Log del Pezzo surfaces with torus action - a searchable database
The ldp-database stores complex log del Pezzo surfaces admitting a non-trivial action by an algebraic torus. So the entries are either toric surfaces or non-toric with a non-trivial \( \mathbb{C}^* \)-action. The ldp-database is searchable through various invariants and properties such as Picard number, Gorenstein index, log canonicity, anticanonical self intersection and more.
The ldp-database contains complete classifications of log del Pezzo surfaces admitting a torus action with the following properties:- Picard number 1 and Gorenstein index at most 200 (271.203 entries, see [1]),
- 1/3-log canonical, (129.904 entries, see [2]),
- toric of Picard number 1 and Picard index at most 10.000 (68.053 entries, see [3]),
- non-toric of Picard number 1 and Picard index at most 2500 (106.355 entries, see [3]),
- full intrinsic quadrics of Picard number at most 2 and Gorenstein index at most 200 (72.081 entries, see [4]),
- full intrinsic quadrics of Picard number 3 and Gorenstein index at most 40 (106.591 entries, see [4]),
In total, the ldp-database contains 692.125 families of surfaces. The ldp-database is redundancy free, that means that different entries are non-isomorphic.
Contributions by: Daniel Hättig, Jürgen Hausen, Katharina Király, Justus Springer and Hendrik Süß.
References
- [1] D. Hättig, B. Hafner, J. Hausen and J. Springer. Del Pezzo surfaces of Picard number one admitting a torus action. arXiv:2207.14790
- [2] D. Hättig, J. Hausen and J. Springer. Classifying log del Pezzo surfaces with torus action. arXiv:2302.03095
- [3] J. Springer. The Picard index of a surface with torus action. arXiv:2308.08879
- [4] J. Hausen and K. Király. Full intrinsic quadrics of dimension two. arXiv:2310.08293
- [5] D. Hättig, J. Hausen and H. Süß. Log del Pezzo \( \mathbb{C}^* \)-surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics. arXiv:2306.03796
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Basic properties:
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The characteristic space of the surface is smooth, i.e. its singularities are at most toric.
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The Cox Ring of the surface is defined by a single quadratic relation.
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Constellation of source and sink of a non-toric \( \mathbb{C}^* \)-surface. "e" = elliptic fixed point, "p" = parabolic fixed point curve.
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The dimension of the family of \( \mathbb{C}^* \)-surfaces.
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e.g. 0, 3, ... | |
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The rank of the Picard group (= rank of the class group).
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e.g. 2, 7, ... | |
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Self intersection number of an anticanonical divisor.
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e.g. 9, 12/17, ... |
Singularities:
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The index of the Picard group inside the class group.
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e.g. 31, 4040, ... | |
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The smallest positive integer \( \iota \) such that \( \iota \) times an anticanonical divisor is Cartier.
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e.g. 5, 124, ... | |
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The maximum of all \( 0 < \varepsilon \leq 1 \) such that the surface is \( \varepsilon \)-log canonical.
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e.g. 1/3, 3/5, ... | |
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The number of singular points.
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e.g. 0, 5, ... | |
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The type of singularity of the elliptic fixed point in the source, if it exists.
"A", "D" and "E" describe the shape of the resolution graph, while the following
number counts the exceptional prime divisors in the minimal resolution of singularities.
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e.g. 7, 12, ... | |
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The type of singularity of the elliptic fixed point in the sink, if it exists.
"A", "D" and "E" describe the shape of the resolution graph, while the following
number counts the exceptional prime divisors in the minimal resolution of singularities.
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e.g. 7, 12, ... |
Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics:
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See [5].
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See [5].
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See [5].
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