1 Introduction
-
Can one explicitly describe a fundamental domain?
-
How many smooth rational curves, elliptic fibrations or projective models are there up to the action of the automorphism group?
-
Can one give generators for the automorphism group?
1.1 Definition of \((\tau , {\bar{\tau }})\)-Generic Enriques Surfaces
No. |
\(\tau (R)\)
|
\(\tau (\overline{R})\)
|
\(\tau ({\widetilde{R}})\)
| exist
|
\(c_{(\tau , {\bar{\tau }})} \)
| rat
| irec
|
---|---|---|---|---|---|---|---|
1 |
\(A_{1}\)
| – | – | 1 | 1 | 96C
| |
2 |
\(2A_{1}\)
| – | – | 1 | 2 | 96C
| |
3 |
\(A_{2}\)
| – | – | 1 | 1 | 96C
| |
4 |
\(3A_{1}\)
| – | – | 1 | 3 | 96C
| |
5 |
\(A_{2}+A_{1}\)
| – | – | 1 | 2 | 96C
| |
6 |
\(A_{3}\)
| – | – | 1 | 1 | 96C
| |
7 |
\(4A_{1}\)
| – | – | 1 | 4 | 96C
| |
8 |
\(4A_{1}\)
|
\(D_{4}\)
|
\(D_{4}\)
| 1 | 4 | 96C
| |
9 |
\(A_{2}+2A_{1}\)
| – | – | 1 | 3 | 96C
| |
10 |
\(A_{3}+A_{1}\)
| – | – | 1 | 2 | 96C
| |
11 |
\(2A_{2}\)
| – | – | 1 | 2 | 96C
| |
12 |
\(A_{4}\)
| – | – | 1 | 1 | 40E
| |
13 |
\(D_{4}\)
| – | – | 1 | 1 | 96A
| |
14 |
\(5A_{1}\)
| – | – | 1 | 5 | 96C
| |
15 |
\(5A_{1}\)
|
\(D_{4}+A_{1}\)
|
\(D_{4}+A_{1}\)
| 1 | 5 | 96C
| |
16 |
\(A_{2}+3A_{1}\)
| – | – | 1 | 4 | 96C
| |
17 |
\(A_{3}+2A_{1}\)
| – | – | 1 | 3 | 96C
| |
18 |
\(A_{3}+2A_{1}\)
|
\(D_{5}\)
|
\(D_{5}\)
| 1 | 3 | 96C
| |
19 |
\(2A_{2}+A_{1}\)
| – | – | 1 | 3 | 96C
| |
20 |
\(A_{4}+A_{1}\)
| – | – | 1 | 2 | 40E
| |
21 |
\(D_{4}+A_{1}\)
| – | – | 1 | 2 | 96A
| |
22 |
\(A_{3}+A_{2}\)
| – | – | 1 | 2 | 96C
| |
23 |
\(A_{5}\)
| – | – | 1 | 1 | 40E
| |
24 |
\(D_{5}\)
| – | – | 1 | 1 | 40A
| |
25 |
\(6A_{1}\)
|
\(D_{4}+2A_{1}\)
|
\(D_{4}+2A_{1}\)
| 1 | 6 | 96C
| |
26 |
\(6A_{1}\)
|
\(D_{6}\)
|
\(D_{6}\)
|
\(\times \)
| 1 | 6 | 96C
|
27 |
\(A_{2}+4A_{1}\)
| – | – | 1 | 5 | 96C
| |
28 |
\(A_{2}+4A_{1}\)
|
\(D_{4}+A_{2}\)
|
\(D_{4}+A_{2}\)
| 1 | 5 | 96C
| |
29 |
\(A_{3}+3A_{1}\)
| – | – | 1 | 4 | 96C
| |
30 |
\(A_{3}+3A_{1}\)
|
\(D_{5}+A_{1}\)
|
\(D_{5}+A_{1}\)
| 1 | 4 | 96C
| |
31 |
\(2A_{2}+2A_{1}\)
| – | – | 1 | 4 | 96C
| |
32 |
\(A_{4}+2A_{1}\)
| – | – | 1 | 3 | 40E
| |
33 |
\(D_{4}+2A_{1}\)
| – | – | 1 | 3 | 96A
| |
34 |
\(D_{4}+2A_{1}\)
|
\(D_{6}\)
|
\(D_{6}\)
| 1 | 3 | 96A
| |
35 |
\(A_{3}+A_{2}+A_{1}\)
| – | – | 1 | 3 | 96C
| |
36 |
\(A_{5}+A_{1}\)
| – | – | 1 | 2 | 40E
| |
37 |
\(A_{5}+A_{1}\)
|
\(E_{6}\)
|
\(E_{6}\)
| 1 | 2 | 40E
| |
38 |
\(D_{5}+A_{1}\)
| – | – | 1 | 2 | 40A
| |
39 |
\(3A_{2}\)
| – | – | 1 | 3 | 96C
| |
40 |
\(3A_{2}\)
|
\(E_{6}\)
|
\(3A_{2}\)
| 1 | 3 | 96C
| |
41 |
\(A_{4}+A_{2}\)
| – | – | 1 | 2 | 40E
| |
42 |
\(D_{4}+A_{2}\)
| – | – | 1 | 2 | 96A
| |
43 |
\(2A_{3}\)
| – | – | 1 | 2 | 96A
| |
44 |
\(2A_{3}\)
|
\(D_{6}\)
|
\(D_{6}\)
| 1 | 2 | 96C
| |
45 |
\(A_{6}\)
| – | – | 1 | 1 | 40C
| |
46 |
\(D_{6}\)
| – | – | 1 | 1 | 40A
| |
47 |
\(E_{6}\)
| – | – | 1 | 1 | 20E
| |
48 |
\(7A_{1}\)
|
\(D_{6}+A_{1}\)
|
\(D_{6}+A_{1}\)
|
\(\times \)
| 1 | 7 | 96C
|
49 |
\(7A_{1}\)
|
\(E_{7}\)
|
\(E_{7}\)
|
\(\times \)
| 1 | 7 | 96A
|
50 |
\(A_{2}+5A_{1}\)
|
\(D_{4}+A_{2}+A_{1}\)
|
\(D_{4}+A_{2}+A_{1}\)
| 1 | 6 | 96C
| |
51 |
\(A_{3}+4A_{1}\)
|
\(D_{5}+2A_{1}\)
|
\(D_{5}+2A_{1}\)
| 1 | 5 | 96C
| |
52 |
\(A_{3}+4A_{1}\)
|
\(D_{4}+A_{3}\)
|
\(D_{4}+A_{3}\)
| 1 | 5 | 96A
| |
53 |
\(A_{3}+4A_{1}\)
|
\(D_{7}\)
|
\(D_{7}\)
|
\(\times \)
| 1 | 5 | 96C
|
54 |
\(2A_{2}+3A_{1}\)
| – | – | 1 | 5 | 96C
| |
55 |
\(A_{4}+3A_{1}\)
| – | – | 1 | 4 | 40E
| |
56 |
\(D_{4}+3A_{1}\)
|
\(D_{6}+A_{1}\)
|
\(D_{6}+A_{1}\)
| 1 | 4 | 96A
| |
57 |
\(D_{4}+3A_{1}\)
|
\(E_{7}\)
|
\(E_{7}\)
|
\(\times \)
| 1 | 4 | 96A
|
58 |
\(A_{3}+A_{2}+2A_{1}\)
| – | – | 1 | 4 | 96C
| |
59 |
\(A_{3}+A_{2}+2A_{1}\)
|
\(D_{5}+A_{2}\)
|
\(D_{5}+A_{2}\)
| 1 | 4 | 96C
| |
60 |
\(A_{5}+2A_{1} \)
| – | – | 1 | 3 | 40E
| |
61 |
\(A_{5}+2A_{1}\)
|
\(E_{6}+A_{1}\)
|
\(E_{6}+A_{1}\)
| 1 | 3 | 40E
| |
62 |
\(D_{5}+2A_{1}\)
| – | – | 1 | 3 | 40A
| |
63 |
\(D_{5}+2A_{1}\)
|
\(D_{7}\)
|
\(D_{7}\)
| 1 | 3 | 40A
| |
64 |
\(3A_{2}+A_{1}\)
| – | – | 1 | 4 | 96C
| |
65 |
\(3A_{2}+A_{1}\)
|
\(E_{6}+A_{1}\)
|
\(3A_{2}+A_{1}\)
| 1 | 4 | 96C
| |
66 |
\(A_{4}+A_{2}+A_{1}\)
| – | – | 1 | 3 | 40E
| |
67 |
\(D_{4}+A_{2}+A_{1}\)
| – | – | 1 | 3 | 96A
| |
68 |
\(2A_{3}+A_{1}\)
| – | – | 1 | 3 | 96A
| |
69 |
\(2A_{3}+A_{1}\)
|
\(D_{6}+A_{1}\)
|
\(D_{6}+A_{1}\)
| 1 | 3 | 96C
| |
70 |
\(2A_{3}+A_{1}\)
|
\(E_{7}\)
|
\(D_{6}+A_{1}\)
| 1 | 3 | 96C
| |
71 |
\(A_{6}+A_{1}\)
| – | – | 1 | 2 | 40C
| |
72 |
\(D_{6}+A_{1}\)
| – | – | 1 | 2 | 40A
| |
73 |
\(D_{6}+A_{1}\)
|
\(E_{7}\)
|
\(E_{7}\)
| 1 | 2 | 40A
| |
74 |
\(E_{6}+A_{1}\)
| – | – | 1 | 2 | 20E
| |
75 |
\(A_{3}+2A_{2}\)
| – | – | 1 | 3 | 96C
| |
76 |
\(A_{5}+A_{2}\)
| – | – | 1 | 2 | 40E
| |
77 |
\(A_{5}+A_{2}\)
|
\(E_{7}\)
|
\(A_{5}+A_{2}\)
| 1 | 2 | 40E
| |
78 |
\(D_{5}+A_{2}\)
| – | – | 1 | 2 | 40A
| |
79 |
\(A_{4}+A_{3}\)
| – | – | 1 | 2 | 40E
| |
80 |
\(D_{4}+A_{3}\)
| – | – | 1 | 2 | 20F
| |
81 |
\(D_{4}+A_{3}\)
|
\(D_{7}\)
|
\(D_{7}\)
| 1 | 2 | 96A
| |
82 |
\(A_{7}\)
| – | – | 1 | 1 | 20D
| |
83 |
\(A_{7}\)
|
\(E_{7}\)
|
\(E_{7}\)
| 1 | 1 | 40C
| |
84 |
\(D_{7}\)
| – | – | 1 | 1 | 20B
| |
85 |
\(E_{7}\)
| – | – | 1 | \(\times \)2 | 20A
| |
86 |
\(8A_{1}\)
|
\(E_{7}+A_{1}\)
|
\(E_{7}+A_{1}\)
|
\(\times \)
| 1 | 8 | 96A
|
87 |
\(8A_{1}\)
|
\(D_{8}\)
|
\(D_{8}\)
|
\(\times \)
| 1 | 8 | 96B
|
88 |
\(8A_{1}\)
|
\(E_{8}\)
|
\(E_{8}\)
|
\(\times \)
| See Remark 1.16 | ||
89 |
\(A_{2}+6A_{1}\)
|
\(D_{6}+A_{2}\)
|
\(D_{6}+A_{2}\)
|
\(\times \)
| 1 | 7 | 96C
|
90 |
\(A_{3}+5A_{1}\)
|
\(D_{7}+A_{1}\)
|
\(D_{7}+A_{1}\)
|
\(\times \)
| 1 | 6 | 96C
|
91 |
\(A_{4}+4A_{1}\)
|
\(D_{4}+A_{4}\)
|
\(D_{4}+A_{4}\)
| 1 | 5 | 40E
| |
92 |
\(D_{4}+4A_{1}\)
|
\(E_{7}+A_{1}\)
|
\(E_{7}+A_{1}\)
|
\(\times \)
| 1 | 5 | 96A
|
93 |
\(D_{4}+4A_{1}\)
|
\(D_{8}\)
|
\(D_{8}\)
|
\(\times \)
| 1 | 5 | 96A
|
94 |
\(D_{4}+4A_{1}\)
|
\(E_{8}\)
|
\(E_{8}\)
|
\(\times \)
| 2 | 5 | 96A
|
95 |
\(A_{3}+A_{2}+3A_{1}\)
|
\(D_{5}+A_{2}+A_{1}\)
|
\(D_{5}+A_{2}+A_{1}\)
| 1 | 5 | 96C
| |
96 |
\(A_{5}+3A_{1}\)
|
\(E_{6}+2A_{1}\)
|
\(E_{6}+2A_{1}\)
| 1 | 4 | 40E
| |
97 |
\(D_{5}+3A_{1}\)
|
\(D_{7}+A_{1} \)
|
\(D_{7}+A_{1} \)
| 1 | 4 | 40A
| |
98 |
\(3A_{2}+2A_{1}\)
|
\(E_{6}+2A_{1}\)
|
\(3A_{2}+2A_{1} \)
| 1 | 5 | 96C
| |
99 |
\(A_{4}+A_{2}+2A_{1}\)
| – | – | 1 | 4 | 40E
| |
100 |
\(D_{4}+A_{2}+2A_{1}\)
|
\(D_{6}+A_{2}\)
|
\(D_{6}+A_{2}\)
| 1 | 4 | 96A
| |
101 |
\(2A_{3}+2A_{1}\)
|
\(E_{7}+A_{1}\)
|
\(D_{6}+2A_{1}\)
| 1 | 4 | 96C
| |
102 |
\(2A_{3}+2A_{1}\)
|
\(D_{5}+A_{3}\)
|
\(D_{5}+A_{3}\)
| 1 | 4 | 96A
| |
103 |
\(2A_{3}+2A_{1}\)
|
\(D_{8}\)
|
\(D_{8}\)
|
\(\times \)
| 1 | 4 | 96C
|
104 |
\(2A_{3}+2A_{1}\)
|
\(E_{8}\)
|
\(D_{8}\)
|
\(\times \)
| 1 | 4 | 96C
|
105 |
\(A_{6}+2A_{1}\)
| – | – | 1 | 3 | 40C
| |
106 |
\(D_{6}+2A_{1}\)
|
\(E_{7}+A_{1}\)
|
\(E_{7}+A_{1}\)
| 1 | 3 | 40A
| |
107 |
\(D_{6}+2A_{1}\)
|
\(D_{8}\)
|
\(D_{8}\)
| 1 | 3 | 40A
| |
108 |
\(D_{6}+2A_{1}\)
|
\(E_{8}\)
|
\(E_{8}\)
|
\(\times \)
| 2 | 3 | 40A
|
109 |
\(E_{6}+2A_{1}\)
| – | – | 1 | 3 | 20E
| |
110 |
\(A_{3}+2A_{2}+A_{1}\)
| – | – | 1 | 4 | 96C
| |
111 |
\(A_{5}+A_{2}+A_{1}\)
| – | – | 1 | 3 | 40E
| |
112 |
\(A_{5}+A_{2}+A_{1}\)
|
\(E_{7}+A_{1} \)
|
\(A_{5}+A_{2}+A_{1}\)
| 1 | 3 | 40E
| |
113 |
\(A_{5}+A_{2}+A_{1}\)
|
\(E_{6}+A_{2}\)
|
\(E_{6}+A_{2}\)
| 1 | 3 | 40E
| |
114 |
\(A_{5}+A_{2}+A_{1}\)
|
\(E_{8}\)
|
\(E_{6}+A_{2}\)
| 1 | 3 | 40E
| |
115 |
\(D_{5}+A_{2}+A_{1}\)
| – | – | 1 | 3 | 40A
| |
116 |
\(A_{4}+A_{3}+A_{1}\)
| – | – | 1 | 3 | 40E
| |
117 |
\(D_{4}+A_{3}+A_{1}\)
|
\(D_{7}+A_{1}\)
|
\(D_{7}+A_{1}\)
| 1 | 3 | 96A
| |
118 |
\(A_{7}+A_{1}\)
| – | – | 1 | 2 | 20D
| |
119 |
\(A_{7}+A_{1}\)
|
\(E_{7}+A_{1}\)
|
\(E_{7}+A_{1}\)
| 1 | 2 | 40C
| |
120 |
\(A_{7}+A_{1}\)
|
\(E_{8}\)
|
\(E_{7}+A_{1}\)
| 1 | 2 | 40C
| |
121 |
\(D_{7}+A_{1}\)
| – | – | 1 | 2 | 20B
| |
122 |
\(E_{7}+A_{1}\)
| – | – | 1 | \(\times \)3 | 20A
| |
123 |
\(E_{7}+A_{1}\)
|
\(E_{8}\)
|
\(E_{8}\)
| 2 | \(\times \)3 | 20A
| |
124 |
\(4A_{2}\)
|
\(E_{6}+A_{2}\)
|
\(4A_{2}\)
| 1 | 4 | 96C
| |
125 |
\(4A_{2}\)
|
\(E_{8}\)
|
\(4A_{2}\)
| 1 | 4 | 96C
| |
126 |
\(A_{4}+2A_{2}\)
| – | – | 1 | 3 | 40E
| |
127 |
\(2A_{3}+A_{2}\)
|
\(D_{6}+A_{2}\)
|
\(D_{6}+A_{2}\)
| 1 | 3 | 96C
| |
128 |
\(A_{6}+A_{2}\)
| – | – | 1 | 2 | 40C
| |
129 |
\(D_{6}+A_{2}\)
| – | – | 1 | 2 | 40A
| |
130 |
\(E_{6}+A_{2}\)
| – | – | 1 | 2 | 20E
| |
131 |
\(E_{6}+A_{2} \)
|
\(E_{8}\)
|
\(E_{6}+A_{2}\)
| 1 | 2 | 20E
| |
132 |
\(A_{5}+A_{3}\)
| – | – | 1 | 2 | 40E
| |
133 |
\(D_{5}+A_{3} \)
| – | – | 1 | 2 | 20F
| |
134 |
\(D_{5}+A_{3}\)
|
\(D_{8}\)
|
\(D_{8}\)
| 1 | 2 | 40A
| |
135 |
\(D_{5}+A_{3}\)
|
\(E_{8}\)
|
\(D_{8}\)
| 1 | 2 | 40A
| |
136 |
\(2A_{4}\)
| – | – | 1 | 2 | 40E
| |
137 |
\(2A_{4}\)
|
\(E_{8}\)
|
\(2A_{4}\)
| 1 | 2 | 40E
| |
138 |
\(D_{4}+A_{4}\)
| – | – | 1 | 2 | 20F
| |
139 |
\(A_{8}\)
| – | – | 1 | 1 | 20D
| |
140 |
\(A_{8}\)
|
\(E_{8}\)
|
\(A_{8}\)
| 1 | 1 | 20D
| |
141 |
\(2D_{4}\)
|
\(D_{8}\)
|
\(D_{8}\)
| 1 | 2 | 20F
| |
142 |
\(2D_{4}\)
|
\(E_{8}\)
|
\(E_{8}\)
|
\(\times \)
| 1 | \(\times \)1 | 96A
|
143 |
\(D_{8}\)
| – | – | 1 | 1 | 12B
| |
144 |
\(D_{8}\)
|
\(E_{8}\)
|
\(E_{8}\)
| 2 | \(\times \)2 | 20B
| |
145 |
\(E_{8}\)
| – | – | 2 | \(\times \)4 | 12A
| |
146 |
\(9A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(\times \)
| See Remark 1.16 | ||
147 |
\(A_{2}+7A_{1}\)
|
\(E_{7}+A_{2}\)
|
\(E_{7}+A_{2}\)
|
\(\times \)
| 1 | 8 | 96A
|
148 |
\(A_{3}+6A_{1}\)
|
\(D_{9}\)
|
\(D_{9}\)
|
\(\times \)
| 1 | 7 | 96B
|
149 |
\(D_{4}+5A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(\times \)
| 2 | 6 | 96A
|
150 |
\(D_{5}+4A_{1}\)
|
\(D_{9}\)
|
\(D_{9}\)
|
\(\times \)
| 1 | 5 | 40A
|
151 |
\(D_{4}+A_{2}+3A_{1}\)
|
\(E_{7}+A_{2}\)
|
\(E_{7}+A_{2}\)
|
\(\times \)
| 1 | 5 | 96A
|
152 |
\(2A_{3}+3A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(D_{8}+A_{1}\)
|
\(\times \)
| 1 | 5 | 96C
|
153 |
\(D_{6}+3A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(\times \)
| 2 | 4 | 40A
|
154 |
\(A_{5}+A_{2}+2A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(E_{6}+A_{2}+A_{1}\)
| 1 | 4 | 40E
| |
155 |
\(A_{4}+A_{3}+2A_{1}\)
|
\(D_{5}+A_{4}\)
|
\(D_{5}+A_{4}\)
| 1 | 4 | 40E
| |
156 |
\(D_{4}+A_{3}+2A_{1}\)
|
\(D_{9}\)
|
\(D_{9}\)
|
\(\times \)
| 1 | 4 | 96A
|
157 |
\(A_{7}+2A_{1} \)
|
\(E_{8}+A_{1}\)
|
\(E_{7}+2A_{1}\)
| 1 | 3 | 40C
| |
158 |
\(D_{7}+2A_{1}\)
|
\(D_{9}\)
|
\(D_{9}\)
| 1 | 3 | 20B
| |
159 |
\(E_{7}+2A_{1}\)
|
\(E_{8}+A_{1} \)
|
\(E_{8}+A_{1}\)
| 2 | \(\times \)4 | 20A
| |
160 |
\(4A_{2}+A_{1}\)
|
\(E_{8}+A_{1} \)
|
\(4A_{2}+A_{1}\)
|
\(\times \)
| 1 | 5 | 40E
|
161 |
\(2A_{3}+A_{2}+A_{1}\)
|
\(E_{7}+A_{2}\)
|
\(D_{6}+A_{2}+A_{1}\)
| 1 | 4 | 96C
| |
162 |
\(A_{6}+A_{2}+A_{1}\)
| – | – | 1 | 3 | 40C
| |
163 |
\(D_{6}+A_{2}+A_{1}\)
|
\(E_{7}+A_{2}\)
|
\(E_{7}+A_{2}\)
| 1 | 3 | 40A
| |
164 |
\(E_{6}+A_{2}+A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(E_{6}+A_{2}+A_{1}\)
| 1 | 3 | 20E
| |
165 |
\(A_{5}+A_{3}+A_{1}\)
|
\(E_{6}+A_{3}\)
|
\(E_{6}+A_{3}\)
| 1 | 3 | 40E
| |
166 |
\(D_{5}+A_{3}+A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(D_{8}+A_{1}\)
| 1 | 3 | 40A
| |
167 |
\(2A_{4}+A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(2A_{4}+A_{1}\)
| 1 | 3 | 40E
| |
168 |
\(A_{8}+A_{1}\)
| – | – | 1 | 2 | 20D
| |
169 |
\(A_{8}+A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(A_{8}+A_{1}\)
| 1 | 2 | 20D
| |
170 |
\(2D_{4}+A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(\times \)
| 1 | \(\times \)2 | 96A
|
171 |
\(D_{8}+A_{1}\)
|
\(E_{8}+A_{1}\)
|
\(E_{8}+A_{1} \)
| 2 | \(\times \)3 | 20B
| |
172 |
\(E_{8}+A_{1}\)
| – | – | 2 | \(\times \)5 | 12A
| |
173 |
\(A_{3}+3A_{2}\)
|
\(E_{6}+A_{3}\)
|
\(A_{3}+3A_{2}\)
| 1 | 4 | 96C
| |
174 |
\(A_{5}+2A_{2} \)
|
\(E_{7}+A_{2}\)
|
\(A_{5}+2A_{2}\)
| 1 | 3 | 40E
| |
175 |
\(A_{7}+A_{2}\)
|
\(E_{7}+A_{2}\)
|
\(E_{7}+A_{2} \)
| 1 | 2 | 40C
| |
176 |
\(E_{7}+A_{2}\)
| – | – | 1 | \(\times \)3 | 20A
| |
177 |
\(3A_{3}\)
|
\(D_{9}\)
|
\(D_{9}\)
|
\(\times \)
| 1 | 3 | 96C
|
178 |
\(D_{6}+A_{3}\)
|
\(D_{9}\)
|
\(D_{9}\)
| 1 | 2 | 40A
| |
179 |
\(E_{6}+A_{3}\)
| – | – | 1 | 2 | 20E
| |
180 |
\(A_{5}+A_{4}\)
| – | – | 1 | 2 | 40E
| |
181 |
\(D_{5}+A_{4}\)
| – | – | 1 | 2 | 20F
| |
182 |
\(A_{9}\)
| – | – | 1 | 1 | 20D
| |
183 |
\(D_{5}+D_{4}\)
|
\(D_{9}\)
|
\(D_{9}\)
| 1 | 2 | 20F
| |
184 |
\(D_{9}\)
| – | – | 1 | \(\times \)2 | 12B
|
1.2 Chambers
1.3 Main Results
rat
of Table 1. Except for the cases marked by \(\times \) in this column, two curves \(C_i\) and \(C_j\) are in the same orbit if and only if the vertices of the dual graph \(\Gamma \) corresponding to \(C_i\) and \(C_j\) belong to the same connected component of \(\Gamma \), and hence \(|{{\mathcal {R}}}(Y)/\mathrm{aut}(Y)|\) is equal to the number of connected components of the Dynkin diagram of type \(\tau \).1.4 The Plan of the Paper
GAP
[34].2 Finite Quadratic Forms, Lattices and Chambers
2.1 Finite Quadratic Forms
2.2 Discriminant Forms and Overlattices
2.3 Faces of a Chamber
2.4 L/M-Chambers
3 The Cone \(\mathrm{Nef}_{Y}\)
3.1 \(S_{X}/S_{Y}(2)\)-Chambers
3.2 Proof of Proposition 1.8
3.3 The Volume of \(\mathrm{Nef}_{Y}/\mathrm{aut}(Y)\)
3.4 Proof of Theorem 1.18
4 Borcherds’ Method
4.1 An Algorithm on a Graph
4.2 17 Primitive Embeddings
infty
, Then each \(L_{26}/L_{10}(2)\)-chamber has only finitely many walls, and they are defined by roots of \(L_{10}\). Moreover the tessellation of \({{\mathcal {P}}}_{10}\) by \(L_{26}/L_{10}(2)\)-chambers is reflexively simple. \(\square \)infty
, and D an \(L_{26}/L_{10}(2)\)-chamber. The automorphism group of D is denoted by4.3 Constructing \(S_{X}\)
-
by Vinberg chambers,
-
by \(L_{26}/S_Y(2)\)-chambers, each of which has only finite number of walls, and
-
by \(S_{X}/S_{Y}(2)\)-chambers, one of which is \(\mathrm{Nef}_{Y}\).
infty
and which has a fixed \(L_{26}/L_{10}(2)\)-chamber \(D_0\), and then proceed to the construction of \(S_{X}\) between \(L_{10}(2)\cong S_{Y}(2)\) and \(L_{26}\) such that the inclusion of \(L_{10}(2)\cong S_{Y}(2)\) into \(S_{X}\) is the embedding \(\pi ^*\), and that the fixed \(L_{26}/L_{10}(2)\)-chamber \(D_0\) is contained in \(\mathrm{Nef}_{Y}\).irec
) of Table 1. If the 5th column (exist
) is not marked by \(\times \), then \(M_p\) satisfies condition (ii).96C
(see Example 4.4). Then the even negative definite lattice \(Q_{\iota }\) contains 2208 vectors v of square-norm \(-4\), and we have 192 pairs \(\alpha =(r, v)\) such that \((r+v)/2\in L_{26}\). Choosing appropriate subsets from these 192 pairs, we can construct \(S_{X}\) for many types \((\tau , {\bar{\tau }})\) (Nos. 1, 2, ...).5 Geometric Algorithms
5.1 Separating Roots
5.2 Splitting Roots
-
\(\mathrm{Nef}_{Y}\cap (r)^{\perp }\) is a wall of \(\mathrm{Nef}_{Y}\) (that is, the hyperplane \((r)^{\perp }\) is disjoint from the interior of \(\mathrm{Nef}_{Y}\)),
-
r splits in \(S_{X}\), and
-
r is the class of a smooth rational curve C on Y.
5.3 Membership Criterion of \(G_Y\) in \( {{\mathrm {O}}}^{{{\mathcal {P}}}}(S_{Y})\)
5.4 Membership Criterion of \(\mathrm{aut}(Y)\) in \( G_Y\)
-
For any ample classes \(a_X\) and \(a_X^\prime \) of X, there exist no root of \(S_{X}\) separating \(a_X^{{\tilde{g}}}\) and \(a_X^\prime \).
-
For any ample classes \(a_Y\) and \(a_Y^\prime \) of Y, any roots of \(S_{Y}\) separating \(a_Y^{g}\) and \(a_Y^\prime \) does not split in \(S_{X}\).
-
There exist ample classes \(a_X\) and \(a_X^\prime \) of X such that there exist no roots of \(S_{X}\) separating \(a_X^{{\tilde{g}}}\) and \(a_X^\prime \).
-
There exist ample classes \(a_Y\) and \(a_Y^\prime \) of Y such that any root of \(S_{Y}\) separating \(a_Y^{g}\) and \(a_Y^\prime \) does not split in \(S_{X}\).
5.5 Criterion for \(\mathrm{aut}(Y)\)-Equivalence
6 Proofs of Main Theorems
6.1 Generators of \(\mathrm{aut}(Y)\) and Representatives of \(\mathrm{Nef}_{Y}/\mathrm{aut}(Y)\)
96C
gives us computational advantage of multiplicative factor the square of \(\mathrm{{vol}}(D_0)=652758220800\).6.2 Calculating \({{\mathcal {R}}}_{\mathrm{temp}}\), \({{\mathcal {E}}}_{\mathrm{temp}}\) and \({{\mathcal {G}}}_X\)
6.3 Rational Curves on Y
-
Let D be an arbitrary element of \(V_{C, 0}\). Then there exists an \(L_{26}/S_Y(2)\)-chamber \(D^\prime \) in \(V_{C^\prime , 0}\) such that \(\mathrm{isoms}(Y, D, D^\prime )\) contains an isometry g such that \([C]^g=[C^\prime ]\).
-
There exist a pair of \(L_{26}/S_Y(2)\)-chambers \(D\in V_{C, 0}\) and \(D^\prime \in V_{C^\prime , 0}\) and an isometry \(g\in \mathrm{isoms}(Y, D, D^\prime )\) such that \([C]^g=[C^\prime ]\).
6.4 Elliptic Fibrations of Y
6.5 Table of Elliptic Fibrations
7 Examples
7.1 An \((E_6, E_6)\)-Generic Enriques Surface
20E
. We see that \(D_0\) is a fundamental domain of the action of \(\mathrm{aut}(Y)\) on \(\mathrm{Nef}_Y\), and hence\(\mathrm{ADE}\)-type | Number | \(\mathrm{ADE}\)-type | Number |
---|---|---|---|
\(E_{6}\) | 1 | \(A_{3}+A_{1}\) | 1 |
\(A_{5}+A_{1}\) | 5 | \(2A_{2}\) | 1 |
\(3A_{2}\) | 1 | \(A_{2}+2A_{1}\) | 1 |
\(D_{5}\) | 1 | \(4A_{1}\) | 5 |
\(A_{5}\) | 1 | \(A_{3}\) | 1 |
\(A_{4}+A_{1}\) | 1 | \(A_{2}+A_{1}\) | 1 |
\(A_{3}+2A_{1}\) | 5 | \(3A_{1}\) | 2 |
\(2A_{2}+A_{1}\) | 1 | \(A_{2}\) | 1 |
\(D_{4}\) | 1 | \(2A_{1}\) | 1 |
\(A_{4}\) | 1 | \(A_{1}\) | 1 |
20E
, we have \(L_{26}/\langle {\Gamma } \rangle ^{\perp }(2)\)-chambers of \({{\mathcal {P}}}_{\langle {\Gamma } \rangle ^{\perp }}\). The intersection \(f_0:={{\mathcal {P}}}_{\langle {\Gamma } \rangle ^{\perp }}\cap D_0\) is one of the \(L_{26}/\langle {\Gamma } \rangle ^{\perp }(2)\)-chambers, and it is the maximal face of \(D_0\) among all the faces f of \(D_0\) such that \(\Gamma (f)=\Gamma \). Let \((V_{\Gamma }, E_{\Gamma })\) be the graph where \(V_{\Gamma }\) is the set of \(L_{26}/\langle {\Gamma } \rangle ^{\perp }(2)\)-chambers on \({{\mathcal {P}}}_{\langle {\Gamma } \rangle ^{\perp }}\) contained in \({{\mathcal {P}}}_{\langle {\Gamma } \rangle ^{\perp }}\cap \mathrm{Nef}_{Y}\) and \(E_{\Gamma }\) is the usual adjacency relation of chambers. Then \(D\mapsto {{\mathcal {P}}}_{\langle {\Gamma } \rangle ^{\perp }}\cap D\) gives a bijection to the set \(V_{\Gamma }\) of vertices from the set of \(L_{26}/S_Y(2)\)-chambers D contained in \(\mathrm{Nef}_{Y}\) such that \({{\mathcal {P}}}_{\langle {\Gamma } \rangle ^{\perp }}\cap D\) is a face of D of dimension \(10-\mu \), or equivalently, such that \({{\mathcal {P}}}_{\langle {\Gamma } \rangle ^{\perp }}\cap D\) contains a non-empty open subset of \({{\mathcal {P}}}_{\langle {\Gamma } \rangle ^{\perp }}\). The group7.2 \((4A_1, 4A_1)\)-Generic and \((4A_1, D_4)\)-Generic Enriques Surfaces
96C
. The complete set \(V_0\) of representatives of orbits of the action of \(\mathrm{aut}(Y)\) on the set of \(L_{26}/S_Y(2)\)-chambers contained in \(\mathrm{Nef}_{Y}\) consists of 5 elements with the orders of stabilizer subgroups 1, 1, 1, 2, 1. Since \(\mathrm{{vol}}(D_0)=1_{\mathrm{{BP}}}/72\), we have96C
. The set \(V_0\) consists of 18 elements with the orders of stabilizer subgroups \(4, \dots , 4\). We have \(|{{\mathcal {R}}}_{\mathrm{temp}}|=154\) and \(|{{\mathcal {E}}}_{\mathrm{temp}}|=21452\).7.3 A \((D_5, D_5)\)-Generic Enriques Surface
40A
to construct \(S_{Y}(2)\hookrightarrow S_{X}\hookrightarrow L_{26}\) for a \((D_5, D_5)\)-generic Enriques surface (No. 24 of Table 1). The set \(V_0\) consists of 6 elements with the orders of stabilizer subgroups \(2, \dots , 2\). In this case, we have \(\mathrm{{vol}}(D_0)=1_{\mathrm{{BP}}}/5760\) and7.4 Enriques Surfaces with Finite Automorphism Group
12A
, and hence \(\mathrm{{vol}}(D_0)=1_{\mathrm{{BP}}}/174182400\) (see [6]). Therefore12B
. We have \(\mathrm{{vol}}(D_0)=1_{\mathrm{{BP}}}/3870720\). Note that \(3870720\cdot |{{\mathfrak {S}}}_4|=|W(R_{D_9})|\). The Enriques surface Y is \((D_9, D_9)\)-generic (No. 184 of Table 1), and we have \({\mathrm {Aut}}(Y)\cong \mathrm{aut}(Y)\cong {{\mathfrak {S}}}_4\). The group \(\mathrm{aut}(Y)\) decomposes \({{\mathcal {R}}}(Y)\) as \(6+6\).