To a branched cover between closed surfaces one can associate a combinatorial datum given by the topological types of and , the degree of , the number of branching points of , and the partitions of given by the local degrees of at the preimages of the branching points. This datum must satisfy the Riemann-Hurwitz condition plus some extra ones if either or both and are non-orientable. A very old question posed by Hurwitz [14] in 1891 asks whether a combinatorial datum satisfying these necessary conditions is actually realizable (namely, associated to some existing ) or not (in which case it is called exceptional). Or, more generally, to count the number of realizations of the datum up to a natural equivalence relation. Many partial answers have been given to the Hurwitz problem over the time, but a complete solution is still missing. In this short course we will report on ancient and recent results and techniques employed to attack the question.
@article{WBLN_2020__7__A2_0, author = {Petronio, Carlo}, title = {The {Hurwitz} existence problem for surface branched covers}, booktitle = {Winter Braids X (Pisa, 2020)}, series = {Winter Braids Lecture Notes}, note = {talk:2}, pages = {1--43}, publisher = {Winter Braids School}, year = {2020}, doi = {10.5802/wbln.34}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/wbln.34/} }
TY - JOUR AU - Petronio, Carlo TI - The Hurwitz existence problem for surface branched covers BT - Winter Braids X (Pisa, 2020) AU - Collectif T3 - Winter Braids Lecture Notes N1 - talk:2 PY - 2020 SP - 1 EP - 43 PB - Winter Braids School UR - http://archive.numdam.org/articles/10.5802/wbln.34/ DO - 10.5802/wbln.34 LA - en ID - WBLN_2020__7__A2_0 ER -
%0 Journal Article %A Petronio, Carlo %T The Hurwitz existence problem for surface branched covers %B Winter Braids X (Pisa, 2020) %A Collectif %S Winter Braids Lecture Notes %Z talk:2 %D 2020 %P 1-43 %I Winter Braids School %U http://archive.numdam.org/articles/10.5802/wbln.34/ %R 10.5802/wbln.34 %G en %F WBLN_2020__7__A2_0
Petronio, Carlo. The Hurwitz existence problem for surface branched covers, in Winter Braids X (Pisa, 2020), Winter Braids Lecture Notes (2020), Talk no. 2, 43 p. doi : 10.5802/wbln.34. http://archive.numdam.org/articles/10.5802/wbln.34/
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