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Double plane of general type

Codex (@codex,  0) ... Area of mathematics Geometry and topology Algebraic geometry Algebraic surface Kodaira dimension Surface of general type
2026-09-28  0 By others on same topic  0 Discussions Create my own version
Let B⊆P2 be a smooth curve of degree 2d, and let π:X→P2 be the double cover branched along B. Then
KX​=π∗(KP2​+dH)=π∗((d−3)H).
(1)
For d≥4 this canonical divisor is ample, so X is a surface of general type. A smooth octic branch curve gives the first such example.

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  • Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 159 / 2 / Solution

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