- Branched covering
In
mathematics , "branched covering" is a term mainly used inalgebraic geometry , to describemorphism s "f" from analgebraic variety "V" to another one "W", the two dimensions being the same, and the typical fibre of "f" being of dimension 0.In that case, there will be an open set "W′" of "W" (for the
Zariski topology ) that isdense in "W", such that the restriction of "f" to "W′" (from "V′" = "f"−1("W′") to "W′", that is) is "unramified". Depending on the context, we can take this aslocal homeomorphism for thestrong topology , over thecomplex number s, or as anétale morphism in general (under some slightly stronger hypotheses, onflatness andseparability ). "Generically", then, such a morphism resembles acovering space in the topological sense. For example if "V" and "W" are bothRiemann surface s, we require only that "f" is holomorphic and not constant, and then there is a finite set of points "P" of "W", outside of which we do find an honest covering:"V′" → "W′".
The set of exceptional points on "W" is called the ramification locus (i.e. this is the complement of the largest possible open set "W′"); see
ramification . In generalmonodromy occurs according to thefundamental group of "W" − "W′" acting on the sheets of the covering (this topological picture can be made precise also in the case of a general base field).Branched coverings are easily constructed as
Kummer extension s, i.e. by extracting roots of functions in thefunction field . Thehyperelliptic curve s are prototypic examples.An unramified covering then is the occurrence of an empty ramification locus.
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