = 11, π = 14, HK = 15, K^{2} = 16, χ = 8, q = 0, N_{6} = 0

Classification S = S_{min} minimal, general type surface H = H_{min}

Syzygies of I_{S}

0←I_{S} ← 4O(−4) 2O(−5)
↖ ⊕ ← 0
O(−6)

Cohomology

↑ i

7 1
h^{i}(I_{S}(p))

4 ⟶ p

Construction E = 2O(−5) ⊕ O(−6), F = 4O(−4) see (3.70) for more details see also (3.70) for the (4, 4) liaison with a Castelnuovo surface Family properties irreducible, unirational, of dimension 70 which equals χ(N_{S|IP^{4}}) = 70

3. Degree 12

12.A.

Invariants d = 12, π = 13, HK = 12, K^{2} = 0, χ = 3, q = 0, N_{6} = 10

Classification S = S_{min} minimal, proper elliptic surface H = H_{min}

Syzygies of I_{S}

3O(−5)

0←I_{S}← ⊕
12O(−6) ↖ 30O(−7) ⟵ 21O(−8) ⟵ 5O(−9) ← 0

Cohomology

↑ i

2

2  h^{i}(I_{S}(p))
4 5

⟶ p

Construction

E = 2O(−1) ⊕ 2Ω^{3}(3), F = ker(15O⟶^{ψ}4O(1))

where ψ is a special morphism; see (4.1) for details see also (4.4) for a (5, 5) liaison with 13.B.

Family properties

12.B1.

Invariants d = 12, π = 14, HK = 14, K^{2} = −5, χ = 3, q = 0, N_{6} = 4

irreducible, unirational, of dimension 49 which differs from χ(N_{S|IP^{4}}) = 42 Classification S = S_{min}(p_{1}, ..., p_{5}) non-minimal, proper elliptic surface H = H_{min} − 2E_{1} − ∑_{2}^{5} E_{i}

Syzygies of I_{S}

0←I_{S}←8O(−5) 7O(−6) O(−7)
↖ ⊕ ⟵ ⊕
4O(−7) 4O(−8) ↖ O(−9) ← 0

Cohomology

↑ i

2

3 2  h^{i}(I_{S}(p))
1 1

⟶ p

Construction

E = 2O(−1) ⊕ 3Ω^{3}(3), F = ker(2Ω^{2}(2) ⊕ Ω^{1}(1)⟶^{ψ}O)

where ψ is a general morphism. see (4.7) for details Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 47

12.B2.

Invariants d = 12, π = 14, HK = 14, K^{2} = −5, χ = 3, q = 0, N_{6} = 4

Classification S = S_{min}(p_{1}, ..., p_{5}) non-minimal, proper elliptic surface H = H_{min} − ∑_{1}^{5} E_{i}

Syzygies of I_{S}

8O(−5) 9O(−6) 2O(−7)
0←I_{S}← ⊕ ⟵ ⊕ ⟵ ⊕
2O(−6) 5O(−7) 4O(−8) ↖ O(−9) ← 0

Cohomology

↑ i

2

3 2  h^{i}(I_{S}(p))
1 1

⟶ p

Construction

E = 2O(−1) ⊕ 3Ω^{3}(3), F = ker(2Ω^{2}(2) ⊕ Ω^{1}(1)⟶^{ψ}O)

4. Degree 13

where ψ is a special morphism. see (4.7) for details Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 47 13.A1.

Invariants d = 13, π = 16, HK = 17, K^{2} = −17, χ = 1, q = 0, N_{6} = 17, [DES]

Classification S = S_{min}(p_{1}, ..., p_{17}) non-minimal, Enriques surface H = H_{min} − ∑_{1}^{17} E_{i}

Syzygies of I_{S}

5O(−5)

0←I_{S}← ⊕
O(−6) ↖ 10O(−7) ⟵ 6O(−8) ⟵ O(−9) ← 0

Cohomology

↑i

6 5 1  h^{i}(I_{S}(p))

1

⟶ p

Construction

E = 16O, F = ker(Syz_{2}(N)⟶^{ψ}O)

where ψ is general and N^{∗} is a generic module with syzygies: see (5.3) for more details

Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 38

\begin{aligned}N^{∗} & ← & R(4) \\ & & ↖ \\ 9R(2) & 10R(1) & R \\ ↖ & ⊕ & ⟵ & ⊕ \\ 16R & 36R(−1) & ↖ & 25R(−2) & ⟵ & 6R(−3) & ← & 0 \end{aligned}

13.A2.

Invariants d = 13, π = 16, HK = 17, K^{2} = −17, χ = 1, q = 0, N_{6} = 17

Classification S = S_{min}(p_{1}, ..., p_{17}) non-minimal, Enriques surface H = H_{min} − ∑_{1}^{17} E_{i}

Syzygies of I_{S}

5O(−5) O(−6)
0←I_{S}← ⊕ ⟵ ⊕
2O(−6) 10O(−7) ↖ 6O(−8) ⟵ O(−9) ← 0

Cohomology

↑i

6 5 1  h^{i}(I_{S}(p))

1

⟶ p

Construction

E = 16O, F = ker(Syz_{2}(N)⟶^{ψ}O)

where ψ is general and N^{∗} is a special module with syzygies: see (5.3) and ff. for more details

Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 38

\begin{aligned}N^{∗} & ← & R(4) \\ & & ↖ \\ 9R(2) & 10R(1) & R \\ ↖ & ⊕ & ⟵ & ⊕ \\ 16R & 36R(−1) & ↖ & 25R(−2) & ⟵ & 6R(−3) & ← & 0 \end{aligned}

Classification S = S_{min}(p_{1}, ..., p_{17}) non-minimal, K3 surface H = H_{min} − 7E_{0} − ∑_{1}^{10} E_{i}

Syzygies of I_{S}

4O(−5)

0←I_{S}← ⊕
5O(−6) ↖ 16O(−7) ← 10O(−8) ← 2O(−9) ← 0

Cohomology

↑ i

1

5 4  h^{i}(I_{S}(p))

2

⟶ p

Construction

E = O(−1) ⊕ 12O, F = ker(Syz_{2}(N)⟶^{ψ}2O)

where ψ is general and N has syzygies of type: see (4.4) for more details see also (4.4) for the (5, 5) liaison with 12.A.

Family properties irreducible, unirational χ(N_{S|IP^{4}}) = 40

N ← 5R(3) ← 21R(2) 30R(1) 12R
↖ ⊕ ⟵ ⊕
2R 15R(−1) ↖ 15R(−2) ← 4R(−3) ← 0

5. Degree 14

14.A.

Invariants d = 14, π = 19, HK = 22, K^{2} = −15, χ = 2, q = 0, N_{6} = 22

Classification S = S_{min}(p_{1}, ..., p_{17}) non-minimal, K3 surface H = H_{min} − 4E_{0} − ∑_{1}^{4} 2E_{i} − ∑_{5}^{14} E_{i}

Syzygies of I_{S}

4O(−5) 2O(−6)
0←I_{S}← ⊕ ⟵ ⊕
4O(−6) 8O(−7) ↖ 3O(−8) ← 0

Cohomology

↑i

1

7 7 3  h^{i}(I_{S}(p))

⟶ p

Construction E = O(−1) ⊕ 15O, F = Syz_{2}(N) where N^{∗} is a special module with syzygies: see (6.1) for more details see also (6.2) for a liaison construction.

Family properties irreducible, unirational, of dimension 39 which differs from χ(N_{S|IP^{4}}) = 38

N←3R(4) 8R(3) 4R(2)
↖ ⊕  ⊕
2R(2) ↖ 5R(1)
⊕
15R ↖ 38R(−1) ← 28R(−2) ← 7R(−3) ← 0

15.A.

Invariants d = 15, π = 21, HK = 25, K^{2} = −25, χ = 0, q = 1, N_{6} = 25, [ADHPR]

Classification S = S_{min}(p_{1}, ..., p_{25}) non-minimal, bielliptic surface H = H_{min} − ∑_{1}^{25} E_{i}

Syzygies of I_{S}

O(−5)

0←I_{S}← ⊕
10O(−6) ↖ 15O(−7) ⟵ 5O(−8) ← 0

Cohomology

↑ i

1 10 10 5  h^{i}(I_{S}(p))

⟶ p

Construction E = 20O, F = Syz_{2}(N) where N^{∗} is a module with syzygies: see [ADHPR] for more details

Family properties irreducible, of dimension 25 which equals χ(N_{S|IP^{4}}) = 25

\begin{aligned}N & ← & 5R(4) & ← & 15R(3) & 10R(2) \\ & & & & ↖ & ⊕ \\ & & & & & R(1) \\ & & & & & ⊕ \\ & & & & & 20R & ↖ & 45R(−1) & ← & 30R(−2) & & 5R(−3) \\ & & & & & & & & & & ↖ & ⊕ & ← & 0 \\ & & & & & & & & & & & R(−4) \end{aligned}

15.B1.

Invariants d = 15, π = 21, HK = 25, K^{2} = −25, χ = 0, q = 2, N_{6} = 25

Classification S = S_{min}(p_{1}, ..., p_{25}) non-minimal, abelian surface H = H_{min} − ∑_{1}^{25} E_{i}

Syzygies of I_{S}

O(−5)

0←I_{S}← ⊕
10O(−6) ↖ 15O(−7) ⟵ 5O(−8) ← 0

Cohomology

↑ i

1

2 10 10 5  h^{i}(I_{S}(p))

⟶ p

Construction E = 20O, F = Syz_{2}(N) where N^{∗} is a module with syzygies: see (7.4) for more details

Family properties irreducible, unirational, of dimension 27 which differs from χ(N_{S|IP^{4}}) = 25

N←5R(4) ← 15R(3) 10R(2)
↖ ⊕
R(1)
⊕
20R ↖ 45R(−1) ← 30R(−2) 5R(−3)
↖ ⊕ ← 0
R(−4)

15.B2.

Invariants d = 15, π = 21, HK = 25, K^{2} = −25, χ = 0, q = 2, N_{6} = 25, [HM],[Au2]

Classification S = S_{min}(p_{1}, ..., p_{25}) non-minimal, abelian surface H = H_{min} − ∑_{1}^{25} E_{i}

Syzygies of I_{S}

3O(−5)

0←I_{S}← ⊕
5O(−6) ↖ 15O(−7) ← 10O(−8) ← 2O(−9) ← 0

Cohomology

↑ i

1

2 10 10 5  h^{i}(I_{S}(p))

⟶ p

Construction

E = O(−1) ⊕ 15O, F = ker(Syz_{2}(N)⟶^{ψ}2O(−1))

where N^{∗} is a module with syzygies: see [HM] and [DES] for more details see also [Au2] for the liaison construction.

Family properties irreducible, of dimension 27 which differs from χ(N_{S|IP^{4}}) = 25

N←5R(4) ← 15R(3) 10R(2)
↖ ⊕
4R(1)  2R(1)
⊕ ↖ ⊕
15R  35R(−1) ← 20R(−2)
↖ 2R(−4) ← 0

15.C.

Invariants d = 15, π = 22, HK = 27, K^{2} = −6, χ = 4, q = 0, N_{6} = 14

Classification S = S_{min}(p_{1}, ..., p_{17}) non-minimal, general type surface H = H_{min} − ∑_{0}^{8} E_{i}

Syzygies of I_{S}

2O(−5)

0←I_{S}← ⊕
7O(−6) ↖ 12O(−7) ⟵ 4O(−8) ← 0

Cohomology

↑i

3

7 8 4  h^{i}(I_{S}(p))

⟶ p

Construction E = 3O(−1) ⊕ 10O, F = Syz_{2}(N) where N^{∗} is a special module with syzygies: see (6, 12) for more details see also (6, 12) for a liaison construction.

Family properties irreducible, of dimension 38 which equals χ(N_{S|IP^{4}}) = 38

\begin{aligned}N & ← & 4R(4) & ← & 12R(3) & 7R(2) \\ & & & & ↖ & ⊕ \\ & & & & & 5R(1) \\ & & & & & ⊕ \\ & & & & & 10R & ↖ & 34R(−1) & ← & 27R(−2) & ← & 7R(−3) & ← & 0 \end{aligned}