Timeline for Existence of holomorphic coverings having small degree
Current License: CC BY-SA 4.0
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| Aug 20, 2019 at 10:44 | comment | added | Jason Starr | Denote by $d$ the least such degree for a closed Riemann surface of genus $g$ of general moduli. By the Riemann-Hurwitz formula, the number of branch points equals $b=2g+2d-2$. The number of moduli of $b$ points on a genus $0$ closed Riemann surface equals $b-3 = 2g+2d-5$. Since the number of moduli of genus $g$ curves equals $3g-3$, it follows that $2g+2d-5 \geq 3g-3$, i.e., $2d\geq g+2$. Therefore the least degree for a genus-$g$ closed Riemann surface satisfies $d\geq \frac{g+2}{2}$. | |
| Aug 20, 2019 at 1:37 | history | edited | Eduardo Longa | CC BY-SA 4.0 |
added 46 characters in body
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| Aug 20, 2019 at 1:10 | history | asked | Eduardo Longa | CC BY-SA 4.0 |