Proceedings of the 44th Symposium on Ring Theory and ...

4. Dolgachev
Proposition 24 ([1]). f(x, y, z)
f Table 1 5 □
Type Class f f t
I I x p 1
+ y p 2
+ z p 3
x p 1
+ y p 2
+ z p 3
(p 1 , p 2 , p 3 ∈ Z ≥2 ) (p 1 , p 2 , p 3 ∈ Z ≥2 )
II II x p 1
+ y p 2
+ yz p 3
p 2 x p 1
+ y p 2
z + z p 3
p 2
(p 1 , p 2 , p 3
p 2
∈ Z ≥2 ) (p 1 , p 2 , p 3
p 2
∈ Z ≥2 )
III IV x p 1
+ zy q2+1 + yz q 3+1
x p 1
+ zy q2+1 + yz q 3+1
(p 1 ∈ Z ≥2 , q 2 , q 3 ∈ Z ≥1 ) (p 1 ∈ Z ≥2 , q 2 , q 3 ∈ Z ≥1 )
IV V x p 1
+ xy p 2
p 1 + yz p 3
p 2 x p 1
y + y p 2
p 1 z + z p 3
p 2
(p 1 , p 3
p 2
∈ Z ≥2 , p 2
p 1
∈ Z ≥1 ) (p 1 , p 3
p 2
∈ Z ≥2 , p 2
p 1
∈ Z ≥1 )
V VII x q 1
y + y q 2
z + z q 3
x zx q 1
+ xy q 2
+ yz q 3
(q 1 , q 2 , q 3 ∈ Z ≥1 ) (q 1 , q 2 , q 3 ∈ Z ≥1 )
Table 1. 3
Table 1 Type [11]
[1] Class
f(x, y, z)
(4.1) C Gf := [ f −1 (0)\{0} /G f
]
X Lf f 0 ∈ C 3
G f 1 C ∗ c f
C Gf Deligne–Mumford
Theorem 25 ([5]). f(x, y, z) C Gf 3
P 1 2
α 1 , α 2 , α 3 α i ≥ 2 i □
Definition 26. Theorem 25 (α 1 , α 2 , α 3 ) (f, G f ) Dolgachev
A Gf .
Theorem 25Orlov Theorem 23 Geigle–Lenzing[6]
Corollary 27 ([12][13]). HMF L f
S
(f) full exceptional collection
□
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