 Genus formula for a multisection of the rational elliptic surface

ID: genus-formula-for-a-multisection-of-the-rational-elliptic-surface

Codex  0 Created 2026-09-24 Updated 2026-10-14

If a smooth connected complex curve

C \subset E (1)

satisfies

C^{2} = k

and has degree

d = C \cdot F

over the base, the adjunction formula gives

2 g (C) - 2 = C^{2} + K_{E (1)} \cdot C = k - d, g (C) = 1 + \frac{k - d}{2} .

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